Numbers and Number Theory


The Arabic numeral system

The Indian numerals discussed in our article Indian numerals form the basis of the European number systems which are now widely used. However they were not transmitted directly from India to Europe but rather came first to the Arabic/Islamic peoples and from them to Europe. The story of this transmission is not, however, a simple one. The eastern and western parts of the Arabic world both saw separate developments of Indian numerals with relatively little interaction between the two. By the western part of the Arabic world we mean the regions comprising mainly North Africa and Spain. Transmission to Europe came through this western Arabic route, coming into Europe first through Spain.

There are other complications in the story, however, for it was not simply that the Arabs took over the Indian number system. Rather different number systems were used simultaneously in the Arabic world over a long period of time. For example there were at least three different types of arithmetic used in Arab countries in the eleventh century: a system derived from counting on the fingers with the numerals written entirely in words, this finger-reckoning arithmetic was the system used for by the business community; the sexagesimal system with numerals denoted by letters of the Arabic alphabet; and the arithmetic of the Indian numerals and fractions with the decimal place-value system.

The first sign that the Indian numerals were moving west comes from a source which predates the rise of the Arab nations. In 662 AD Severus Sebokht, a Nestorian bishop who lived in Keneshra on the Euphrates river, wrote:-

I will omit all discussion of the science of the Indians, … , of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians, and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs. If those who believe, because they speak Greek, that they have arrived at the limits of science, would read the Indian texts, they would be convinced, even if a little late in the day, that there are others who know something of value.

This passage clearly indicates that knowledge of the Indian number system was known in lands soon to become part of the Arab world as early as the seventh century. The passage itself, of course, would certainly suggest that few people in that part of the world knew anything of the system. Severus Sebokht as a Christian bishop would have been interested in calculating the date of Easter (a problem to Christian churches for many hundreds of years). This may have encouraged him to find out about the astronomy works of the Indians and in these, of course, he would find the arithmetic of the nine symbols.

By 776 AD the Arab empire was beginning to take shape and we have another reference to the transmission of Indian numerals. We quote from a work of al-Qifti Chronology of the scholarswritten around the end the 12th century but quoting much earlier sources:-

… a person from India presented himself before the Caliph al-Mansur in the year [ 776 AD] who was well versed in the siddhanta method of calculation related to the movment of the heavenly bodies, and having ways of calculating equations based on the half-chord [essentially the sine] calculated in half-degrees … This is all contained in a work … from which he claimed to have taken the half-chord calculated for one minute. Al-Mansur ordered this book to be translated into Arabic, and a work to be written, based on the translation, to give the Arabs a solid base for calculating the movements of the planets …

Now in [1] (where a longer quote is given) Ifrah tries to determine which Indian work is referred to. He concludes that the work was most likely to have been Brahmagupta‘s Brahmasphutasiddhanta (The Opening of the Universe) which was written in 628. Irrespective of whether Ifrah is right, since all Indian texts after Aryabhata I‘s Aryabhatiya used the Indian number system of the nine signs, certainly from this time the Arabs had a translation into Arabic of a text written in the Indian number system.

It is often claimed that the first Arabic text written to explain the Indian number system was written by al-Khwarizmi. However there are difficulties here which many authors tend to ignore. The Arabic text is lost but a twelfth century Latin translation, Algoritmi de numero Indorum (in English Al-Khwarizmi on the Hindu Art of Reckoning) gave rise to the word algorithm deriving from his name in the title. Unfortunately the Latin translation is known to be much changed from al-Khwarizmi‘s original text (of which even the title is unknown). The Latin text certainly describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The first use of zero as a place holder in positional base notation is considered by some to be due to al-Khwarizmi in this work. The difficulty which arises is that al-Baghdadi refers to the Arabic original which, contrary to what was originally thought, seems not to be a work on Indian numerals but rather a work on finger counting methods. This becomes clear from the references by al-Baghdadi to the lost work. However the numerous references to al-Khwarizmi‘s book on the Indian nine symbols must mean that he did write such a work. Some degree of mystery still remains.

At first the Indian methods were used by the Arabs with a dust board. In fact in the western part of the Arabic world the Indian numerals came to be known as Guba (or Gubar or Ghubar) numerals from the Arabic word meaning “dust”. A dust board was used because the arithmetical methods required the moving of numbers around in the calculation and rubbing some out some of them as the calculation proceeded. The dust board allowed this in the same sort of way that one can use a blackboard, chalk and a blackboard eraser. Any student who has attended lectures where the lecturer continually changes and replaces parts of the mathematics as the demonstration progresses will understand the disadvantage of the dust board!

Around the middle of the tenth century al-Uqlidisi wrote Kitab al-fusul fi al-hisab al-Hindi which is the earliest surviving book that presents the Indian system. In it al-Uqlidisi argues that the system is of practical value:-

Most arithmeticians are obliged to use it in their work: since it is easy and immediate, requires little memorisation, provides quick answers, demands little thought … Therefore, we say that it is a science and practice that requires a tool, such as a writer, an artisan, a knight needs to conduct their affairs; since if the artisan has difficulty in finding what he needs for his trade, he will never succeed; to grasp it there is no difficulty, impossibility or preparation.

In the fourth part of this book al-Uqlidisi showed how to modify the methods of calculating with Indian symbols, which had required a dust board, to methods which could be carried out with pen and paper. Certainly the fact that the Indian system required a dust board had been one of the main obstacles to its acceptance. For example As-Suli, after praising the Indian system for its great simplicity, wrote in the first half of the tenth century:-

Official scribes nevertheless avoid using [the Indian system] because it requires equipment [like a dust board] and they consider that a system that requires nothing but the members of the body is more secure and more fitting to the dignity of a leader.

Al-Uqlidisi‘s work is therefore important in attempting to remove one of the obstacles to acceptance of the Indian nine symbols. It is also historically important as it is the earliest known text offering a direct treatment of decimal fractions.

Despite many scholars finding calculating with Indian symbols helpful in their work, the business community continued to use their finger arithmetic throughout the tenth century. Abu’l-Wafa, who was himself an expert in the use of Indian numerals, nevertheless wrote a text on how to use finger-reckoning arithmetic since this was the system used by the business community and teaching material aimed at these people had to be written using the appropriate system. Let us give a little information about the Arab letter numerals which are contained in Abu’l-Wafa‘s work.

The numbers were represented by letters but not in the dictionary order. The system was known as huruf al jumal which meant “letters for calculating” and also sometimes as abjad which is just the first four numbers (1 = a, 2 = b, j = 3, d = 4). The numbers from 1 to 9 were represented by letters, then the numbers 10, 20, 30, …, 90 by the next nine letters (10 = y, 20 = k, 30 = l, 40 = m, …), then 100, 200, 300, … , 900 by the next letters (100 = q, 200 = r, 300 = sh, 400 = ta, …). There were 28 Arabic letters and so one was left over which was used to represent 1000.

Arabic astronomers used a base 60 version of Arabic letter system. Although Arabic is written from right to left, we shall give an example writing in the left to right style that we use in writing English. A number, say 43° 21′ 14″, would have been written as “mj ka yd” in this base 60 version of the “abjad” letters for calculating.

A contemporary of al-Baghdadi, writing near the beginning of the eleventh century, was ibn Sina (better known in the West as Avicenna). We know many details of his life for he wrote an autobiography. Certainly ibn Sina was a remarkable child, with a memory and an ability to learn which amazed the scholars who met in his father’s home. A group of scholars from Egypt came to his father’s house in about 997 when ibn Sina was ten years old and they taught him Indian arithmetic. He also tells of being taught Indian calculation and algebra by a seller of vegetables. All this shows that by the beginning of the eleventh century calculation with the Indian symbols was fairly widespread and, quite significantly, was know to a vegetable trader.

What of the numerals themselves. We have seen in the article Indian numerals that the form of the numerals themselves varied in different regions and changed over time. Exactly the same happened in the Arabic world.

Here is an example of an early form of Indian numerals being used in the eastern part of the Arabic empire. It comes from a work of al-Sijzi, not an original work by him but rather the work of another mathematician which al-Sijzi copied at Shiraz and dated his copy 969.

The numerals from al-Sizji’s treatise of 969

The numerals had changed their form somewhat 100 years later when this copy of one of al-Biruni‘s astronomical texts was made. Here are the numerals as they appear in a 1082 copy.

The numerals from al-Biruni’s treatise copied in 1082

In fact a closer look will show that between 969 and 1082 the biggest change in the numerals was the fact that the 2 and the 3 have been rotated through 90°. There is a reason for this change which came about due to the way that scribes wrote, for they wrote on a scroll which they wound from right to left across their bodies as they sat cross-legged. The scribes therefore, instead of writing from right to left (the standard way that Arabic was written) wrote in lines from top to bottom. The script was rotated when the scroll was read and the characters when then in the correct orientation.

Here is an example of how the text was written

Perhaps because scribes did not have much experience at writing Indian numerals, they wrote 2 and 3 the correct way round instead of writing them rotated by 90° so that they would appear correctly when the scroll was rotated to be read.

Here is an example of what the scribe should write

and here is what the scribe actually wrote

The form of the numerals in the west of the Arabic empire look more familiar to those using European numerals today which is not surprising since it is from these numerals that the Indian number system reach Europe.

al-Banna al-Marrakushi’s form of the numerals

He gave this form of the numerals in his practical arithmetic book written around the beginning of the fourteenth century. He lived most of his life in Morocco which was in close contact with al-Andalus, or Andalusia, which was the Arab controlled region in the south of Spain.

The first surviving example of the Indian numerals in a document in Europe was, however, long before the time of al-Banna. The numerals appear in the Codex Vigilanus copied by a monk in Spain in 976. However the main part of Europe was not ready at this time to accept new ideas of any kind. Acceptance was slow, even as late as the fifteenth century when European mathematics began its rapid development which continues today. We will not examine the many contributions to bringing the Indian number system to Europe in this article but we will end with just one example which, however, is a very important one. Fibonacciwrites in his famous book Liber abaci published in Pisa in 1202:-

When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians’ nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms.

Babylonian numerals

The Babylonian civilisation in Mesopotamia replaced the Sumerian civilisation and the Akkadian civilisation. We give a little historical background to these events in our article Babylonian mathematics. Certainly in terms of their number system the Babylonians inherited ideas from the Sumerians and from the Akkadians. From the number systems of these earlier peoples came the base of 60, that is the sexagesimal system. Yet neither the Sumerian nor the Akkadian system was a positional system and this advance by the Babylonians was undoubtedly their greatest achievement in terms of developing the number system. Some would argue that it was their biggest achievement in mathematics.

Often when told that the Babylonian number system was base 60 people’s first reaction is: what a lot of special number symbols they must have had to learn. Now of course this comment is based on knowledge of our own decimal system which is a positional system with nine special symbols and a zero symbol to denote an empty place. However, rather than have to learn 10 symbols as we do to use our decimal numbers, the Babylonians only had to learn two symbols to produce their base 60 positional system.

Now although the Babylonian system was a positional base 60 system, it had some vestiges of a base 10 system within it. This is because the 59 numbers, which go into one of the places of the system, were built from a ‘unit’ symbol and a ‘ten’ symbol.

Here are the 59 symbols built from these two symbols

Now given a positional system one needs a convention concerning which end of the number represents the units. For example the decimal 12345 represents

1 × 104 + 2 × 103 + 3 × 102 + 4 × 10 + 5.

If one thinks about it this is perhaps illogical for we read from left to right so when we read the first digit we do not know its value until we have read the complete number to find out how many powers of 10 are associated with this first place. The Babylonian sexagesimal positional system places numbers with the same convention, so the right most position is for the units up to 59, the position one to the left is for 60 × n where 1 ≤ n ≤ 59, etc. Now we adopt a notation where we separate the numerals by commas so, for example, 1,57,46,40 represents the sexagesimal number

1 × 603 + 57 × 602 + 46 × 60 + 40

which, in decimal notation is 424000.

Here is 1,57,46,40 in Babylonian numerals

Now there is a potential problem with the system. Since two is represented by two characters each representing one unit, and 61 is represented by the one character for a unit in the first place and a second identical character for a unit in the second place then the Babylonian sexagesimal numbers 1,1 and 2 have essentially the same representation. However, this was not really a problem since the spacing of the characters allowed one to tell the difference. In the symbol for 2 the two characters representing the unit touch each other and become a single symbol. In the number 1,1 there is a space between them.

A much more serious problem was the fact that there was no zero to put into an empty position. The numbers sexagesimal numbers 1 and 1,0, namely 1 and 60 in decimals, had exactly the same representation and now there was no way that spacing could help. The context made it clear, and in fact despite this appearing very unsatisfactory, it could not have been found so by the Babylonians. How do we know this? Well if they had really found that the system presented them with real ambiguities they would have solved the problem – there is little doubt that they had the skills to come up with a solution had the system been unworkable. Perhaps we should mention here that later Babylonian civilisations did invent a symbol to indicate an empty place so the lack of a zero could not have been totally satisfactory to them.

An empty place in the middle of a number likewise gave them problems. Although not a very serious comment, perhaps it is worth remarking that if we assume that all our decimal digits are equally likely in a number then there is a one in ten chance of an empty place while for the Babylonians with their sexagesimal system there was a one in sixty chance. Returning to empty places in the middle of numbers we can look at actual examples where this happens.

Here is an example from a cuneiform tablet (actually AO 17264 in the Louvre collection in Paris) in which the calculation to square 147 is carried out. In sexagesimal 147 = 2,27 and squaring gives the number 21609 = 6,0,9.

Here is the Babylonian example of 2,27 squared

Perhaps the scribe left a little more space than usual between the 6 and the 9 than he would have done had he been representing 6,9.

Now if the empty space caused a problem with integers then there was an even bigger problem with Babylonian sexagesimal fractions. The Babylonians used a system of sexagesimal fractions similar to our decimal fractions. For example if we write 0.125 then this is 1/10 + 2/100 + 5/1000 = 1/8. Of course a fraction of the form a/b, in its lowest form, can be represented as a finite decimal fraction if and only if b has no prime divisors other than 2 or 5. So 1/3 has no finite decimal fraction. Similarly the Babylonian sexagesimal fraction 0;7,30 represented 7/60+ 30/3600which again written in our notation is 1/8.

Since 60 is divisible by the primes 2, 3 and 5 then a number of the form a/b, in its lowest form, can be represented as a finite decimal fraction if and only if b has no prime divisors other than 2, 3 or 5. More fractions can therefore be represented as finite sexagesimal fractions than can as finite decimal fractions. Some historians think that this observation has a direct bearing on why the Babylonians developed the sexagesimal system, rather than the decimal system, but this seems a little unlikely. If this were the case why not have 30 as a base? We discuss this problem in some detail below.

Now we have already suggested the notation that we will use to denote a sexagesimal number with fractional part. To illustrate 10,12,5;1,52,30 represents the number

10 × 602 + 12 × 60 + 5 + 1/60 + 52/602 + 30/603

which in our notation is 36725 1/32. This is fine but we have introduced the notation of the semicolon to show where the integer part ends and the fractional part begins. It is the “sexagesimal point” and plays an analogous role to a decimal point. However, the Babylonians has no notation to indicate where the integer part ended and the fractional part began. Hence there was a great deal of ambiguity introduced and “the context makes it clear” philosophy now seems pretty stretched. If I write 10,12,5,1,52,30 without having a notation for the “sexagesimal point” then it could mean any of:

0;10,12, 5, 1,52,30
  10;12, 5, 1,52,30
  10,12; 5, 1,52,30
  10,12, 5; 1,52,30
  10,12, 5, 1;52,30
  10,12, 5, 1,52;30
  10,12, 5, 1,52,30

in addition, of course, to 10, 12, 5, 1, 52, 30, 0 or 0 ; 0, 10, 12, 5, 1, 52, 30 etc.

Finally we should look at the question of why the Babylonians had a number system with a base of 60. The easy answer is that they inherited the base of 60 from the Sumerians but that is no answer at all. It only leads us to ask why the Sumerians used base 60. The first comment would be that we do not have to go back further for we can be fairly certain that the sexagesimal system originated with the Sumerians. The second point to make is that modern mathematicians were not the first to ask such questions. Theon of Alexandria tried to answer this question in the fourth century AD and many historians of mathematics have offered an opinion since then without any coming up with a really convincing answer.

Theon‘s answer was that 60 was the smallest number divisible by 1, 2, 3, 4, and 5 so the number of divisors was maximised. Although this is true it appears too scholarly a reason. A base of 12 would seem a more likely candidate if this were the reason, yet no major civilisation seems to have come up with that base. On the other hand many measures do involve 12, for example it occurs frequently in weights, money and length subdivisions. For example in old British measures there were twelve inches in a foot, twelve pennies in a shilling etc.

Neugebauer proposed a theory based on the weights and measures that the Sumerians used. His idea basically is that a decimal counting system was modified to base 60 to allow for dividing weights and measures into thirds. Certainly we know that the system of weights and measures of the Sumerians do use 1/3 and 2/3 as basic fractions. However although Neugebauer may be correct, the counter argument would be that the system of weights and measures was a consequence of the number system rather than visa versa.

Several theories have been based on astronomical events. The suggestion that 60 is the product of the number of months in the year (moons per year) with the number of planets (Mercury, Venus, Mars, Jupiter, Saturn) again seems far fetched as a reason for base 60. That the year was thought to have 360 days was suggested as a reason for the number base of 60 by the historian of mathematics Moritz Cantor. Again the idea is not that convincing since the Sumerians certainly knew that the year was longer than 360 days. Another hypothesis concerns the fact that the sun moves through its diameter 720 times during a day and, with 12 Sumerian hours in a day, one can come up with 60.

Some theories are based on geometry. For example one theory is that an equilateral triangle was considered the fundamental geometrical building block by the Sumerians. Now an angle of an equilateral triangle is 60° so if this were divided into 10, an angle of 6° would become the basic angular unit. Now there are sixty of these basic units in a circle so again we have the proposed reason for choosing 60 as a base. Notice this argument almost contradicts itself since it assumes 10 as the basic unit for division!

I [EFR] feel that all of these reasons are really not worth considering seriously. Perhaps I’ve set up my own argument a little, but the phrase “choosing 60 as a base” which I just used is highly significant. I just do not believe that anyone ever chose a number base for any civilisation. Can you imagine the Sumerians setting set up a committee to decide on their number base – no things just did not happen in that way. The reason has to involve the way that counting arose in the Sumerian civilisation, just as 10 became a base in other civilisations who began counting on their fingers, and twenty became a base for those who counted on both their fingers and toes.

Here is one way that it could have happened. One can count up to 60 using your two hands. On your left hand there are three parts on each of four fingers (excluding the thumb). The parts are divided from each other by the joints in the fingers. Now one can count up to 60 by pointing at one of the twelve parts of the fingers of the left hand with one of the five fingers of the right hand. This gives a way of finger counting up to 60 rather than to 10. Anyone convinced?

A variant of this proposal has been made by others. Perhaps the most widely accepted theory proposes that the Sumerian civilisation must have come about through the joining of two peoples, one of whom had base 12 for their counting and the other having base 5. Although 5 is nothing like as common as 10 as a number base among ancient peoples, it is not uncommon and is clearly used by people who counted on the fingers of one hand and then started again. This theory then supposes that as the two peoples mixed and the two systems of counting were used by different members of the society trading with each other then base 60 would arise naturally as the system everyone understood.

I have heard the same theory proposed but with the two peoples who mixed to produce the Sumerians having 10 and 6 as their number bases. This version has the advantage that there is a natural unit for 10 in the Babylonian system which one could argue was a remnant of the earlier decimal system. One of the nicest things about these theories is that it may be possible to find written evidence of the two mixing systems and thereby give what would essentially amount to a proof of the conjecture. Do not think of history as a dead subject. On the contrary our views are constantly changing as the latest research brings new evidence and new interpretations to light.

Egyptian numerals

The Egyptians had a writing system based on hieroglyphs from around 3000 BC. Hieroglyphs are little pictures representing words. It is easy to see how they would denote the word “bird” by a little picture of a bird but clearly without further development this system of writing cannot represent many words. The way round this problem adopted by the ancient Egyptians was to use the spoken sounds of words. For example, to illustrate the idea with an English sentence, we can see how “I hear a barking dog” might be represented by:

“an eye”, “an ear”, “bark of tree” + “head with crown”, “a dog”.

Of course the same symbols might mean something different in a different context, so “an eye” might mean “see” while “an ear” might signify “sound”.

The Egyptians had a bases 10 system of hieroglyphs for numerals. By this we mean that they has separate symbols for one unit, one ten, one hundred, one thousand, one ten thousand, one hundred thousand, and one million.

Here are the numeral hieroglyphs.

To make up the number 276, for example, fifteen symbols were required: two “hundred” symbols, seven “ten” symbols, and six “unit” symbols. The numbers appeared thus:

276 in hieroglyphs.

Here is another example:

4622 in hieroglyphs.

Note that the examples of 276 and 4622 in hieroglyphs are seen on a stone carving from Karnak, dating from around 1500 BC, and now displayed in the Louvre in Paris.

As can easily be seen, adding numeral hieroglyphs is easy. One just adds the individual symbols, but replacing ten copies of a symbol by a single symbol of the next higher value. Fractions to the ancient Egyptians were limited to unit fractions (with the exception of the frequently used 2/3 and less frequently used 3/4). A unit fraction is of the form 1/n where n is an integer and these were represented in numeral hieroglyphs by placing the symbol representing a “mouth”, which meant “part”, above the number. Here are some examples:

Notice that when the number contained too many symbols for the “part” sign to be placed over the whole number, as in 1/249 , then the “part” symbol was just placed over the “first part” of the number. [It was the first part for here the number is read from right to left.]

We should point out that the hieroglyphs did not remain the same throughout the two thousand or so years of the ancient Egyptian civilisation. This civilisation is often broken down into three distinct periods:

Old Kingdom – around 2700 BC to 2200 BC
Middle Kingdom – around 2100 BC to 1700 BC
New Kingdom – around 1600 BC to 1000 BC

Numeral hieroglyphs were somewhat different in these different periods, yet retained a broadly similar style.

Another number system, which the Egyptians used after the invention of writing on papyrus, was composed of hieratic numerals. These numerals allowed numbers to be written in a far more compact form yet using the system required many more symbols to be memorised. There were separate symbols for

1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 20, 30, 40, 50, 60, 70, 80, 90,
100, 200, 300, 400, 500, 600, 700, 800, 900,
1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000

Here are versions of the hieratic numerals

With this system numbers could be formed of a few symbols. The number 9999 had just 4 hieratic symbols instead of 36 hieroglyphs. One major difference between the hieratic numerals and our own number system was the hieratic numerals did not form a positional system so the particular numerals could be written in any order.

Here is one way the Egyptians wrote 2765 in hieratic numerals

Here is a second way of writing 2765 in hieratic numerals with the order reversed

Like the hieroglyphs, the hieratic symbols changed over time but they underwent more changes with six distinct periods. Initially the symbols that were used were quite close to the corresponding hieroglyph but their form diverged over time. The versions we give of the hieratic numerals date from around 1800 BC. The two systems ran in parallel for around 2000 years with the hieratic symbols being used in writing on papyrus, as for example in the Rhind papyrus and the Moscow papyrus, while the hieroglyphs continued to be used when carved on stone.

Greek number systems

There were no single Greek national standards in the first millennium BC. since the various island states prided themselves on their independence. This meant that they each had their own currency, weights and measures etc. These in turn led to small differences in the number system between different states since a major function of a number system in ancient times was to handle business transactions. However we will not go into sufficient detail in this article to examine the small differences between the system in separate states but rather we will look at its general structure. We should say immediately that the ancient Greeks had different systems for cardinal numbers and ordinal numbers so we must look carefully at what we mean by Greek number systems. Also we shall look briefly at some systems proposed by various Greek mathematicians but not widely adopted.

The first Greek number system we examine is their acrophonic system which was use in the first millennium BC. ‘Acrophonic’ means that the symbols for the numerals come from the first letter of the number name, so the symbol has come from an abreviation of the word which is used for the number. Here are the symbols for the numbers 5, 10, 100, 1000, 10000.

Acrophonic 5, 10, 100, 1000, 10000.

We have omitted the symbol for ‘one’, a simple ‘|’, which was an obvious notation not coming from the initial letter of a number. For 5, 10, 100, 1000, 10000 there will be only one puzzle for the reader and that is the symbol for 5 which should by P if it was the first letter of Pente. However this is simply a consequence of changes to the Greek alphabet after the numerals coming from these letters had been fixed. By that time the symbols were probably not thought of as coming from the letters so there was no move to change them with changes to the symbols for the letters. The original form of π was G and so Pente was originally Gente.

Now the system was based on the additive principle in a similar way to Roman numerals. This means that 8 is simply V|||, the symbol for five followed by three symbols for one. Here is 1-10 in Greek acrophonic numbers.

1-10 in Greek acrophonic numbers.

If base 10 is used with an additive system without intermediate symbols then many characters are required to express certain numbers. The number 9999 would require 36 symbols in such a system and this is very cumbersome. We have already seen that that Greek acrophonic numbers had a special symbol for 5. This is not surprising for it cuts down the characters required and also presumably arises from counting on fingers. We have 10 fingers but there is 5 on each hand. What is slightly more surprising is that the system had intermediate symbols for 50, 500, 5000, and 50000 but they were not new characters, rather they were composite symbols made from 5 and the symbols for 10, 100, 1000, 10000 respectively. Here is how the composites were formed.

Combining acrophonic numerals.

Notice that since there was no positional aspect of the system, there was no need for zero as an empty place holder. The symbol H represented 100 as no problem is created in the representation by the number having no tens or units.

Now this is not the only way in which such composite symbols were created. We have already mentioned that different states used variants of the number system and, although we are not going to examine these in detail, let us at least give some indication by showing some forms of 50 that have been found. Most of these forms are older than the main form of the numerals we have considered being more typical of the period 1500 BC to 1000 BC.

Different forms of 50 in different Greek States.

The next point worth noting is that this number system did not really consist of abstract numbers in the way we think of numbers today. Today the number 2 is applied to any collection of two objects and 2 is thought of as an abstract property that all such collections of two objects have in common. We know that the ancient Greeks had a somewhat different idea because the numbers were used in slightly different forms depending to what the number referred. The most frequent use of this particular number system was for sums of money. The basic unit of money was the drachma with a larger unit being the talent worth 6000 drachmas. The drachma was subdivided into smaller units, namely the obol which was 1/6 of a drachma, and the chalkos which was 1/8 of an obol. Half and quarter obols were also used. Notice that this system of currency was not based on the decimal system although the number system had 10 as a base and 5 as a secondary base.

The different units of currency were denoted by modifying the notation for the units in the number.

5678 drachmas would be written in this way:

The form of the units would denote drachmas.

3807 talents would be written as:

The units would now appear as T (T for talent). A sum of money involving both drachmas and obols would be written as:

3807 drachmas and 3 obols:

This acrophonic system was used for more than money. A very similar system was also used in dealing with weights and measures which is not surprising since the value of money would certainly have evolved from a system of weights. This is confirmed by the fact that the drachma was also the name of the unit of weight.

We now look at a second ancient Greek number system, the alphabetical numerals, or as it is sometimes called, the ‘learned’ system. As the name ‘alphabetical’ suggests the numerals are based on giving values to the letters of the alphabet. It is worth noting that the Greeks were one of the first to adopt a system of writing based on an alphabet. They were not the inventors of this form of writing, for the Phoenicians had such a system in place first. The Greek alphabet used to write words was taken over from the Phoenician system and was quite close to it. We will not examine the forms of the Greek letters themselves, but it is certainly worth stressing how important this form of writing was to be in advancing knowledge. It is fundamental to our ways of communicating in most countries today, although some peoples do prefer to use other forms of writing.

There are 24 letters in the classical Greek alphabet and these were used together with 3 older letters which have fallen out of use. These 27 letters are

Of these we have given both the upper case and lower case versions of the 24 classical letters. The letters digamma, koppa, and san are the obsolete ones. Although we have not given their symbols in the above table their symbols appear in the numeral tables below. The first nine of these letters were taken as the symbols for 1, 2, … , 9.

alphabetical 1-9.

Notice that 6 is represented by the symbol for the obsolete letter digamma.

The next nine letters were taken as the symbols for 10, 20, … , 90.

alphabetical 10-90.

Notice that 90 is represented by the symbol for the obsolete letter koppa.

The remaining nine letters were taken as the symbols for 100, 200, … , 900.

alphabetical 100-900.

Notice that 900 is represented by the symbol for the obsolete letter san.

Sometimes when these letters are written to represent numbers, a bar was put over the symbol to distinguish it from the corresponding letter.

Now numbers were formed by the additive principle. For example 11, 12, … , 19 were written:

alphabetical 11-19.

Larger numbers were constructed in the same sort of way. For example here is 269.

alphabetical 269.

Now this number system is compact but without modification is has the major drawback of not allowing numbers larger than 999 to be expressed. Composite symbols were created to overcome this problem. The numbers between 1000 and 9000 were formed by adding a subscript or superscript iota to the symbols for 1 to 9.

First form of 1000, …, 9000.

Second form of 1000, …, 9000.

How did the Greeks represent numbers greater than 9999? Well they based their numbers larger than this on the myriad which was 10000. The symbol M with small numerals for a number up to 9999 written above it meant that the number in small numerals was multiplied by 10000. Hence writing β above the M represented 20000:

The number 20000.

Similarly written above the M represented 1230000:

The number 1230000.

Of course writing a large number above the M was rather difficult so often in such cases the small numeral number was written in front of the M rather than above it. An example from Aristarchus:

Aristarchus wrote the number 71755875 as:

For most purposes this number system could represent all the numbers which might arise in normal day to day life. In fact numbers as large as 71755875 would be unlikely to arise very often. On the other hand mathematicians did see the need to extend the number system and we now look at two such proposals, first one by Apollonius and then briefly one by Archimedes(although in fact historically Archimedes made his proposal nearly 50 years before Apollonius).

Although we do not have first hand knowledge of the proposal by Apollonius we do know of it through a report by Pappus. The system we have described above works with products by a myriad. The idea which Apolloniusused to extend the system to larger numbers was to work with powers of the myriad. An M with an α above it represented 10000, M with β above it represented M2, namely 100000000, etc. The number to be multiplied by 10000, 10000000, etc is written after the M symbol and is written between the parts of the number, a word which is best interpreted as ‘plus’. As an example here is the way that Apollonius would have written 587571750269.

Apollonius’s representation of 587571750269.

Archimedes designed a similar system but rather than use 10000 = 104 as the basic number which was raised to various powers he used 100000000 = 108 raised to powers. The first octet for Archimedes consisted of numbers up to 108 while the second octet was the numbers from 108 up to 1016. Using this system Archimedes calculated that the number of grains of sand which could be fitted into the universe was of the order of the eighth octet, that is of the order of 1064.

Mathematics of the Incas

It is often thought that mathematics can only develop after a civilisation has developed some form of writing. Although not easy for us to understand today, many civilisations reached highly advanced states without ever developing written records. Now of course it is difficult for us to know much about such civilisations since there is no written record to be studied today. This article looks at the mathematical achievements of one such civilisation.

The civilisation we discuss, which does not appear to have found a need to develop writing, is that of the Incas. The Inca empire which existed in 1532, before the Spanish conquest, was vast. It spread over an area which stretched from what is now the northern border of Ecuador to Mendoza in west-central Argentina and to the Maule River in central Chile. The Inca people numbered around 12 million but they were from many different ethnic groups and spoke about 20 different languages. The civilisation had reached a high level of sophistication with a remarkable system of roads, agriculture, textile design, and administration. Of course even if writing is not required to achieve this level, counting and recording of numerical information is necessary. The Incas had developed a method of recording numerical information which did not require writing. It involved knots in strings called quipu.

The quipu was not a calculator, rather it was a storage device. Remember that the Incas had no written records and so the quipu played a major role in the administration of the Inca empire since it allowed numerical information to be kept. Let us first describe the basic quipu, with its positional number system, and then look at the ways that it was used in Inca society.

The quipu consists of strings which were knotted to represent numbers. A number was represented by knots in the string, using a positional base 10 representation. If the number 586 was to be recorded on the string then six touching knots were placed near the free end of the string, a space was left, then eight touching knots for the 10s, another space, and finally 5 touching knots for the 100s.

586 on a quipu.

For larger numbers more knot groups were used, one for each power of 10, in the same way as the digits of the number system we use here are occur in different positions to indicate the number of the corresponding power of 10 in that position.

Now it is not quite true that the same knots were used irrespective of the position as would be the case in a true positional system. There seems only one exception, namely the unit position, where different styles of knots were used from those in the other positions. In fact two different styles were used in the units position, one style if the unit were a 1 and a second style if the unit were greater than one. Both these styles differed from the standard knot used for all other positions. The system had a zero position, for this would be represented as no knots in that position. This meant that the spacing had to be highly regular so that zero positions would be clear.

There are many drawings and descriptions of quipus made by the Spanish invaders. Garcilaso de la Vega, whose mother was an Inca and whose father was Spanish, wrote :-

According to their position, the knots signified units, tens, hundreds, thousands, ten thousands and, exceptionally, hundred thousands, and they are all well aligned on their different cords as the figures that an accountant sets down, column by column, in his ledger.

Now of course recording a number on a string would, in itself, not be that useful. A quipu had many strings and there had to be some way that the string carrying the record of a particular number could be identified. The primary way this was done was by the use of colour. Numbers were recorded on strings of a particular colour to identify what that number was recording. For example numbers of cattle might be recorded on green strings while numbers of sheep might be recorded on white strings. The colours each had several meanings, some of which were abstract ideas, some concrete as in the cattle and sheep example. White strings had the abstract meaning of “peace” while red strings had the abstract meaning of “war”.

As well as the colour coding, another way of distinguishing the strings was to make some strings subsidiary ones, tied to the middle of a main string rather than being tied to the main horizontal cord.

Quipu with subsidiary cords.

We quote Garcilaso de la Vega again:-

The ordinary judges gave a monthly account of the sentences they imposed to their superiors, and they in turn reported to their immediate superiors, and so on finally to the Inca or those of his Supreme Council. The method of making these reports was by means of knots, made of various colours, where knots of such and such colours denote that such and such crimes had been punished. Smaller threads attached to thicker cords were of different colours to signify the precise nature of the punishment that had been inflicted.

It was not only judges who sent quipus to be kept in a central record. The Inca king appointed quipucamayocs, or keepers of the knots, to each town. Larger towns might have had up to thirty quipucamayocs who were essentially government statisticians, keeping official census records of the population, records of the produce of the town, its animals and weapons. This and other information was sent annually to the capital Cuzco. There was even an official delivery service to take to quipus to Cuzco which consisted of relay runners who passed the quipus on to the next runner at specially constructed staging posts. The terrain was extremely difficult yet the Incas had constructed roads to make the passing of information by quipus surprisingly rapid.

Much information on the quipus comes from a letter of the Peruvian Felipe Guaman Poma de Ayala to the King of Spain, written about eighty years after the Spanish conquest of the Incas. This remarkable letter contains 1179 pages and there are several drawings which show quipus. A fascinating aspect of one of these drawing is a picture of a counting board in the bottom left hand corner of one of them. This is called the yupana and is presumed to be the counting board of the Incas.

This is what the yupana looked like.

Interpretations of how this counting board, or Peruvian abacus, might have been used have been given by several authors,. However some historians are less certain that this really is a Peruvian abacus.

It is unclear from Poma’s commentary whether it is his version of a device associated with Spanish activities analogous to those of the person depicted or whether he is implying its association with the Incas. In either case, his commentary makes interpretation of the configuration and the meaning of the unfilled and filled holes highly speculative.

It is a difficult task to gain further insights into the mathematical understanding of the Incas. The book by Urton is interesting for it examines the concept of number as understood by the Inca people. As one might expect, their concept of number was a very concrete one, unlike our concept of number which is a highly abstract one (although this is not really understood by many people). The concrete way of conceiving numbers is illustrated by different words used when describing properties of numbers. Now the ideas of an even number, say, relies on having an abstract concept of number which is independent of the objects being counted. However, the Peruvian languages had different words which applied to different types of objects.

… the two together that make a pair …… the one together with its mate …

… two – in reference to one thing that is divided into two parts …

… a pair of two separate things bound intimately together, such as two bulls yoked together for ploughing …


This is a fascinating topic and one which deserves much further research. One wonders whether the Incas applied their number system to solve mathematical problems. Was it merely for recording? If the yupana really was an abacus then it must have been used to solve problems and this prompts the intriguing question of what these problems were. A tantalising glimpse may be contained in the writings of the Spanish priest José de Acosta who lived among the Incas from 1571 to 1586. He writes in his book Historia Natural Moral de las Indias which was published in Madrid in 1596:-

To see them use another kind of calculator, with maize kernels, is a perfect joy. In order to carry out a very difficult computation for which an able computer would require pen and paper, these Indians make use of their kernels. They place one here, three somewhere else and eight, I know not where. They move one kernel here and there and the fact is that they are able to complete their computation without making the smallest mistake. as a matter of fact, they are better at practical arithmetic than we are with pen and ink. Whether this is not ingenious and whether these people are wild animals let those judge who will! What I consider as certain is that in what they undertake to do they are superior to us.

What a pity that de Acosta did not have the mathematical skills to give a precise description which would have allowed us to understand this method of calculation by the Incas.

Indian numerals

It is worth beginning this article with the same quote from Laplace which we give in the article Overview of Indian mathematics. Laplace wrote:-

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.

The purpose of this article is to attempt the difficult task of trying to describe how the Indians developed this ingenious system. We will examine two different aspects of the Indian number systems in this article. First we will examine the way that the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 evolved into the form which we recognise today. Of course it is important to realise that there is still no standard way of writing these numerals. The different fonts on this computer can produce many forms of these numerals which, although recognisable, differ markedly from each other. Many hand-written versions are even hard to recognise.

The second aspect of the Indian number system which we want to investigate here is the place value system which, as Laplace comments in the quote which we gave at the beginning of this article, seems “so simple that its significance and profound importance is no longer appreciated.” We should also note the fact, which is important to both aspects, that the Indian number systems are almost exclusively base 10, as opposed to the Babylonian base 60 systems.

Beginning with the numerals themselves, we certainly know that today’s symbols took on forms close to that which they presently have in Europe in the 15th century. It was the advent of printing which motivated the standardisation of the symbols. However we must not forget that many countries use symbols today which are quite different from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and unless one learns these symbols they are totally unrecognisable as for example the Greek alphabet is to someone unfamiliar with it.

One of the important sources of information which we have about Indian numerals comes from al-Biruni. During the 1020s al-Biruni made several visits to India. Before he went there al-Biruni already knew of Indian astronomy and mathematics from Arabic translations of some Sanskrit texts. In India he made a detailed study of Hindu philosophy and he also studied several branches of Indian science and mathematics. Al-Biruni wrote 27 works on India and on different areas of the Indian sciences. In particular his account of Indian astronomy and mathematics is a valuable contribution to the study of the history of Indian science. Referring to the Indian numerals in a famous book written about 1030 he wrote:-

Whilst we use letters for calculation according to their numerical value, the Indians do not use letters at all for arithmetic. And just as the shape of the letters that they use for writing is different in different regions of their country, so the numerical symbols vary.

It is reasonable to ask where the various symbols for numerals which al-Biruni saw originated. Historians trace them all back to the Brahmi numerals which came into being around the middle of the third century BC. Now these Brahmi numerals were not just symbols for the numbers between 1 and 9. The situation is much more complicated for it was not a place-value system so there were symbols for many more numbers. Also there were no special symbols for 2 and 3, both numbers being constructed from the symbol for 1.

Here is the Brahmi one, two, three.

There were separate Brahmi symbols for 4, 5, 6, 7, 8, 9 but there were also symbols for 10, 100, 1000, … as well as 20, 30, 40, … , 90 and 200, 300, 400, …, 900.

The Brahmi numerals have been found in inscriptions in caves and on coins in regions near Poona, Bombay, and Uttar Pradesh. Dating these numerals tells us that they were in use over quite a long time span up to the 4thcentury AD. Of course different inscriptions differ somewhat in the style of the symbols.

Here is one style of the Brahmi numerals.

We should now look both forward and backward from the appearance of the Brahmi numerals. Moving forward leads to many different forms of numerals but we shall choose to examine only the path which has led to our present day symbols. First, however, we look at a number of different theories concerning the origin of the Brahmi numerals.

There is no problem in understanding the symbols for 1, 2, and 3. However the symbols for 4, … , 9 appear to us to have no obvious link to the numbers they represent. There have been quite a number of theories put forward by historians over many years as to the origin of these numerals.

  1. The Brahmi numerals came from the Indus valley culture of around 2000 BC.
  2. The Brahmi numerals came from Aramaean numerals.
  3. The Brahmi numerals came from the Karoshthi alphabet.
  4. The Brahmi numerals came from the Brahmi alphabet.
  5. The Brahmi numerals came from an earlier alphabetic numeral system, possibly due to Panini.
  6. The Brahmi numerals came from Egypt.

Basically these hypotheses are of two types. One is that the numerals came from an alphabet in a similar way to the Greek numerals which were the initial letters of the names of the numbers. The second type of hypothesis is that they derive from an earlier number system of the same broad type as Roman numerals. For example the Aramaean numerals of hypothesis 2 are based on I (one) and X (four):


Ifrah examines each of the six hypotheses in turn and rejects them, although one would have to say that in some cases it is more due to lack of positive evidence rather than to negative evidence.

Ifrah proposes a theory of his own , namely that:-

… the first nine Brahmi numerals constituted the vestiges of an old indigenous numerical notation, where the nine numerals were represented by the corresponding number of vertical lines … To enable the numerals to be written rapidly, in order to save time, these groups of lines evolved in much the same manner as those of old Egyptian Pharonic numerals. Taking into account the kind of material that was written on in India over the centuries (tree bark or palm leaves) and the limitations of the tools used for writing (calamus or brush), the shape of the numerals became more and more complicated with the numerous ligatures, until the numerals no longer bore any resemblance to the original prototypes.

It is a nice theory, and indeed could be true, but there seems to be absolutely no positive evidence in its favour. The idea is that they evolved from:

One might hope for evidence such as discovering numerals somewhere on this evolutionary path. However, it would appear that we will never find convincing proof for the origin of the Brahmi numerals.

If we examine the route which led from the Brahmi numerals to our present symbols (and ignore the many other systems which evolved from the Brahmi numerals) then we next come to the Gupta symbols. The Gupta period is that during which the Gupta dynasty ruled over the Magadha state in northeastern India, and this was from the early 4th century AD to the late 6thcentury AD. The Gupta numerals developed from the Brahmi numerals and were spread over large areas by the Gupta empire as they conquered territory.

The Gupta numerals evolved into the Nagari numerals, sometimes called the Devanagari numerals. This form evolved from the Gupta numerals beginning around the 7th century AD and continued to develop from the 11th century onward. The name literally means the “writing of the gods” and it was the considered the most beautiful of all the forms which evolved. For example al-Biruni writes:-

What we [the Arabs] use for numerals is a selection of the best and most regular figures in India.

These “most regular figures” which al-Biruni refers to are the Nagari numerals which had, by his time, been transmitted into the Arab world. The way in which the Indian numerals were spread to the rest of the world between the 7th to the 16th centuries in examined in detail in [7]. In this paper, however, Gupta claims that Indian numerals had reached Southern Europe by the end of the 5th century but his argument is based on the Geometry of Boethius which is now known to be a forgery dating from the first half of the 11th century. It would appear extremely unlikely that the Indian numerals reach Europe as early as Gupta suggests.

We now turn to the second aspect of the Indian number system which we want to examine in this article, namely the fact that it was a place-value system with the numerals standing for different values depending on their position relative to the other numerals. Although our place-value system is a direct descendant of the Indian system, we should note straight away that the Indians were not the first to develop such a system. The Babylonians had a place-value system as early as the 19th century BC but the Babylonian systems were to base 60. The Indians were the first to develop a base 10 positional system and, considering the date of the Babylonian system, it came very late indeed.

The oldest dated Indian document which contains a number written in the place-value form used today is a legal document dated 346 in the Chhedi calendar which translates to a date in our calendar of 594 AD. This document is a donation charter of Dadda III of Sankheda in the Bharukachcha region. The only problem with it is that some historians claim that the date has been added as a later forgery. Although it was not unusual for such charters to be modified at a later date so that the property to which they referred could be claimed by someone who was not the rightful owner, there seems no conceivable reason to forge the date on this document. Therefore, despite the doubts, we can be fairly sure that this document provides evidence that a place-value system was in use in India by the end of the 6th century.

Many other charters have been found which are dated and use of the place-value system for either the date or some other numbers within the text. These include:

  1. a donation charter of Dhiniki dated 794 in the Vikrama calendar which translates to a date in our calendar of 737 AD.
  2. an inscription of Devendravarman dated 675 in the Shaka calendar which translates to a date in our calendar of 753 AD.
  3. a donation charter of Danidurga dated 675 in the Shaka calendar which translates to a date in our calendar of 737 AD.
  4. a donation charter of Shankaragana dated 715 in the Shaka calendar which translates to a date in our calendar of 793 AD.
  5. a donation charter of Nagbhata dated 872 in the Vikrama calendar which translates to a date in our calendar of 815 AD.
  6. an inscription of Bauka dated 894 in the Vikrama calendar which translates to a date in our calendar of 837 AD.

All of these are claimed to be forgeries by some historians but some, or all, may well be genuine.

The first inscription which is dated and is not disputed is the inscription at Gwalior dated 933 in the Vikrama calendar which translates to a date in our calendar of 876 AD. Further details of this inscription is given in the article on zero.

There is indirect evidence that the Indians developed a positional number system as early as the first century AD. The evidence is found from inscriptions which, although not in India, have been found in countries which were assimilating Indian culture. Another source is the Bakhshali manuscript which contains numbers written in place-value notation. The problem here is the dating of this manuscript, a topic which is examined in detail in our article on the Bakhshali manuscript.

We are left, of course, with asking the question of why the Indians developed such an ingenious number system when the ancient Greeks, for example, did not. A number of theories have been put forward concerning this question. Some historians believe that the Babylonian base 60 place-value system was transmitted to the Indians via the Greeks. We have commented in the article on zero about Greek astronomers using the Babylonian base 60 place-value system with a symbol o similar to our zero. The theory here is that these ideas were transmitted to the Indians who then combined this with their own base 10 number systems which had existed in India for a very long time.

A second hypothesis is that the idea for place-value in Indian number systems came from the Chinese. In particular the Chinese had pseudo-positional number rods which, it is claimed by some, became the basis of the Indian positional system. This view is put forward by, for example, Lay Yong Lam; . Lam argues that the Chinese system already contained what he calls the:-

… three essential features of our numeral notation system: (i) nine signs and the concept of zero, (ii) a place value system and (iii) a decimal base.

A third hypothesis is put forward by Joseph . His idea is that the place-value in Indian number systems is something which was developed entirely by the Indians. He has an interesting theory as to why the Indians might be pushed into such an idea. The reason, Joseph believes, is due to the Indian fascination with large numbers. Freudenthal is another historian of mathematics who supports the theory that the idea came entirely from within India.

To see clearly this early Indian fascination with large numbers, we can take a look at the Lalitavistara which is an account of the life of Gautama Buddha. It is hard to date this work since it underwent continuous development over a long period but dating it to around the first or second century AD is reasonable. In Lalitavistara Gautama, when he is a young man, is examined on mathematics. He is asked to name all the numerical ranks beyond a koti which is 107. He lists the powers of 10 up to 1053. Taking this as a first level he then carries on to a second level and gets eventually to 10421. Gautama’s examiner says:-

You, not I, are the master mathematician.

It is stories such as this, and many similar ones, which convince Joseph that the fascination of the Indians with large numbers must have driven them to invent a system in which such numbers are easily expressed, namely a place-valued notation. He writes :-

The early use of such large numbers eventually led to the adoption of a series of names for successive powers of ten. The importance of these number names cannot be exaggerated. The word-numeral system, later replaced by an alphabetic notation, was the logical outcome of proceeding by multiples of ten. … The decimal place-value system developed when a decimal scale came to be associated with the value of the places of the numbers arranged left to right or right to left. and this was precisely what happened in India …

However, the same story in Lalitavistara convinces Kaplan that the Indians’ ideas of numbers came from the Greeks, for to him the story is an Indian version of Archimedes‘ Sand-reckoner. All that we know is that the place-value system of the Indians, however it arose, was transmitted to the Arabs and later into Europe to have, in the words of Laplace, profound importance on the development of mathematics.

Mayan mathematics

Hernán Cortés, excited by stories of the lands which Columbus had recently discovered, sailed from Spain in 1505 landing in Hispaniola which is now Santo Domingo. After farming there for some years he sailed with Velázquez to conquer Cuba in 1511. He was twice elected major of Santiago then, on 18 February 1519, he sailed for the coast of Yucatán with a force of 11 ships, 508 soldiers, 100 sailors, and 16 horses. He landed at Tabasco on the northern coast of the Yucatán peninsular. He met with little resistance from the local population and they presented him with presents including twenty girls. He married Malinche, one of these girls.

The people of the Yucatán peninsular were descendants of the ancient Mayan civilisation which had been in decline from about 900 AD. It is the mathematical achievements of this civilisation which we are concerned with in this article. However, before describing these, we should note that Cortés went on to conquer the Aztec peoples of Mexico. He captured Tenochtitlán before the end of 1519 (the city was rebuilt as Mexico City in 1521) and the Aztec empire fell to Cortés before the end of 1521. Malinche, who acted as interpreter for Cortés, played an important role in his ventures.

In order to understand how knowledge of the Mayan people has reached us we must consider another Spanish character in this story, namely Diego de Landa. He joined the Franciscan Order in 1541 when about 17 years old and requested that he be sent to the New World as a missionary. Landa helped the Mayan peoples in the Yucatán peninsular and generally tried his best to protect them from their new Spanish masters. He visited the ruins of the great cities of the Mayan civilisation and learnt from the people about their customs and history.

However, despite being sympathetic to the Mayan people, Landa abhorred their religious practices. To the devote Christian that Landa was, the Mayan religion with its icons and the Mayan texts written in hieroglyphics appeared like the work of the devil. He ordered all Mayan idols be destroyed and all Mayan books be burned. Landa seems to have been surprised at the distress this caused the Mayans.

Nobody can quite understand Landa’s feelings but perhaps he regretted his actions or perhaps he tried to justify them. Certainly what he then did was to write a book Relación de las cosas de Yucatán (1566) which describes the hieroglyphics, customs, temples, religious practices and history of the Mayans which his own actions had done so much to eradicate. The book was lost for many years but rediscovered in Madrid three hundred years later in 1869.

A small number of Mayan documents survived destruction by Landa. The most important are: the Dresden Codex now kept in the Sächsische Landesbibliothek Dresden; the Madrid Codex now kept in the American Museum in Madrid; and the Paris Codex now in the Bibliothèque nationale in Paris. The Dresden Codex is a treatise on astronomy, thought to have been copied in the eleventh century AD from an original document dating from the seventh or eighth centuries AD.

The Dresden codex:

Knowledge of the Mayan civilisation has been greatly increased in the last thirty years . Modern techniques such as high resolution radar images, aerial photography and satellite images have changed conceptions of the Maya civilisation. We are interested in the Classic Period of the Maya which spans the period 250 AD to 900 AD, but this classic period was built on top of a civilisation which had lived in the region from about 2000 BC.

The Maya of the Classic Period built large cities, around fifteen have been identified in the Yucatán peninsular, with recent estimates of the population of the city of Tikal in the Southern Lowlands being around 50000 at its peak. Tikal is probably the largest of the cities and recent studies have identified about 3000 separate constructions including temples, palaces, shrines, wood and thatch houses, terraces, causeways, plazas and huge reservoirs for storing rainwater. The rulers were astronomer priests who lived in the cities who controlled the people with their religious instructions. Farming with sophisticated raised fields and irrigation systems provided the food to support the population.

A common culture, calendar, and mythology held the civilisation together and astronomy played an important part in the religion which underlay the whole life of the people. Of course astronomy and calendar calculations require mathematics and indeed the Maya constructed a very sophisticated number system. We do not know the date of these mathematical achievements but it seems certain that when the system was devised it contained features which were more advanced than any other in the world at the time.

The Maya number system was a base twenty system.

Here are the Mayan numerals.

Almost certainly the reason for base 20 arose from ancient people who counted on both their fingers and their toes. Although it was a base 20 system, called a vigesimal system, one can see how five plays a major role, again clearly relating to five fingers and toes. In fact it is worth noting that although the system is base 20 it only has three number symbols (perhaps the unit symbol arising from a pebble and the line symbol from a stick used in counting). Often people say how impossible it would be to have a number system to a large base since it would involve remembering so many special symbols. This shows how people are conditioned by the system they use and can only see variants of the number system in close analogy with the one with which they are familiar. Surprising and advanced features of the Mayan number system are the zero, denoted by a shell for reasons we cannot explain, and the positional nature of the system. However, the system was not a truly positional system as we shall now explain.

In a true base twenty system the first number would denote the number of units up to 19, the next would denote the number of 20’s up to 19, the next the number of 400’s up to 19, etc. However although the Maya number system starts this way with the units up to 19 and the 20’s up to 19, it changes in the third place and this denotes the number of 360’s up to 19 instead of the number of 400’s. After this the system reverts to multiples of 20 so the fourth place is the number of 18 × 202, the next the number of 18 × 203 and so on. For example [ 8;14;3;1;12 ] represents

12 + 1 × 20 + 3 × 18 × 20 + 14 × 18 × 202 + 8 × 18 × 203 = 1253912.

As a second example [ 9;8;9;13;0 ] represents

0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 202 + 9 × 18 × 203=1357100.

Both these examples are found in the ruins of Mayan towns and we shall explain their significance below.

Now the system we have just described is used in the Dresden Codex and it is the only system for which we have any written evidence.Ifrah argues that the number system we have just introduced was the system of the Mayan priests and astronomers which they used for astronomical and calendar calculations. This is undoubtedly the case and that it was used in this way explains some of the irregularities in the system as we shall see below. It was the system used for calendars. However Ifrah also argues for a second truly base 20 system which would have been used by the merchants and was the number system which would also have been used in speech. This, he claims had a circle or dot (coming from a cocoa bean currency according to some, or a pebble used for counting according to others) as its unity, a horizontal bar for 5 and special symbols for 20, 400, 8000 etc. Ifrah writes :-

Even though no trace of it remains, we can reasonably assume that the Maya had a number system of this kind, and that intermediate numbers were figured by repeating the signs as many times as was needed.

Let us say a little about the Maya calendar before returning to their number systems, for the calendar was behind the structure of the number system. Of course, there was also an influence in the other direction, and the base of the number system 20 played a major role in the structure of the calendar.

The Maya had two calendars. One of these was a ritual calendar, known as the Tzolkin, composed of 260 days. It contained 13 “months” of 20 days each, the months being named after 13 gods while the twenty days were numbered from 0 to 19. The second calendar was a 365-day civil calendar called the Haab. This calendar consisted of 18 months, named after agricultural or religious events, each with 20 days (again numbered 0 to 19) and a short “month” of only 5 days that was called the Wayeb. The Wayeb was considered an unlucky period and Landa wrote in his classic text that the Maya did not wash, comb their hair or do any hard work during these five days. Anyone born during these days would have bad luck and remain poor and unhappy all their lives.

Why then was the ritual calendar based on 260 days? This is a question to which we have no satisfactory answer. One suggestion is that since the Maya lived in the tropics the sun was directly overhead twice every year. Perhaps they measured 260 days and 105 days as the successive periods between the sun being directly overhead (the fact that this is true for the Yucatán peninsular cannot be taken to prove this theory). A second theory is that the Maya had 13 gods of the “upper world”, and 20 was the number of a man, so giving each god a 20 day month gave a ritual calendar of 260 days.

At any rate having two calendars, one with 260 days and the other with 365 days, meant that the two would calendars would return to the same cycle after lcm(260, 365) = 18980 days. Now this is after 52 civil years (or 73 ritual years) and indeed the Maya had a sacred cycle consisting of 52 years. Another major player in the calendar was the planet Venus. The Mayan astronomers calculated its synodic period (after which it has returned to the same position) as 584 days. Now after only two of the 52 years cycles Venus will have made 65 revolutions and also be back to the same position. This remarkable coincidence would have meant great celebrations by the Maya every 104 years.

Now there was a third way that the Mayan people had of measuring time which was not strictly a calendar. It was an absolute timescale which was based on a creation date and time was measured forward from this. What date was the Mayan creation date? The date most often taken is 12 August 3113 BC but we should say straightaway that not all historians agree that this was the zero of this so-called “Long Count”. Now one might expect that this measurement of time would either give the number of ritual calendar years since creation or the number of civil calendar years since creation. However it does neither.

The Long Count is based on a year of 360 days, or perhaps it is more accurate to say that it is just a count of days with then numbers represented in the Mayan number system. Now we see the probable reason for the departure of the number system from a true base 20 system. It was so that the system approximately represented years. Many inscriptions are found in the Mayan towns which give the date of erection in terms of this long count. Consider the two examples of Mayan numbers given above. The first

[ 8;14;3;1;12 ]

is the date given on a plate which came from the town of Tikal. It translates to

12 + 1 × 20 + 3 × 18 × 20 + 14 × 18 × 202 + 8 × 18 × 203

which is 1253912 days from the creation date of 12 August 3113 BC so the plate was carved in 320 AD.

The second example

[ 9;8;9;13;0 ]

is the completion date on a building in Palenque in Tabasco, near the landing site of Cortés. It translates to

0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 202 + 9 × 18 × 203

which is 1357100 days from the creation date of 12 August 3113 BC so the building was completed in 603 AD.

We should note some properties (or more strictly non-properties) of the Mayan number system. The Mayans appear to have had no concept of a fraction but, as we shall see below, they were still able to make remarkably accurate astronomical measurements. Also since the Mayan numbers were not a true positional base 20 system, it fails to have the nice mathematical properties that we expect of a positional system. For example

[ 9;8;9;13;0 ] = 0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 202 + 9 × 18 × 203 = 1357100


[ 9;8;9;13 ] = 13 + 9 × 20 + 8 × 18 × 20 + 9 × 18 × 202 = 67873.

Moving all the numbers one place left would multiply the number by 20 in a true base 20 positional system yet 20 × 67873 = 1357460 which is not equal to 1357100. For when we multiple [ 9;8;9;13 ] by 20 we get 9 × 400 where in [ 9;8;9;13;0 ] we have 9 × 360.

We should also note that the Mayans almost certainly did not have methods of multiplication for their numbers and definitely did not use division of numbers. Yet the Mayan number system is certainly capable of being used for the operations of multiplication and division as the authors of [15] demonstrate.

Finally we should say a little about the Mayan advances in astronomy. Rodriguez writes in [19]:-

The Mayan concern for understanding the cycles of celestial bodies, particularly the Sun, the Moon and Venus, led them to accumulate a large set of highly accurate observations. An important aspect of their cosmology was the search for major cycles, in which the position of several objects repeated.

The Mayans carried out astronomical measurements with remarkable accuracy yet they had no instruments other than sticks. They used two sticks in the form of a cross, viewing astronomical objects through the right angle formed by the sticks. The Caracol building in Chichén Itza is thought by many to be a Mayan observatory. Many of the windows of the building are positioned to line up with significant lines of sight such as that of the setting sun on the spring equinox of 21 March and also certain lines of sight relating to the moon.

The Caracol building in Chichén Itza:

With such crude instruments the Maya were able to calculate the length of the year to be 365.242 days (the modern value is 365.242198 days). Two further remarkable calculations are of the length of the lunar month. At Copán (now on the border between Honduras and Guatemala) the Mayan astronomers found that 149 lunar months lasted 4400 days. This gives 29.5302 days as the length of the lunar month. At Palenque in Tabasco they calculated that 81 lunar months lasted 2392 days. This gives 29.5308 days as the length of the lunar month. The modern value is 29.53059 days. Was this not a remarkable achievement?

There are, however, very few other mathematical achievements of the Maya. Groemer [14] describes seven types of frieze ornaments occurring on Mayan buildings from the period 600 AD to 900 AD in the Puuc region of the Yucatán. This area includes the ruins at Kabah and Labna. Groemer gives twenty-five illustrations of friezes which show Mayan inventiveness and geometric intuition in such architectural decorations.

Fermat’s last theorem

Pierre de Fermat died in 1665. Today we think of Fermatas a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague’s book.

There is a statue of Fermat and his muse in his home town of Toulouse:
(Click it to see a larger version)

Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat‘s letters and other mathematical papers, comments written in books, etc. with the object of publishing his father’s mathematical ideas. In this way the famous ‘Last theorem’ came to be published. It was found by Samuel written as a marginal note in his father’s copy of Diophantus‘s Arithmetica.

Fermat’s Last Theorem states that

xn + yn = zn

has no non-zero integer solutions for x, y and z when n > 2. Fermat wrote

I have discovered a truly remarkable proof which this margin is too small to contain.

Fermat almost certainly wrote the marginal note around 1630, when he first studied Diophantus‘s Arithmetica. It may well be that Fermat realised that his remarkable proof was wrong, however, since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians. Although the special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat.

In fact in all the mathematical work left by Fermat there is only one proof. Fermat proves that the area of a right triangle cannot be a square. Clearly this means that a rational triangle cannot be a rational square. In symbols, there do not exist integers x, y, z with
x2 + y2 = z2 such that xy/2 is a square. From this it is easy to deduce the n = 4 case of Fermat’s theorem.

It is worth noting that at this stage it remained to prove Fermat’s Last Theorem for odd primes n only. For if there were integers x, y, z with xn + yn = zn then if n = pq,

(xq)p + (yq)p = (zq)p.

Euler wrote to Goldbach on 4 August 1753 claiming he had a proof of Fermat’s Theorem when n = 3. However his proof in Algebra (1770) contains a fallacy and it is far from easy to give an alternative proof of the statement which has the fallacious proof. There is an indirect way of mending the whole proof using arguments which appear in other proofs of Euler so perhaps it is not too unreasonable to attribute the n = 3 case to Euler.

Euler‘s mistake is an interesting one, one which was to have a bearing on later developments. He needed to find cubes of the form

p2 + 3q2

and Euler shows that, for any a, b if we put

p = a3 – 9ab2, q = 3(a2bb3) then
p2 + 3q2 = (a2 + 3b2)3.

This is true but he then tries to show that, if p2 + 3q2 is a cube then an a and b exist such that p and q are as above. His method is imaginative, calculating with numbers of the form a + b√-3. However numbers of this form do not behave in the same way as the integers, which Euler did not seem to appreciate.

The next major step forward was due to Sophie Germain. A special case says that if n and 2n + 1 are primes then xn+ yn = zn implies that one of x, y, z is divisible by n. Hence Fermat’s Last Theorem splits into two cases.

Case 1: None of x, y, z is divisible by n.
Case 2: One and only one of x, y, z is divisible by n.

Sophie Germain proved Case 1 of Fermat’s Last Theorem for all n less than 100 and Legendre extended her methods to all numbers less than 197. At this stage Case 2 had not been proved for even n = 5 so it became clear that Case 2 was the one on which to concentrate. Now Case 2 for n = 5 itself splits into two. One of x, y, z is even and one is divisible by 5. Case 2(i) is when the number divisible by 5 is even; Case 2(ii) is when the even number and the one divisible by 5 are distinct.

Case 2(i) was proved by Dirichlet and presented to the Paris Académie des Sciences in July 1825. Legendre was able to prove Case 2(ii) and the complete proof for n = 5 was published in September 1825. In fact Dirichlet was able to complete his own proof of the n = 5 case with an argument for Case 2(ii) which was an extension of his own argument for Case 2(i).

In 1832 Dirichlet published a proof of Fermat’s Last Theorem for n = 14. Of course he had been attempting to prove the n = 7 case but had proved a weaker result. The n = 7 case was finally solved by Lamé in 1839. It showed why Dirichlet had so much difficulty, for although Dirichlet‘s n = 14 proof used similar (but computationally much harder) arguments to the earlier cases, Lamé had to introduce some completely new methods. Lamé‘s proof is exceedingly hard and makes it look as though progress with Fermat’s Last Theorem to larger n would be almost impossible without some radically new thinking.

The year 1847 is of major significance in the study of Fermat’s Last Theorem. On 1 March of that year Laméannounced to the Paris Académie that he had proved Fermat’s Last Theorem. He sketched a proof which involved factorizing xn + yn = zn into linear factors over the complex numbers. Lamé acknowledged that the idea was suggested to him by Liouville. However Liouvilleaddressed the meeting after Lamé and suggested that the problem of this approach was that uniqueness of factorisation into primes was needed for these complex numbers and he doubted if it were true. Cauchysupported Lamé but, in rather typical fashion, pointed out that he had reported to the October 1847 meeting of the Académie an idea which he believed might prove Fermat’s Last Theorem.

Much work was done in the following weeks in attempting to prove the uniqueness of factorization. Wantzel claimed to have proved it on 15 March but his argument

It is true for n = 2, n = 3 and n = 4 and one easily sees that the same argument applies for n > 4

was somewhat hopeful.
[Wantzel is correct about n = 2 (ordinary integers), n = 3 (the argument Euler got wrong) and n = 4 (which was proved by Gauss).]

On 24 May Liouville read a letter to the Académie which settled the arguments. The letter was from Kummer, enclosing an off-print of a 1844 paper which proved that uniqueness of factorization failed but could be ‘recovered’ by the introduction of ideal complex numbers which he had done in 1846. Kummer had used his new theory to find conditions under which a prime is regular and had proved Fermat’s Last Theorem for regular primes. Kummer also said in his letter that he believed 37 failed his conditions.

By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime p is regular (and so Fermat’s Last Theorem is true for that prime) if p does not divide the numerators of any of the Bernoulli numbersB2 , B4 , …, Bp-3 . The Bernoulli number Bi is defined by

x/(ex – 1) = Bixi /i!

Kummer shows that all primes up to 37 are regular but 37 is not regular as 37 divides the numerator of B32 .

The only primes less than 100 which are not regular are 37, 59 and 67. More powerful techniques were used to prove Fermat’s Last Theorem for these numbers. This work was done and continued to larger numbers by Kummer, Mirimanoff, Wieferich, Furtwängler, Vandiverand others. Although it was expected that the number of regular primes would be infinite even this defied proof. In 1915 Jensen proved that the number of irregular primes is infinite.

Despite large prizes being offered for a solution, Fermat’s Last Theorem remained unsolved. It has the dubious distinction of being the theorem with the largest number of published false proofs. For example over 1000 false proofs were published between 1908 and 1912. The only positive progress seemed to be computing results which merely showed that any counter-example would be very large. Using techniques based on Kummer‘s work, Fermat’s Last Theorem was proved true, with the help of computers, for n up to 4,000,000 by 1993.

In 1983 a major contribution was made by Gerd Faltingswho proved that for every n > 2 there are at most a finite number of coprime integers x, y, z with xn + yn = zn. This was a major step but a proof that the finite number was 0 in all cases did not seem likely to follow by extending Faltings‘ arguments.

The final chapter in the story began in 1955, although at this stage the work was not thought of as connected with Fermat’s Last Theorem. Yutaka Taniyama asked some questions about elliptic curves, i.e. curves of the form y2= x3 + ax + b for constants a and b. Further work by Weiland Shimura produced a conjecture, now known as the Shimura-Taniyama–Weil Conjecture. In 1986 the connection was made between the Shimura-Taniyama– Weil Conjecture and Fermat’s Last Theorem by Frey at Saarbrücken showing that Fermat’s Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space.

Further work by other mathematicians showed that a counter-example to Fermat’s Last Theorem would provide a counter -example to the Shimura-Taniyama–Weil Conjecture. The proof of Fermat’s Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. Wiles gave a series of three lectures at the Isaac NewtonInstitute in Cambridge, England the first on Monday 21 June, the second on Tuesday 22 June. In the final lecture on Wednesday 23 June 1993 at around 10.30 in the morning Wiles announced his proof of Fermat‘s Last Theorem as a corollary to his main results. Having written the theorem on the blackboard he said I will stop here and sat down. In fact Wiles had proved the Shimura-Taniyama–Weil Conjecture for a class of examples, including those necessary to prove Fermat‘s Last Theorem.

This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wilesessentially withdrew his claim to have a proof. He states

The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semisquare case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures.

In March 1994 Faltings, writing in Scientific American, said

If it were easy, he would have solved it by now. Strictly speaking, it was not a proof when it was announced.

Weil, also in Scientific American, wrote

I believe he has had some good ideas in trying to construct the proof but the proof is not there. To some extent, proving Fermat’s Theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest.

In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work. They tried a new approach with a similar lack of success. In August 1994 Wiles addressed the International Congress of Mathematicians but was no nearer to solving the difficulties.

Taylor suggested a last attempt to extend Flach’s method in the way necessary and Wiles, although convinced it would not work, agreed mainly to enable him to convince Taylor that it could never work. Wiles worked on it for about two weeks, then suddenly inspiration struck.

In a flash I saw that the thing that stopped it [the extension of Flach’s method] working was something that would make another method I had tried previously work.

On 6 October Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one. Faltings sent a simplification of part of the proof.

No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat’s Last Theorem.

Perfect numbers

It is not known when perfect numbers were first studied and indeed the first studies may go back to the earliest times when numbers first aroused curiosity. It is quite likely, although not certain, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked, see for example [17] where detailed justification for this idea is given. Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties. Before we begin to look at the history of the study of perfect numbers, we define the concepts which are involved.

Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the ‘aliquot parts’ of a number.

An aliquot part of a number is a proper quotient of the number. So for example the aliquot parts of 10 are 1, 2 and 5. These occur since 1 = 10/10, 2 = 10/5, and 5 = 10/2. Note that 10 is not an aliquot part of 10 since it is not a proper quotient, i.e. a quotient different from the number itself. A perfect number is defined to be one which is equal to the sum of its aliquot parts.

The four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries.

6 = 1 + 2 + 3,
28 = 1 + 2 + 4 + 7 + 14,
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064

The first recorded mathematical result concerning perfect numbers which is known occurs in Euclid‘s Elements written around 300BC. It may come as a surprise to many people to learn that there are number theory results in Euclid‘s Elements since it is thought of as a geometry book. However, although numbers are represented by line segments and so have a geometrical appearance, there are significant number theory results in the Elements. The result which is if interest to us here is Proposition 36 of Book IX of the Elements which states :-

If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.

Here ‘double proportion’ means that each number of the sequence is twice the preceding number. To illustrate this Proposition consider 1 + 2 + 4 = 7 which is prime. Then

(the sum) × (the last) = 7 × 4 = 28,

which is a perfect number. As a second example, 1 + 2 + 4 + 8 + 16 = 31 which is prime. Then 31 × 16 = 496 which is a perfect number.

Now Euclid gives a rigorous proof of the Proposition and we have the first significant result on perfect numbers. We can restate the Proposition in a slightly more modern form by using the fact, known to the Pythagoreans, that

1 + 2 + 4 + … + 2k-1 = 2k – 1.

The Proposition now reads:-

If, for some k > 1, 2k – 1 is prime then 2k-1(2k – 1) is a perfect number.

The next significant study of perfect numbers was made by Nicomachus of Gerasa. Around 100 AD Nicomachuswrote his famous text Introductio Arithmetica which gives a classification of numbers based on the concept of perfect numbers. Nicomachus divides numbers into three classes: the superabundant numbers which have the property that the sum of their aliquot parts is greater than the number; deficient numbers which have the property that the sum of their aliquot parts is less than the number; and perfect numbers which have the property that the sum of their aliquot parts is equal to the number

Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect. And those which are said to be opposite to each other, the superabundant and the deficient, are divided in their condition, which is inequality, into the too much and the too little.

However Nicomachus has more than number theory in mind for he goes on to show that he is thinking in moral terms in a way that might seem extraordinary to mathematicians today :-

In the case of the too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is produced wanting, defaults, privations and insufficiencies. And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort – of which the most exemplary form is that type of number which is called perfect.

Now satisfied with the moral considerations of numbers, Nicomachus goes on to provide biological analogies in which he describes superabundant numbers as being like an animal with (see [8], or [1]):-

… ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands….

Deficient numbers are compared to animals with:-

a single eye, … one armed or one of his hands has less than five fingers, or if he does not have a tongue…

Nicomachus goes on to describe certain results concerning perfect numbers. All of these are given without any attempt at a proof. Let us state them in modern notation.

(1) The nth perfect number has n digits.
(2) All perfect numbers are even.
(3) All perfect numbers end in 6 and 8 alternately.
(4) Euclid‘s algorithm to generate perfect numbers will give all perfect numbers i.e. every perfect number is of the form 2k-1(2k – 1), for some k > 1, where 2k – 1 is prime.
(5) There are infinitely many perfect numbers.

We will see how these assertions have stood the test of time as we carry on with our discussions, but let us say at this point that assertions (1) and (3) are false while, as stated, (2), (4) and (5) are still open questions. However, since the time of Nicomachus we do know a lot more about his five assertions than the simplistic statement we have just made.

There exists an elegant and sure method of generating these numbers, which does not leave out any perfect numbers and which does not include any that are not; and which is done in the following way. First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096; and then they must be totalled each time there is a new term, and at each totalling examine the result, if you find that it is prime and non-composite, you must multiply it by the quantity of the last term that you added to the line, and the product will always be perfect. If, otherwise, it is composite and not prime, do not multiply it, but add on the next term, and again examine the result, and if it is composite leave it aside, without multiplying it, and add on the next term. If, on the other hand, it is prime, and non-composite, you must multiply it by the last term taken for its composition, and the number that results will be perfect, and so on as far as infinity.

As we have seen this algorithm is precisely that given by Euclid in the Elements. However, it is probable that this methods of generating perfect numbers was part of the general mathematical tradition handed down from before Euclid‘s time and continuing till Nicomachus wrote his treatise. Whether the five assertions of Nicomachuswere based on any more than this algorithm and the fact the there were four perfect numbers known to him 6, 28, 496 and 8128, it is impossible to say, but it does seem unlikely that anything more lies behind the unproved assertions.

… only one is found among the units, 6, only one other among the tens, 28, and a third in the rank of the hundreds, 496 alone, and a fourth within the limits of the thousands, that is, below ten thousand, 8128. And it is their accompanying characteristic to end alternately in 6 or 8, and always to be even.When these have been discovered, 6 among the units and 28 in the tens, you must do the same to fashion the next. … the result is 496, in the hundreds; and then comes 8128 in the thousands, and so on, as far as it is convenient for one to follow.

Despite the fact that Nicomachus offered no justification of his assertions, they were taken as fact for many years. Of course there was the religious significance that we have not mentioned yet, namely that 6 is the number of days taken by God to create the world, and it was believed that the number was chosen by him because it was perfect. Again God chose the next perfect number 28 for the number of days it takes the Moon to travel round the Earth. Saint Augustine (354-430) writes in his famous text The City of God :-

Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect…

The Arab mathematicians were also fascinated by perfect numbers and Thabit ibn Qurra wrote the Treatise on amicable numbers in which he examined when numbers of the form 2np, where p is prime, can be perfect. Ibn al-Haytham proved a partial converse to Euclid‘s proposition in the unpublished work Treatise on analysis and synthesis when he showed that perfect numbers satisfying certain conditions had to be of the form 2k-1(2k – 1) where 2k – 1 is prime.

Among the many Arab mathematicians to take up the Greek investigation of perfect numbers with great enthusiasm was Ismail ibn Ibrahim ibn Fallus (1194-1239) who wrote a treatise based on the Introduction to arithmetic by Nicomachus. He accepted Nicomachus‘s classification of numbers but the work is purely mathematical, not containing the moral comments of Nicomachus. Ibn Fallus gave, in his treatise, a table of ten numbers which were claimed to be perfect, the first seven are correct and are in fact the first seven perfect numbers, the remaining three numbers are incorrect.

At the beginning of the renaissance of mathematics in Europe around 1500 the assertions of Nicomachus were taken as truths, nothing further being known concerning perfect numbers not even the work of the Arabs. Some even believed the further unjustified and incorrect result that 2k-1(2k – 1) is a perfect number for every odd k. Pacioli certainly seems to have believed in this fallacy. Charles de Bovelles, a theologian and philosopher, published a book on perfect numbers in 1509. In it he claimed that Euclid‘s formula 2k-1(2k – 1) gives a perfect number for all odd integers k, see [10]. Yet, rather remarkably, although unknown until comparatively recently, progress had been made.

The fifth perfect number has been discovered again (after the unknown results of the Arabs) and written down in a manuscript dated 1461. It is also in a manuscript which was written by Regiomontanus during his stay at the University of Vienna, which he left in 1461, see [14]. It has also been found in a manuscript written around 1458, while both the fifth and sixth perfect numbers have been found in another manuscript written by the same author probably shortly after 1460. All that is known of this author is that he lived in Florence and was a student of Domenico d’Agostino Vaiaio.

In 1536, Hudalrichus Regius made the first breakthrough which was to become common knowledge to later mathematicians, when he published Utriusque Arithmetices in which he gave the factorisation 211 – 1 = 2047 = 23 . 89. With this he had found the first prime p such that 2p-1(2p – 1) is not a perfect number. He also showed that 213 – 1 = 8191 is prime so he had discovered (and made his discovery known) the fifth perfect number 212(213 – 1) = 33550336. This showed that Nicomachus‘s first assertion is false since the fifth perfect number has 8 digits. Nicomachus‘s claim that perfect numbers ended in 6 and 8 alternately still stood however. It is perhaps surprising that Regius, who must have thought he had made one of the major breakthroughs in mathematics, is virtually unheard of today.

J Scheybl gave the sixth perfect number in 1555 in his commentary to a translation of Euclid‘s Elements. This was not noticed until 1977 and therefore did not influence progress on perfect numbers.

The next step forward came in 1603 when Cataldi found the factors of all numbers up to 800 and also a table of all primes up to 750 (there are 132 such primes). Cataldiwas able use his list of primes to show that 217– 1 = 131071 is prime (since 7502 = 562500 > 131071 he could check with a tedious calculation that 131071 had no prime divisors). From this Cataldi now knew the sixth perfect number, namely 216(217 – 1) = 8589869056. This result by Cataldi showed that Nicomachus‘s assertion that perfect numbers ended in 6 and 8 alternately was false since the fifth and sixth perfect numbers both ended in 6. Cataldialso used his list of primes to check that 219 – 1 = 524287 was prime (again since 7502 = 562500 > 524287) and so he had also found the seventh perfect number, namely 218(219 – 1) = 137438691328.

As the reader will have already realised, the history of perfect numbers is littered with errors and Cataldi, despite having made the major advance of finding two new perfect numbers, also made some false claims. He writes in Utriusque Arithmetices that the exponents p = 2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37 give perfect numbers 2p-1(2p– 1). He is, of course, right for p = 2, 3, 5, 7, 13, 17, 19 for which he had a proof from his table of primes, but only one of his further four claims 23, 29, 31, 37 is correct.

Many mathematicians were interested in perfect numbers and tried to contribute to the theory. For example Descartes, in a letter to Mersenne in 1638, wrote [8]:-

… I think I am able to prove that there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number. But I can see nothing which would prevent one from finding numbers of this sort. For example, if 22021 were prime, in multiplying it by 9018009 which is a square whose root is composed of the prime numbers 3, 7, 11, 13, one would have 198585576189, which would be a perfect number. But, whatever method one might use, it would require a great deal of time to look for these numbers…

The next major contribution was made by Fermat. He told Roberval in 1636 that he was working on the topic and, although the problems were very difficult, he intended to publish a treatise on the topic. The treatise would never be written, partly because Fermat never got round to writing his results up properly, but also because he did not achieve the substantial results on perfect numbers he had hoped. In June 1640 Fermat wrote to Mersennetelling him about his discoveries concerning perfect numbers. He wrote:-

… here are three propositions I have discovered, upon which I hope to erect a great structure. The numbers less by one than the double progression, like

1  2  3   4   5   6    7    8    9    10    11    12    13
1  3  7  15  31  63  127  255  511  1023  2047  4095  8191

let them be called the radicals of perfect numbers, since whenever they are prime, they produce them. Put above these numbers in natural progression 1, 2, 3, 4, 5, etc., which are called their exponents. This done, I say

  1. When the exponent of a radical number is composite, its radical is also composite. Just as 6, the exponent of 63, is composite, I say that 63 will be composite.
  2. When the exponent is a prime number, I say that its radical less one is divisible by twice the exponent. Just as 7, the exponent of 127, is prime, I say that 126 is a multiple of 14.
  3. When the exponent is a prime number, I say that its radical cannot be divisible by any other prime except those that are greater by one than a multiple of double the exponent…

Here are three beautiful propositions which I have found and proved without difficulty, I shall call them the foundations of the invention of perfect numbers. I don’t doubt that Frenicle de Bessy got there earlier, but I have only begun and without doubt these propositions will pass as very lovely in the minds of those who have not become sufficiently hypocritical of these matters, and I would be very happy to have the opinion of M Roberval.

Shortly after writing this letter to Mersenne, Fermatwrote to Frenicle de Bessy on 18 October 1640. In this letter he gave a generalisation of results in the earlier letter stating the result now known as Fermat‘s Little Theorem which shows that for any prime p and an integer a not divisible by p, ap-1– 1 is divisible by p. Certainly Fermat found his Little Theorem as a consequence of his investigations into perfect numbers.

Using special cases of his Little Theorem, Fermat was able to disprove two of Cataldi‘s claims in his June 1640 letter to Mersenne. He showed that 223 – 1 was composite (in fact 223 – 1 = 47 × 178481) and that 237 – 1 was composite (in fact 237 – 1 = 223 × 616318177). Frenicle de Bessy had, earlier in that year, asked Fermat (in correspondence through Mersenne) if there was a perfect number between 1020 and 1022. In fact assuming that perfect numbers are of the form 2p-1(2p – 1) where p is prime, the question readily translates into asking whether 237 – 1 is prime. Fermat not only states that 237– 1 is composite in his June 1640 letter, but he tells Mersenne how he factorised it.

Fermat used three theorems:-

(i) If n is composite, then 2n – 1 is composite.(ii) If n is prime, then 2n – 2 is a multiple of 2n.

(iii) If n is prime, p a prime divisor of 2n– 1, then p – 1 is a multiple of n.

Note that (i) is trivial while (ii) and (iii) are special cases of Fermat‘s Little Theorem. Fermat proceeds as follows: If p is a prime divisor of 237 – 1, then 37 divides p – 1. As p is odd, it is a prime of the form 2 × 37m+1, for some m. The first case to try is p = 149 and this fails (a test division is carried out). The next case to try is 223 (the case m = 3) which succeeds and 237 – 1 = 223 × 616318177.

Mersenne was very interested in the results that Fermatsent him on perfect numbers and soon produced a claim of his own which was to fascinate mathematicians for a great many years. In 1644 he published Cogitata physica mathematica in which he claimed that 2p – 1 is prime (and so 2p-1(2p – 1) is a perfect number) for

p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257

and for no other value of p up to 257. Now certainly Mersenne could not have checked these results and he admitted this himself saying:-

… to tell if a given number of 15 or 20 digits is prime, or not, all time would not suffice for the test.

The remarkable fact is that Mersenne did very well if this was no more than a guess. There are 47 primes p greater than 19 yet less than 258 for which 2p – 1 might have been either prime or composite. Mersenne got 42 right and made 5 mistakes.

Primes of the form 2p– 1 are called Mersenne primes.

The next person to make a major contribution to the question of perfect numbers was Euler. In 1732 he proved that the eighth perfect number was 230(231 – 1) = 2305843008139952128. It was the first new perfect number discovered for 125 years. Then in 1738 Eulersettled the last of Cataldi‘s claims when he proved that 229 – 1 was not prime (so Cataldi‘s guesses had not been very good). Now it should be noticed (as it was at the time) that Mersenne had been right on both counts, since p = 31 appears in his list but p = 29 does not.

In two manuscripts which were unpublished during his life, Euler proved the converse of Euclid‘s result by showing that every even perfect number had to be of the form 2p-1(2p – 1). This verifies the fourth assertion of Nicomachus at least in the case of even numbers. It also leads to an easy proof that all even perfect numbers end in either a 6 or 8 (but not alternately). Euler also tried to make some headway on the problem of whether odd perfect numbers existed. He was able to prove the assertion made by Descartes in his letter to Mersenne in 1638 from which we quoted above. He went a little further and proved that any odd perfect number had to have the form


where 4n+1 is prime. However, as with most others whose contribution we have examined, Euler made predictions about perfect numbers which turned out to be wrong. He claimed that 2p-1(2p – 1) was perfect for p = 41 and p = 47 but Euler does have the distinction of finding his own error, which he corrected in 1753.

The search for perfect numbers had now become an attempt to check whether Mersenne was right with his claims in Cogitata physica mathematica. In fact Euler‘s results had made many people believe that Mersennehad some undisclosed method which would tell him the correct answer. In fact Euler‘s perfect number 230(231 – 1) remained the largest known for over 150 years. Mathematicians such as Peter Barlow wrote in his book Theory of Numbers published in 1811, that the perfect number 230(231 – 1):-

… is the greatest that ever will be discovered; for as they are merely curious, without being useful, it is not likely that any person will ever attempt to find one beyond it.

This, of course, turned out to be yet one more false assertion about perfect numbers!

The first error in Mersenne‘s list was discovered in 1876 by Lucas. He was able to show that 267 – 1 is not a prime although his methods did not allow him to find any factors of it. Lucas was also able to verify that one of the numbers in Mersenne‘s list was correct when he showed that 2127 – 1 is a Mersenne prime and so 2126(2127– 1) is indeed a perfect number. Lucas made another important advance which, as modified by Lehmer in 1930, is the basis of computer searches used today to find Mersenneprimes, and so to find perfect numbers. Following the announcement by Lucas that p = 127 gave the Mersenneprime 2p – 1, Catalan conjectured that, if m = 2p – 1 is prime then 2m – 1 is also prime. This Catalan sequence is 2p – 1 where

p = 3, 7, 127, 170141183460469231731687303715884105727, …

Of course if this conjecture were true it would solve the still open question of whether there are an infinite number of Mersenne primes (and also solve the still open question of whether there are infinitely many perfect numbers). However checking whether the fourth term of this sequence, namely 2p – 1 for p = 170141183460469231731687303715884105727, is prime is well beyond what is possible.

In 1883 Pervusin showed that 260(261– 1) is a perfect number. This was shown independently three years later by Seelhoff. Many mathematicians leapt to defend Mersenne saying that the number 67 in his list was a misprint for 61.

In 1903 Cole managed to factorise 267 – 1, the number shown to be composite by Lucas, but for which no factors were known up to that time. In October 1903 Colepresented a paper On the factorisation of large numbers to a meeting of the American Mathematical Society. In one of the strangest ‘talks’ ever given, Cole wrote on the blackboard

267 – 1 = 147573952589676412927.

Then he wrote 761838257287 and underneath it 193707721. Without speaking a work he multiplied the two numbers together to get 147573952589676412927 and sat down to applause from the audience. [It is worth remarking that the computer into which I (EFR) am typing this article gave this factorisation of 267 – 1 in about a second – times have changed!]

Further mistakes made by Mersenne were found. In 1911 Powers showed that 288(289 – 1) was a perfect number, then a few years later he showed that 2107– 1 is a prime and so 2106(2107– 1) is a perfect number. In 1922 Kraitchik showed that Mersenne was wrong in his claims for his largest prime of 257 when he showed that 2257– 1 is not prime.

We have followed the progress of finding even perfect numbers but there was also attempts to show that an odd perfect number could not exist. The main thrust of progress here has been to show the minimum number of distinct prime factors that an odd perfect number must have. Sylvester worked on this problem and wrote (see [20]):-

… the existence of [ an odd perfect number] – its escape, so to say, from the complex web of conditions which hem it in on all sides – would be little short of a miracle.

In fact Sylvester proved in 1888 that any odd perfect number must have at least 4 distinct prime factors. Later in the same year he improved his result to five factors and, over the years, this has been steadily improved until today we know that an odd perfect number would have to have at least eight distinct prime factors, and at least 29 prime factors which are not necessarily distinct. It is also known that such a number would have more than 300 digits and a prime divisor greater than 106. The problem of whether an odd perfect number exists, however, remains unsolved.

Today 46 perfect numbers are known, 288(289– 1) being the last to be discovered by hand calculations in 1911 (although not the largest found by hand calculations), all others being found using a computer. In fact computers have led to a revival of interest in the discovery of Mersenne primes, and therefore of perfect numbers. At the moment the largest known Mersenne prime is 243112609 – 1 (which is also the largest known prime) and the corresponding largest known perfect number is 243112608(243112609 – 1). It was discovered in August 2008 and this, the 45th such prime to be discovered, contains more than 10 million digits. If you wonder why we have not included the number in decimal form, then let me say that it contains about 150 times as many characters as this whole article on perfect numbers. Also worth noting is the fact that although this is the 45th to be discovered, it is not be the 45th largest perfect number as not all smaller cases had been ruled out. A month after this discovery the 46th (but smaller) Mersenne prime was discovered.

Prime numbers

Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.

The mathematicians of Pythagoras‘s school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.
A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.
A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.
You can see more about these numbers in the History topics article Perfect numbers.

By the time Euclid‘s Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.

Euclid also showed that if the number 2n – 1 is prime then the number 2n-1(2n – 1) is a perfect number. The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form. It is not known to this day whether there are any odd perfect numbers.

In about 200 BC the Greek Eratosthenes devised an algorithm for calculating primes called the Sieve of Eratosthenes.

There is then a long gap in the history of prime numbers during what is usually called the Dark Ages.

The next important developments were made by Fermatat the beginning of the 17th Century. He proved a speculation of Albert Girard that every prime number of the form 4 n + 1 can be written in a unique way as the sum of two squares and was able to show how any number could be written as a sum of four squares.
He devised a new method of factorising large numbers which he demonstrated by factorising the number 2027651281 = 44021 × 46061.
He proved what has come to be known as Fermat’s Little Theorem (to distinguish it from his so-called Last Theorem).
This states that if p is a prime then for any integer a we have ap = a modulo p.
This proves one half of what has been called the Chinese hypothesis which dates from about 2000 years earlier, that an integer n is prime if and only if the number 2n – 2 is divisible by n. The other half of this is false, since, for example, 2341 – 2 is divisible by 341 even though 341 = 31 × 11 is composite. Fermat‘s Little Theorem is the basis for many other results in Number Theory and is the basis for methods of checking whether numbers are prime which are still in use on today’s electronic computers.

Fermat corresponded with other mathematicians of his day and in particular with the monk Marin Mersenne. In one of his letters to Mersenne he conjectured that the numbers 2n + 1 were always prime if n is a power of 2. He had verified this for n = 1, 2, 4, 8 and 16 and he knew that if n were not a power of 2, the result failed. Numbers of this form are called Fermat numbers and it was not until more than 100 years later that Euler showed that the next case 232 + 1 = 4294967297 is divisible by 641 and so is not prime.

Number of the form 2n – 1 also attracted attention because it is easy to show that if unless n is prime these number must be composite. These are often called Mersenne numbersMn because Mersenne studied them.

Not all numbers of the form 2n – 1 with n prime are prime. For example 211 – 1 = 2047 = 23 × 89 is composite, though this was first noted as late as 1536.
For many years numbers of this form provided the largest known primes. The number M19 was proved to be prime by Cataldi in 1588 and this was the largest known prime for about 200 years until Euler proved that M31is prime. This established the record for another century and when Lucas showed that M127 (which is a 39 digit number) is prime that took the record as far as the age of the electronic computer.
In 1952 the Mersenne numbers M521, M607, M1279, M2203 and M2281 were proved to be prime by Robinson using an early computer and the electronic age had begun.

By 2005 a total of 42 Mersenne primes have been found. The largest is M25964951 which has 7816230 decimal digits.

Euler‘s work had a great impact on number theory in general and on primes in particular.
He extended Fermat‘s Little Theorem and introduced the Euler φ-function. As mentioned above he factorised the 5thFermat Number 232 + 1, he found 60 pairs of the amicable numbers referred to above, and he stated (but was unable to prove) what became known as the Law of Quadratic Reciprocity.
He was the first to realise that number theory could be studied using the tools of analysis and in so-doing founded the subject of Analytic Number Theory. He was able to show that not only is the so-called Harmonic series ∑ (1/n) divergent, but the series

1/2 + 1/3 + 1/5 + 1/7 + 1/11 + …

formed by summing the reciprocals of the prime numbers, is also divergent. The sum to n terms of the Harmonic series grows roughly like log(n), while the latter series diverges even more slowly like log[ log(n) ]. This means, for example, that summing the reciprocals of all the primes that have been listed, even by the most powerful computers, only gives a sum of about 4, but the series still diverges to ∞.

At first sight the primes seem to be distributed among the integers in rather a haphazard way. For example in the 100 numbers immediately before 10 000 000 there are 9 primes, while in the 100 numbers after there are only 2 primes. However, on a large scale, the way in which the primes are distributed is very regular. Legendre and Gauss both did extensive calculations of the density of primes. Gauss (who was a prodigious calculator) told a friend that whenever he had a spare 15 minutes he would spend it in counting the primes in a ‘chiliad’ (a range of 1000 numbers). By the end of his life it is estimated that he had counted all the primes up to about 3 million. Both Legendre and Gauss came to the conclusion that for large n the density of primes near n is about 1/log(n). Legendregave an estimate for π(n) the number of primes ≤ n of

π(n) = n/(log(n) – 1.08366)

while Gauss‘s estimate is in terms of the logarithmic integral

π(n) = ∫ (1/log(t) dt where the range of integration is 2 to n.

You can see the Legendre estimate and the Gauss estimate and can compare them.

The statement that the density of primes is 1/log(n) is known as the Prime Number Theorem. Attempts to prove it continued throughout the 19th Century with notable progress being made by Chebyshev and Riemann who was able to relate the problem to something called the Riemann Hypothesis: a still unproved result about the zeros in the Complex plane of something called the Riemann zeta-function. The result was eventually proved (using powerful methods in Complex analysis) by Hadamard and de la Vallée Poussin in 1896.

There are still many open questions (some of them dating back hundreds of years) relating to prime numbers.
Some unsolved problems

  1. The Twin Primes Conjecture that there are infinitely many pairs of primes only 2 apart.
  2. Goldbach’s Conjecture (made in a letter by C Goldbachto Euler in 1742) that every even integer greater than 2 can be written as the sum of two primes.
  3. Are there infinitely many primes of the form n2 + 1 ?
    (Dirichlet proved that every arithmetic progression : {a + bn | nN} with a, b coprime contains infinitely many primes.)
  4. Is there always a prime between n2 and (n + 1)2 ?
    (The fact that there is always a prime between n and 2n was called Bertrand‘s conjecture and was proved by Chebyshev.)
  5. Are there infinitely many prime Fermat numbers? Indeed, are there any prime Fermat numbers after the fourth one?
  6. Is there an arithmetic progression of consecutive primes for any given (finite) length? e.g. 251, 257, 263, 269 has length 4. The largest example known has length 10.
  7. Are there infinitely many sets of 3 consecutive primes in arithmetic progression. (True if we omit the word consecutive.)
  8. n2n + 41 is prime for 0 ≤ n ≤ 40. Are there infinitely many primes of this form? The same question applies to n2 – 79 n + 1601 which is prime for 0 ≤ n ≤ 79.
  9. Are there infinitely many primes of the form n# + 1? (where n# is the product of all primes ≤ n.)
  10. Are there infinitely many primes of the form n# – 1?
  11. Are there infinitely many primes of the form n! + 1?
  12. Are there infinitely many primes of the form n! – 1?
  13. If p is a prime, is 2p – 1 always square free? i.e. not divisible by the square of a prime.
  14. Does the Fibonacci sequence contain an infinite number of primes?

Here are the latest prime records that we know.

The largest known prime (found by GIMPS [Great Internet Mersenne Prime Search] in August 2008) was the 45thMersenne prime: M43112609 which has 1209780189 decimal digits. The most recently discovered Mersenne prime (September 2008) is M37156667. See the Official announcement

The largest known twin primes are 2003663613 × 2195000 ± 1. They have 58711 digits and were announced by Vautier, McKibbon and Gribenko in 2007.

The largest known factorial prime (prime of the form n! ± 1) is 34790! – 1. It is a number of 142891 digits and was announced by Marchal, Carmody and Kuosa in 2002.

The largest known primorial prime (prime of the form n# ± 1 where n# is the product of all primes ≤ n) is 392113# + 1. It is a number of 169966 digits and was announced by Heuer in 2001.

A history of Zero

One of the commonest questions which the readers of this archive ask is: Who discovered zero? Why then have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form. If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it. The historical record, however, shows quite a different path towards the concept. Zero makes shadowy appearances only to vanish again almost as if mathematicians were searching for it yet did not recognise its fundamental significance even when they saw it.

The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name “zero” derives ultimately from the Arabic sifr which also gives us the word “cipher”.)

Neither of the above uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as ‘real’ problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 “things” and then to the abstract idea of “five”. If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.

One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 ” 6.

The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 ” 6 we never find 216 ”. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.

If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer “It’s three fifty” means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.

We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.

Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid‘s Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.

Now there were exceptions to what we have just stated. The exceptions were the mathematicians who were involved in recording astronomical data. Here we find the first use of the symbol which we recognise today as the notation for zero, for Greek astronomers began to use the symbol O. There are many theories why this particular notation was used. Some historians favour the explanation that it is omicron, the first letter of the Greek word for nothing namely “ouden”. Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number – it represented 70 (the Greek number system was based on their alphabet). Other explanations offered include the fact that it stands for “obol”, a coin of almost no value, and that it arises when counters were used for counting on a sand board. The suggestion here is that when a counter was removed to leave an empty column it left a depression in the sand which looked like O.

Ptolemy in the Almagest written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O. By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself. The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics.

The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers. As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far. For example Mukherjee  claims:-

… the mathematical conception of zero … was also present in the spiritual form from 17 000 years back in India.

What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.

In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word “kha” for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.

We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.

We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.

Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:-

The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.

Subtraction is a little harder:-

A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.

Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:-

A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.

In 830, around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita Sara Samgraha which was designed as an updating of Brahmagupta‘s book. He correctly states that:-

… a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it.

However his attempts to improve on Brahmagupta‘s statements on dividing by zero seem to lead him into error. He writes:-

A number remains unchanged when divided by zero.

Since this is clearly incorrect my use of the words “seem to lead him into error” might be seen as confusing. The reason for this phrase is that some commentators on Mahavira have tried to find excuses for his incorrect statement.

Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-

A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.

So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and √0 = 0.

Perhaps we should note at this point that there was another civilisation which developed a place-value number system with a zero. This was the Maya people who lived in central America, occupying the area which today is southern Mexico, Guatemala, and northern Belize. This was an old civilisation but flourished particularly between 250 and 900. We know that by 665 they used a place-value number system to base 20 with a symbol for zero. However their use of zero goes back further than this and was in use before they introduced the place-valued number system. This is a remarkable achievement but sadly did not influence other peoples.

You can see a separate article about Mayan mathematics.

The brilliant work of the Indian mathematicians was transmitted to the Islamic and Arabic mathematicians further west. It came at an early stage for al-Khwarizmiwrote Al’Khwarizmi on the Hindu Art of Reckoning which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This work was the first in what is now Iraq to use zero as a place holder in positional base notation. Ibn Ezra, in the 12th century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe. The Book of the Number describes the decimal system for integers with place values from left to right. In this work ibn Ezra uses zero which he calls galgal (meaning wheel or circle). Slightly later in the 12th century al-Samawalwas writing:-

If we subtract a positive number from zero the same negative number remains. … if we subtract a negative number from zero the same positive number remains.

The Indian ideas spread east to China as well as west to the Islamic countries. In 1247 the Chinese mathematician Ch’in Chiu-Shao wrote Mathematical treatise in nine sections which uses the symbol O for zero. A little later, in 1303, Zhu Shijie wrote Jade mirror of the four elements which again uses the symbol O for zero.

Fibonacci was one of the main people to bring these new ideas about the number system to Europe.

An important link between the Hindu-Arabic number system and the European mathematics is the Italian mathematician Fibonacci.

In Liber Abaci he described the nine Indian symbols together with the sign 0 for Europeans in around 1200 but it was not widely used for a long time after that. It is significant that Fibonacci is not bold enough to treat 0 in the same way as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the “sign” zero while the other symbols he speaks of as numbers. Although clearly bringing the Indian numerals to Europe was of major importance we can see that in his treatment of zero he did not reach the sophistication of the Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and Islamic mathematicians such as al-Samawal.

One might have thought that the progress of the number systems in general, and zero in particular, would have been steady from this time on. However, this was far from the case. Cardan solved cubic and quartic equations without using zero. He would have found his work in the 1500’s so much easier if he had had a zero but it was not part of his mathematics. By the 1600’s zero began to come into widespread use but still only after encountering a lot of resistance.

Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21stcentury begin on 1 January 2001. Zero is still causing problems!

A history of Pi

A little known verse of the Bible reads

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23)

The same verse can be found in II Chronicles 4, 2. It occurs in a list of specifications for the great temple of Solomon, built around 950 BC and its interest here is that it gives π = 3. Not a very accurate value of course and not even very accurate in its day, for the Egyptian and Mesopotamian values of 25/8 = 3.125 and √10 = 3.162 have been traced to much earlier dates: though in defence of Solomon’s craftsmen it should be noted that the item being described seems to have been a very large brass casting, where a high degree of geometrical precision is neither possible nor necessary. There are some interpretations of this which lead to a much better value.

The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable. The earliest values of π including the ‘Biblical’ value of 3, were almost certainly found by measurement. In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there is good evidence for 4 × (8/9)2 = 3.16 as a value for π.

The first theoretical calculation seems to have been carried out by Archimedes of Syracuse (287-212 BC). He obtained the approximation

223/71 < π < 22/7.

Before giving an indication of his proof, notice that very considerable sophistication involved in the use of inequalities here. Archimedes knew, what so many people to this day do not, that π does not equal 22/7, and made no claim to have discovered the exact value. If we take his best estimate as the average of his two bounds we obtain 3.1418, an error of about 0.0002.

Here is Archimedes‘ argument.

Consider a circle of radius 1, in which we inscribe a regular polygon of 3 × 2n-1 sides, with semiperimeter bn, and superscribe a regular polygon of 3 × 2n-1 sides, with semiperimeter an.

The diagram for the case n = 2 is on the right.

The effect of this procedure is to define an increasing sequence

b1 , b2 , b3 , …

and a decreasing sequence

a1 , a2 , a3 , …

such that both sequences have limit π.

Using trigonometrical notation, we see that the two semiperimeters are given by

an = K tan(π/K), bn = K sin(π/K),

where K = 3 × 2n-1. Equally, we have

an+1 = 2K tan(π/2K), bn+1 = 2K sin(π/2K),

and it is not a difficult exercise in trigonometry to show that

(1/an + 1/bn) = 2/an+1   . . . (1)an+1bn = (bn+1)2       . . . (2)

Archimedes, starting from a1 = 3 tan(π/3) = 3√3 and b1 = 3 sin(π/3) = 3√3/2, calculated a2 using (1), then b2 using (2), then a3 using (1), then b3 using (2), and so on until he had calculated a6and b6. His conclusion was that

b6 < π < a6 .

It is important to realise that the use of trigonometry here is unhistorical: Archimedes did not have the advantage of an algebraic and trigonometrical notation and had to derive (1) and (2) by purely geometrical means. Moreover he did not even have the advantage of our decimal notation for numbers, so that the calculation of a6 and b6 from (1) and (2) was by no means a trivial task. So it was a pretty stupendous feat both of imagination and of calculation and the wonder is not that he stopped with polygons of 96 sides, but that he went so far.

For of course there is no reason in principle why one should not go on. Various people did, including:

Ptolemy (c. 150 AD) 3.1416
Zu Chongzhi (430-501 AD) 355/113
al-Khwarizmi (c. 800 ) 3.1416
al-Kashi (c. 1430) 14 places
Viète (1540-1603) 9 places
Roomen (1561-1615) 17 places
Van Ceulen (c. 1600) 35 places

Except for Zu Chongzhi, about whom next to nothing is known and who is very unlikely to have known about Archimedes‘ work, there was no theoretical progress involved in these improvements, only greater stamina in calculation. Notice how the lead, in this as in all scientific matters, passed from Europe to the East for the millennium 400 to 1400 AD.

Al-Khwarizmi lived in Baghdad, and incidentally gave his name to ‘algorithm’, while the words al jabr in the title of one of his books gave us the word ‘algebra’. Al-Kashi lived still further east, in Samarkand, while Zu Chongzhi, one need hardly add, lived in China.

The European Renaissance brought about in due course a whole new mathematical world. Among the first effects of this reawakening was the emergence of mathematical formulae for π. One of the earliest was that of Wallis(1616-1703)

2/π = ( …)/( …)

and one of the best-known is

π/4 = 1 – 1/3 + 1/51/7 + ….

This formula is sometimes attributed to Leibniz (1646-1716) but is seems to have been first discovered by James Gregory (1638- 1675).

These are both dramatic and astonishing formulae, for the expressions on the right are completely arithmetical in character, while π arises in the first instance from geometry. They show the surprising results that infinite processes can achieve and point the way to the wonderful richness of modern mathematics.

From the point of view of the calculation of π, however, neither is of any use at all. In Gregory‘s series, for example, to get 4 decimal places correct we require the error to be less than 0.00005 = 1/20000, and so we need about 10000 terms of the series. However, Gregory also showed the more general result

tan-1x = xx3/3 + x5/5 – … (-1 ≤ x ≤ 1)   . . . (3)

from which the first series results if we put x = 1. So using the fact that

tan-1(1/√3) = π/6 we getπ/6 = (1/√3)(1 – 1/(3.3) + 1/(5.3.3) – 1/( + …

which converges much more quickly. The 10th term is 1/(19 × 39√3), which is less than 0.00005, and so we have at least 4 places correct after just 9 terms.

An even better idea is to take the formula

π/4 = tan-1(1/2) + tan-1(1/3)   . . . (4)

and then calculate the two series obtained by putting first 1/2 and the 1/3 into (3).

Clearly we shall get very rapid convergence indeed if we can find a formula something like

π/4 = tan-1(1/a) + tan-1(1/b)

with a and b large. In 1706 Machin found such a formula:

π/4 = 4 tan-1(1/5) – tan-1(1/239)   . . . (5)

Actually this is not at all hard to prove, if you know how to prove (4) then there is no real extra difficulty about (5), except that the arithmetic is worse. Thinking it up in the first place is, of course, quite another matter.

With a formula like this available the only difficulty in computing π is the sheer boredom of continuing the calculation. Needless to say, a few people were silly enough to devote vast amounts of time and effort to this tedious and wholly useless pursuit. One of them, an Englishman named Shanks, used Machin‘s formula to calculate π to 707 places, publishing the results of many years of labour in 1873. Shanks has achieved immortality for a very curious reason which we shall explain in a moment.
Here is a summary of how the improvement went:

1699: Sharp used Gregory‘s result to get 71 correct digits
1701: Machin used an improvement to get 100 digits and the following used his methods:
1719: de Lagny found 112 correct digits
1789: Vega got 126 places and in 1794 got 136
1841: Rutherford calculated 152 digits and in 1853 got 440
1873: Shanks calculated 707 places of which 527 were correct

A more detailed Chronology is available.

Shanks knew that π was irrational since this had been proved in 1761 by Lambert. Shortly after Shanks‘ calculation it was shown by Lindemann that π is transcendental, that is, π is not the solution of any polynomial equation with integer coefficients. In fact this result of Lindemann showed that ‘squaring the circle’ is impossible. The transcendentality of π implies that there is no ruler and compass construction to construct a square equal in area to a given circle.

Very soon after Shanks‘ calculation a curious statistical freak was noticed by De Morgan, who found that in the last of 707 digits there was a suspicious shortage of 7’s. He mentions this in his Budget of Paradoxes of 1872 and a curiosity it remained until 1945 when Ferguson discovered that Shanks had made an error in the 528thplace, after which all his digits were wrong. In 1949 a computer was used to calculate π to 2000 places. In this and all subsequent computer expansions the number of 7’s does not differ significantly from its expectation, and indeed the sequence of digits has so far passed all statistical tests for randomness.

You can see 2000 places of π.

We should say a little of how the notation π arose. Oughtred in 1647 used the symbol d/π for the ratio of the diameter of a circle to its circumference. David Gregory(1697) used π/r for the ratio of the circumference of a circle to its radius. The first to use π with its present meaning was an Welsh mathematician William Jones in 1706 when he states “3.14159 andc. = π”. Euler adopted the symbol in 1737 and it quickly became a standard notation.

We conclude with one further statistical curiosity about the calculation of π, namely Buffon‘s needle experiment. If we have a uniform grid of parallel lines, unit distance apart and if we drop a needle of length k < 1 on the grid, the probability that the needle falls across a line is 2k/π. Various people have tried to calculate π by throwing needles. The most remarkable result was that of Lazzerini (1901), who made 34080 tosses and got

π = 355/113 = 3.1415929

which, incidentally, is the value found by Zu Chongzhi. This outcome is suspiciously good, and the game is given away by the strange number 34080 of tosses. Kendall and Moran comment that a good value can be obtained by stopping the experiment at an optimal moment. If you set in advance how many throws there are to be then this is a very inaccurate way of computing π. Kendall and Moran comment that you would do better to cut out a large circle of wood and use a tape measure to find its circumference and diameter.

Still on the theme of phoney experiments, Gridgeman, in a paper which pours scorn on Lazzerini and others, created some amusement by using a needle of carefully chosen length k = 0.7857, throwing it twice, and hitting a line once. His estimate for π was thus given by

2 × 0.7857 / π = 1/2

from which he got the highly creditable value of π = 3.1428. He was not being serious!

It is almost unbelievable that a definition of π was used, at least as an excuse, for a racial attack on the eminent mathematician Edmund Landau in 1934. Landau had defined π in this textbook published in Göttingen in that year by the, now fairly usual, method of saying that π/2 is the value of x between 1 and 2 for which cos x vanishes. This unleashed an academic dispute which was to end in Landau‘s dismissal from his chair at Göttingen. Bieberbach, an eminent number theorist who disgraced himself by his racist views, explains the reasons for Landau‘s dismissal:-

Thus the valiant rejection by the Göttingen student body which a great mathematician, Edmund Landau, has experienced is due in the final analysis to the fact that the un-German style of this man in his research and teaching is unbearable to German feelings. A people who have perceived how members of another race are working to impose ideas foreign to its own must refuse teachers of an alien culture.

G H Hardy replied immediately to Bieberbach in a published note about the consequences of this un-German definition of π

There are many of us, many Englishmen and many Germans, who said things during the War which we scarcely meant and are sorry to remember now. Anxiety for one’s own position, dread of falling behind the rising torrent of folly, determination at all cost not to be outdone, may be natural if not particularly heroic excuses. Professor Bieberbach‘s reputation excludes such explanations of his utterances, and I find myself driven to the more uncharitable conclusion that he really believes them true.

Not only in Germany did π present problems. In the USA the value of π gave rise to heated political debate. In the State of Indiana in 1897 the House of Representatives unanimously passed a Bill introducing a new mathematical truth.

Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square of one side.
(Section I, House Bill No. 246, 1897)

The Senate of Indiana showed a little more sense and postponed indefinitely the adoption of the Act!

Open questions about the number π

  1. Does each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each occur infinitely often in π?
  2. Brouwer‘s question: In the decimal expansion of π, is there a place where a thousand consecutive digits are all zero?
  3. Is π simply normal to base 10? That is does every digit appear equally often in its decimal expansion in an asymptotic sense?
  4. Is π normal to base 10? That is does every block of digits of a given length appear equally often in its decimal expansion in an asymptotic sense?
  5. Is π normal ? That is does every block of digits of a given length appear equally often in the expansion in every base in an asymptotic sense? The concept was introduced by Borel in 1909.
  6. Another normal question! We know that π is not rational so there is no point from which the digits will repeat. However, if π is normal then the first million digits 314159265358979… will occur from some point. Even if π is not normal this might hold! Does it? If so from what point? Note: Up to 200 million the longest to appear is 31415926 and this appears twice.

As a postscript, here is a mnemonic for the decimal expansion of π. Each successive digit is the number of letters in the corresponding word.

How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard…:3.14159265358979323846264…

You can see more about the history of π in the History topic: Squaring the circle and you can see a Chronology of how calculations of π have developed over the years.


The Golden ratio

Euclid, in The Elements, says that the line AB is divided in extreme and mean ratio by C if AB:AC = AC:CB.

Although Euclid does not use the term, we shall call this the golden ratio. The definition appears in Book VI but there is a construction given in Book II, Theorem 11, concerning areas which is solved by dividing a line in the golden ratio. As well as constructions to divide a line in the golden ratio, Euclid gives applications such as the construction of a regular pentagon, an icosahedron and a dodecahedron. Here is how the golden ratio comes into the construction of a pentagon.

First construct an isosceles triangle whose base angles are double the vertex angle. This is done by taking a line AB and marking C on the line in the golden ratio. Then draw a circle with centre A radius AB. Mark D on the circle so that AC = CD = BD. The triangle ABD has the property that its base angles are double its vertex angle.

Now starting with such a triangle ABD draw a circle through A, B and D. Then bisect the angle ADB with the line DE meeting the circle at E. Note that the line passes through C, the point dividing AB in the golden ratio. Similarly construct F and draw the pentagon AEBDF.

Of course nobody believes that Euclid‘s Elements represents original work so there is the question of who studied the golden ratio before Euclid. Now some historians believe that Book II of The Elements covers material originally studied by Theodorus of Cyrene while others attribute the material to Pythagoras, or at least to the Pythagoreans. Proclus, writing in the fifth century AD, claims:-

Eudoxus … multiplied the number of propositions concerning the section which had their origin in Plato, employing the method of analysis for their solution.

Many believe that by ‘section’ Proclus means ‘golden ratio’. Eudoxuscertainly attended lectures by Plato so it is entirely reasonable that he might work on topics suggested during these lectures. Heath writes in his edition of Euclid‘s Elements:-

This idea that Plato began the study of [the golden ratio] as a subject in itself is not in the least inconsistent with the supposition that the problem of Euclid II, 11 was solved by the Pythagoreans.

Heath claims later in the same work that the construction of a pentagon using the isosceles triangle method referred to above was known to the Pythagoreans so there is a fair amount of evidence to suggest that this is where the study of the golden ratio began.

Hypsicles, around 150 BC, wrote on regular polyhedra. He is the author of what has been called Book XIV of Euclid‘s Elements, a work which deals with inscribing regular solids in a sphere. The golden ratio enters into the constructions.

Up to this time the golden ratio seems to have been considered as a geometrical property and there is no obvious sign that any attempt was made to associate a number with the ratio. Of course if AB has length 1 and AC = x where C divides AB in the golden ratio, then we can use simple algebra to find x.

1/x = x/(1 – x) gives x2 + x – 1 = 0 so x = (√5-1)/2.Then the golden ratio is 1/x = (√5 + 1)/2 = 1.6180339887498948482…

Heron certainly begins to compute approximate ratios, and in his work he gives approximate values for the ratio of the area of the pentagon to the area of the square of one side. With Ptolemytrigonometric tables, at least in terms of chords of circles, begin to be computed. He calculates the side of a regular pentagon in terms of the radius of the circumscribed circle.

With the development of algebra by the Arabs one might expect to find the quadratic equation (or a related one) to that which we have given above. Al-Khwarizmi does indeed give several problems on dividing a line of length 10 into two parts and one of these does find a quadratic equation for the length of the smaller part of the line of length 10 divided in the golden ratio. There is no mention of the golden ratio, however, and it is unclear whether al-Khwarizmi is thinking of this particular problem.

Abu Kamil gives similar equations which arise from dividing a line of length 10 in various ways. Two of these ways are related to the golden ratio but again it is unclear whether Abu Kamil is aware of this. However, when Fibonacci produced Liber Abaci he used many Arabic sources and one of them was the problems of Abu Kamil. Fibonacci clearly indicates that he is aware of the connection between Abu Kamil‘s two problems and the golden ratio. In Liber Abaci he gives the lengths of the segments of a line of length 10 divided in the golden ratio as √125 -5 and 15 – √125.

Pacioli wrote Divina proportione (Divine proportion) which is his name for the golden ratio. The book contains little new on the topic, collecting results from Euclid and other sources on the golden ratio. He states (without any attempt at a proof or a reference) that the golden ratio cannot be rational. He also states the result given in Liber Abaci on the lengths of the segments of a line of length 10 divided in the golden ratio. There is little new in Pacioli‘s book which merely restates (usually without proof) results which had been published by other authors. Of course the title is interesting and Pacioli writes:

… it seems to me that the proper title for this treatise must be Divine Proportion. This is because there are very many similar attributes which I find in our proportion – all befitting God himself – which is the subject of our very useful discourse.

He gives five such attributes, perhaps the most interesting being:-

… just like God cannot be properly defined, nor can be understood through words, likewise this proportion of ours cannot ever be designated through intelligible numbers, nor can it be expressed through any rational quantity, but always remains occult and secret, and is called irrational by the mathematicians.

Cardan, Bombelli and others included problems in their texts on finding the golden ratio using quadratic equations. A surprising piece of information is contained in a copy of the 1509 edition of Pacioli‘s Euclid’s Elements. Someone has written a note which clearly shows that they knew that the ratio of adjacent terms in the Fibonacci sequence tend to the golden number. Handwriting experts date the note as early 16thcentury so there is the intriguing question as to who wrote it. See [6] for further details.

The first known calculation of the golden ratio as a decimal was given in a letter written in 1597 by Michael Maestlin, at the University of Tübingen, to his former student Kepler. He gives “about 0.6180340” for the length of the longer segment of a line of length 1 divided in the golden ratio. The correct value is 0.61803398874989484821… . The mystical feeling for the golden ratio was of course attractive to Kepler, as was its relation to the regular solids. His writings on the topic are a mixture of good mathematics and magic. He, like the annotator of Pacioli‘s Euclid, knows that the ratio of adjacent terms of the Fibonacci sequence tends to the golden ratio and he states this explicitly in a letter he wrote in 1609.

The result that the quotients of adjacent terms of the Fibonacci sequence tend to the golden ratio is usually attributed to Simson who gave the result in 1753. We have just seen that he was not the first give the result and indeed Albert Girard also discovered it independently of Kepler. It appears in a publication of 1634 which appeared two years after Albert Girard‘s death.

In this article we have used the term golden ratio but this term was never used by any of the mathematicians who we have noted above contributed to its development. We commented that “section” was possibly used by Proclus although some historians dispute that his reference to section means the golden ratio. The common term used by early writers was simply “division in extreme and mean ratio”. Paciolicertainly introduced the term “divine proportion” and some later writers such as Ramus and Clavius adopted this term. Clavius also used the term “proportionally divided” and similar expressions appear in the works of other mathematicians. The term “continuous proportion” was also used.

The names now used are golden ratio, golden number or golden section. These terms are modern in the sense that they were introduced later than any of the work which we have discussed above. The first known use of the term appears in a footnote in Die reine Elementar-Matematik by Martin Ohm (the brother of Georg Simon Ohm):-

One is also in the habit of calling this division of an arbitrary line in two such parts the golden section; one sometimes also says in this case: the line r is divided in continuous proportion.

The first edition of Martin Ohm’s book appeared in 1826. The footnote just quoted does not appear and the text uses the term “continuous proportion”. Clearly sometime between 1826 and 1835 the term “golden section” began to be used but its origin is a puzzle. It is fairly clear from Ohm’s footnote that the term “golden section” is not due to him. Fowler, in [9], examines the evidence and reaches the conclusion that 1835 marks the first appearance of the term.

The golden ratio has been famed throughout history for its aesthetic properties and it is claimed that the architecture of Ancient Greece was strongly influenced by its use. The article [11] discusses whether the golden section is a universal natural phenomenon, to what extent it has been used by architects and painters, and whether there is a relationship with aesthetics.

The number e

One of the first articles which we included in the “History Topics” section of our web archive was on the history of π. It is a very popular article and has prompted many to ask for a similar article about the number e. There is a great contrast between the historical developments of these two numbers and in many ways writing a history of e is a much harder task than writing one for π. The number e is, compared to π, a relative newcomer on the mathematics scene.

The number e first comes into mathematics in a very minor way. This was in 1618 when, in an appendix to Napier‘s work on logarithms, a table appeared giving the natural logarithms of various numbers. However, that these were logarithms to base e was not recognised since the base to which logarithms are computed did not arise in the way that logarithms were thought about at this time. Although we now think of logarithms as the exponents to which one must raise the base to get the required number, this is a modern way of thinking. We will come back to this point later in this essay. This table in the appendix, although carrying no author’s name, was almost certainly written by Oughtred. A few years later, in 1624, again e almost made it into the mathematical literature, but not quite. In that year Briggs gave a numerical approximation to the base 10 logarithm of e but did not mention e itself in his work.

The next possible occurrence of e is again dubious. In 1647 Saint-Vincentcomputed the area under a rectangular hyperbola. Whether he recognised the connection with logarithms is open to debate, and even if he did there was little reason for him to come across the number e explicitly. Certainly by 1661 Huygens understood the relation between the rectangular hyperbola and the logarithm. He examined explicitly the relation between the area under the rectangular hyperbola yx = 1 and the logarithm. Of course, the number e is such that the area under the rectangular hyperbola from 1 to e is equal to 1. This is the property that makes e the base of natural logarithms, but this was not understood by mathematicians at this time, although they were slowly approaching such an understanding.

Huygens made another advance in 1661. He defined a curve which he calls “logarithmic” but in our terminology we would refer to it as an exponential curve, having the form y = kax. Again out of this comes the logarithm to base 10 of e, which Huygens calculated to 17 decimal places. However, it appears as the calculation of a constant in his work and is not recognised as the logarithm of a number (so again it is a close call but e remains unrecognised).

Further work on logarithms followed which still does not see the number e appear as such, but the work does contribute to the development of logarithms. In 1668 Nicolaus Mercator published Logarithmotechnia which contains the series expansion of log(1+x). In this work Mercator uses the term “natural logarithm” for the first time for logarithms to base e. The number e itself again fails to appear as such and again remains elusively just round the corner.

Perhaps surprisingly, since this work on logarithms had come so close to recognising the number e, when e is first “discovered” it is not through the notion of logarithm at all but rather through a study of compound interest. In 1683 Jacob Bernoulli looked at the problem of compound interest and, in examining continuous compound interest, he tried to find the limit of (1 + 1/n)n as n tends to infinity. He used the binomial theorem to show that the limit had to lie between 2 and 3 so we could consider this to be the first approximation found to e. Also if we accept this as a definition of e, it is the first time that a number was defined by a limiting process. He certainly did not recognise any connection between his work and that on logarithms.

We mentioned above that logarithms were not thought of in the early years of their development as having any connection with exponents. Of course from the equation x = at, we deduce that t = log x where the log is to base a, but this involves a much later way of thinking. Here we are really thinking of log as a function, while early workers in logarithms thought purely of the log as a number which aided calculation. It may have been Jacob Bernoulli who first understood the way that the log function is the inverse of the exponential function. On the other hand the first person to make the connection between logarithms and exponents may well have been James Gregory. In 1684 he certainly recognised the connection between logarithms and exponents, but he may not have been the first.

As far as we know the first time the number e appears in its own right is in 1690. In that year Leibniz wrote a letter to Huygens and in this he used the notation b for what we now call e. At last the number e had a name (even if not its present one) and it was recognised. Now the reader might ask, not unreasonably, why we have not started our article on the history of e at the point where it makes its first appearance. The reason is that although the work we have described previously never quite managed to identify e, once the number was identified then it was slowly realised that this earlier work is relevant. Retrospectively, the early developments on the logarithm became part of an understanding of the number e.

We mentioned above the problems arising from the fact that log was not thought of as a function. It would be fair to say that Johann Bernoullibegan the study of the calculus of the exponential function in 1697 when he published Principia calculi exponentialium seu percurrentium. The work involves the calculation of various exponential series and many results are achieved with term by term integration.

So much of our mathematical notation is due to Euler that it will come as no surprise to find that the notation e for this number is due to him. The claim which has sometimes been made, however, that Euler used the letter e because it was the first letter of his name is ridiculous. It is probably not even the case that the e comes from “exponential”, but it may have just be the next vowel after “a” and Euler was already using the notation “a” in his work. Whatever the reason, the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731. He made various discoveries regarding e in the following years, but it was not until 1748 when Euler published Introductio in Analysin infinitorum that he gave a full treatment of the ideas surrounding e. He showed that

e = 1 + 1/1! + 1/2! + 1/3! + …

and that e is the limit of (1 + 1/n)n as n tends to infinity. Euler gave an approximation for e to 18 decimal places,

e = 2.718281828459045235

without saying where this came from. It is likely that he calculated the value himself, but if so there is no indication of how this was done. In fact taking about 20 terms of 1 + 1/1! + 1/2! + 1/3! + … will give the accuracy which Euler gave. Among other interesting results in this work is the connection between the sine and cosine functions and the complex exponential function, which Euler deduced using De Moivre‘s formula.

Interestingly Euler also gave the continued fraction expansion of e and noted a pattern in the expansion. In particular he gave


Euler did not give a proof that the patterns he spotted continue (which they do) but he knew that if such a proof were given it would prove that e is irrational. For, if the continued fraction for (e – 1)/2 were to follow the pattern shown in the first few terms, 6, 10, 14, 18, 22, 26, … (add 4 each time) then it will never terminate so (e – 1)/2 (and so e) cannot be rational. One could certainly see this as the first attempt to prove that e is not rational.

The same passion that drove people to calculate to more and more decimal places of π never seemed to take hold in quite the same way for e. There were those who did calculate its decimal expansion, however, and the first to give e to a large number of decimal places was Shanks in 1854. It is worth noting that Shanks was an even more enthusiastic calculator of the decimal expansion of π. Glaisher showed that the first 137 places of Shanks calculations for e were correct but found an error which, after correction by Shanks, gave e to 205 places. In fact one needs about 120 terms of 1 + 1/1! + 1/2! + 1/3! + … to obtain e correct to 200 places.

In 1864 Benjamin Peirce had his picture taken standing in front of a blackboard on which he had written the formula ii = √(eπ). In his lectures he would say to his students:-

Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important.

Most people accept Euler as the first to prove that e is irrational. Certainly it was Hermite who proved that e is not an algebraic number in 1873. It is still an open question whether ee is algebraic, although of course all that is lacking is a proof – no mathematician would seriously believe that ee is algebraic! As far as we are aware, the closest that mathematicians have come to proving this is a recent result that at least one of ee and e to the power e2 is transcendental.

Further calculations of decimal expansions followed. In 1884 Boorman calculated e to 346 places and found that his calculation agreed with that of Shanks as far as place 187 but then became different. In 1887 Adams calculated the logarithm of e to the base 10 to 272 places.

Pell’s equation

We will discuss below whether Pell‘s equation is properly named. By this we mean simply: did Pell contribute at all to the study of Pell‘s equation? There is no doubt that the equation had been studied in depth for hundreds of years before Pell was born. In fact the first contribution by Brahmagupta was made around 1000 years before Pell‘s time and it is with Brahmagupta‘s contribution that we begin our historical study.

First let us say what Pell‘s equation is. We are talking about the indeterminate quadratic equation

nx2 + 1 = y2

which we can also write as

y2nx2 = 1

where n is a given integer and we are looking for integer solutions (x, y).

Now, although it is fair to say that Brahmagupta was the first to study this equation, it is equally possible to see that earlier authors had studied problems related to Pell‘s equation. To mention some briefly: Diophantusexamines problems related to Pell‘s equation and we can reduce Archimedes‘ “cattle problem” to solving Pell‘s equation although there is no evidence that Archimedesmade this connection.

Let us first note that

(b2na2)(d2nc2) = (bd + nac)2n(bc + ad)2


(b2na2)(d2nc2) = (bdnac)2n(bcad)2

From this we see that if

b2na2 = 1 and d2nc2 = 1


(bd + nac)2n(bc + ad)2 = 1


(bdnac)2n(bcad)2 = 1.

In other words, if (a, b) and (c, d) are solutions to Pell‘s equation then so are

(bc + ad, bd + nac) and (bcad, bdnac).

This fundamentally important fact generalises easily to give Brahmagupta‘s lemma, namely that if (a, b) and (c, d) are integer solutions of ‘Pell type equations’ of the form

na2 + k = b2 and nc2 + k‘ = d2


(bc + ad, bd + nac) and (bcad, bdnac)

are both integer solutions of the ‘Pell type equation’

nx2 + kk‘ = y2.

The proof that we have given is due to European mathematicians in the 17th Century (and we shall comment further on it later in this article), but Brahmagupta‘s lemma was discovered by Brahmagupta in 628 AD. The method was called samasa by the Indian mathematicians but we shall call it the ‘method of composition’. In fact this method of composition allowed Brahmagupta to make a number of fundamental discoveries regarding Pell‘s equation.

One property that he deduced was that if (a, b) satisfies Pell‘s equation so does (2ab, b2 + na2). This follows immediately applying the method of composition to (a, b) and (a, b). Now of course the method of composition can be applied again to (a, b) and (2ab, b2+ na2) to get another solution and Brahmagupta immediately saw that from one solution of Pell‘s equation he could generate many solutions.

This was not the only way that Brahmagupta used the method of composition. He also noted that, using a similar argument to what we have just given, if x = a, y = b is a solution of nx2 + k = y2then applying the method of composition to (a, b) and (a, b) gave (2ab, b2 + na2) as a solution of nx2 + k2 = y2 and so, dividing through by k2, gives

x = 2ab/k, y = (b2 + na2)/k

as a solution of Pell‘s equation nx2 + 1 = y2.

How does this help? These values of x and y do not look like integers. Well if k = 2 then, since (a, b) is a solution of nx2 + k = y2 we have na2 = b2 – 2. Thus

x = 2ab/2 = ab,y = (b2 + na2)/2 = (2b2 – 2)/2 = b2 – 1

and this is an integer solution to Pell‘s equation. If k = -2 then essentially the same argument works while if k = 4 or k = -4 then a more complicated method, still based on the method of composition, shows that integer solutions to Pell‘s equation can be found. So Brahmaguptawas able to show that if he could find (a, b) which “nearly” satisfied Pell‘s equation in the sense that na2 + k = b2 where k = 1, -1, 2, -2, 4, or -4 then he could find one, and therefore many, integer solutions to Pell‘s equation. Often he could find trial solutions which worked for one of these values of k and so in many cases he was able to give solutions.

For example, if we attempt to solve 23x2 + 1 = y2 we see that a = 1, b = 5 satisfies 23a2 + 2 = b2 so, by the above argument, x = 5, y = 24 satisfies Pell‘s equation. Applying the method of composition to the pair (5, 24) gives

x = 2×5×24 = 240,  y = 242 + 23×52 = 1151

as another solution. Applying the method of composition again yields

x = 11515,  y = 55224

and yet again gives

x = 552480,  y = 2649601

and so on.

Among the examples Brahmagupta gives himself is a solution of Pell‘s equation

83x2 + 1 = y2

where he notes that the pair (1, 9) satisfies

83×12 – 2 = 92

and applies his method to find the solution

x = 9,  y = 82.

We can now generate a sequence of solutions (x,y):

(9, 82),
(1476, 13447),
(242055, 2205226),
(39695544, 361643617),
(6509827161, 59307347962),
(1067571958860, 9726043422151),
(175075291425879, 1595011813884802)

and so on.

One can only marvel at this brilliant work done by Brahmagupta in 628 AD.

The next step forward was taken by Bhaskara II in 1150. He discovered the cyclic method, called chakravala by the Indians, which was an algorithm to produce a solution to Pell‘s equation nx2 + 1 = y2 starting off from any “close” pair (a, b) with na2 + k = b2. We can assume that a and b are coprime, for otherwise we could divide each by their gcd and get a “closer” solution with smaller k. Clearly a, k are then also coprime.

The method relies on a simple observation, namely that, for any m, (1, m) satisfies the ‘Pell type equation’

n.12 + (m2n) = m2.

Bhaskara II now applies the method of composition to the pairs (a, b) and (1, m) to get

n(am + b)2 + (m2n)k = (bm + na)2.

Dividing by k we note that

x = (am + b)/k, y = (bm + na)/k

is a solution to

nx2 + (m2n)/k = y2.

Since a, k are coprime we can chose m such that am + b is divisible by k. Bhaskara II now knows (but he gives no proof) that when m is chosen so that am + b is divisible by k then m2n and bm + na are also divisible by k. With such a choice of m he therefore has integer solutions

x = (am + b)/k, y = (bm + na)/k

to the ‘Pell type equation’ nx2 + (m2n)/k = y2 where (m2n)/k is also an integer.

Next Bhaskara II knows that there are infinitely many m such that am + b is divisible by k. He chooses the one which makes m2n as small as possible in absolute value. If (m2n)/k is one of 1, -1, 2, -2, 4, -4 then we can apply Brahmagupta‘s method to find a solution to Pell‘s equation nx2+ 1 = y2. If (m2n)/k is not one of these values then repeat the process starting this time with the solution x = (am + b)/k, y = (bm + na)/k to the ‘Pell type equation’ nx2 + (m2n)/k = y2 in exactly the same way as we applied the process to na2 + k = b2. Bhaskara II knows (almost certainly by experience rather than by having a proof) that the process will end after a finite number of steps. This happens when an equation of the form nx2 + t = y2 is reached where t is one of 1, -1, 2, -2, 4, -4.

Bhaskara II gives examples in Bijaganita and the first we look at is

61x2 + 1 = y2.

Using the above method he chooses m so the (m + 8)/3 is an integer, making sure that m2 – 61 is as small as possible. Taking m = 7 he gets

x = 5,  y = 39

as a solution to the ‘Pell type equation’ nx2 – 4 = y2. But this is an equation which Brahmagupta‘s method solves giving

x = 226153980,  y = 1766319049

as the smallest solution to 61x2 + 1 = y2.

Why do we suspect that Bhaskara II had no proof of the method? Well there are at least two reasons. Firstly the proof is long and difficult and would appear to be well beyond 12th Century mathematics. Secondly the algorithm always reaches a solution of Pell‘s equation after a finite number of steps without stopping when an equation of the type nx2 + k = y2 where k = -1, 2, -2, 4, or -4 is reached and then applying Brahmagupta‘s method. If experience of the algorithm is only via examples then, knowing how to proceed when k = -1, 2, -2, 4, or -4 is reached, it is natural to switch to Brahmagupta‘s method at that point. However, when one writes down a proof it should become clear that the algorithm switching to Brahmagupta‘s method is never necessary (although can reach the solution more quickly).

The next contribution to Pell‘s equation was made by Narayana who, in the 14th Century, wrote a commentary on Bhaskara II‘s Bijaganita.Narayana gave some new examples of the cyclic method. Here are two of his examples:

103x2 + 1 = y2.

Choosing a = 1, b = 10 Narayana obtains

103×12 – 3 = 102.

Choose m so that m + 10 is divisible by -3 with m2 – 103 as small as possible leads to m = 11 and we obtain

103×72 – 6 = 712.

Next we must choose m so that 7m + 71 is divisible by -6 and m2 – 103 as small as possible. Take m = 7 to get the equation

103×202 + 9 = 2032.

Continuing, choose m so that 20m+203 is divisible by 9 and m2 – 103 as small as possible. Take m = 11 to get the equation

103×472 + 2 = 4772.

Now Narayana applies Brahmagupta‘s method, in the form we gave above for equations with k = 2, to obtain the solutions

x = 22419,  y = 227528.

His next example is a solution of Pell‘s equation

97x2 + 1 = y2

which leads successively, by applying the cyclic method, to the equations

97×12 + 3 = 10297×72 + 8 = 69297×202 + 9 = 1972

97×532 + 11 = 5222

97×862 – 3 = 8472

97×5692 – 1 = 56042

Finally Narayana applies Brahmagupta‘s method to this last equation to get the solution

x = 6377352,  y = 62809633

Now the brilliant ideas of Brahmagupta, Bhaskara II and Narayana were completely unknown to the European mathematicians in the 17thCentury. The European interest began in 1657 when Fermat issued a challenge to the mathematicians of Europe and England. Fermat wrote:-

We await these solutions, which, if England or Belgic or Celtic Gaul do not produce, then Narbonese Gaul will.

Narbonese Gaul, of course, was the area around Toulouse where Fermatlived! One of Fermat‘s challenge problems was the same example of Pell’s equation which had been studied by Bhaskara II 500 years earlier, namely to find solutions to

61x2 + 1 = y2.

Several mathematicians participated in Fermat‘s challenge, in particular Frenicle de Bessy, Brouncker and Wallis. There followed an exchange of letters between these mathematicians during 1657-58 which Wallispublished in Commercium epistolicum in 1658. Brouncker discovered a method of solution which is essentially the same as the method of continued fractions which was later developed rigorously by Lagrange. Frenicle de Bessy tabulated the solutions of Pell‘s equation for all n up to 150, although this was never published and his efforts have been lost. He challenged Brouncker who was claiming to be able to solve any example of Pell‘s equation to solve

313x2 + 1 = y2.

Brouncker found the smallest solutions, using his method, which is

x = 1819380158564160,  y = 32188120829134849

and he sent these to Frenicle de Bessy claiming that they had only taken him “an hour or two” to find.

In Commercium epistolicum Wallis gave two methods of proving Brahmagupta‘s lemma which are both essentially equivalent to the argument we gave at the beginning of this article based on the result

(b2na2)(d2nc2) = (bd + nac)2n(bc + ad)2.

In 1658 Rahn published an algebra book which contained an example of Pell‘s equation. This book was written with help from Pell and it is the only known connection between Pell and the equation which has been named after him.

Wallis published Treatise on Algebra in 1685 and Chapter 98 of that work is devoted to giving methods to solve Pell‘s equation based on the exchange of letters he had published in Commercium epistolicum in 1658. However, in his algebra text Wallis put all the methods into a standard form.

We should note that by this time several mathematicians had claimed that Pell‘s equation nx2 + 1 = y2 had solutions for any n. Wallis, describing Brouncker‘s method, had made that claim, as had Fermat when commenting on the solutions proposed to his challenge. In fact Fermatclaimed, correctly of course, that for any n Pell‘s equation had infinitely many solutions.

Euler gave Brahmagupta‘s lemma and its proof in a similar form to that which we have given above. He was, of course, aware of the work of Brouncker on Pell‘s equation as presented by Wallis, but he was totally unaware of the contributions of the Indian mathematicians. He gave the basis for the continued fractions approach to solving Pell‘s equation which was put into a polished form by Lagrange in 1766. The other major contribution of Euler was in naming the equation “Pell‘s equation” and it is generally believed that he gave it that name because he confused Brouncker and Pell, thinking that the major contributions which Wallishad reported on as due to Brouncker were in fact the work of Pell.

Lagrange published his Additions to Euler‘s Elements of algebra in 1771 and this contains his rigorous version of Euler‘s continued fraction approach to Pell‘s equation. This established rigorously the fact that for every n Pell‘s equation had infinitely many solutions. The solution depends on the continued fraction expansion of √n. In the continued fraction of the square root of an integer the same denominators recur periodically. Moreover, the pattern in most of the recurring sequence is “palindromic”. i.e. up to the last element, the second half of the periodic sequence is the first half in reverse. The last number in the repeating sequence is double the integer part of the square root.

For example √19 has the continued fraction expansion

which recurs with length 6. The convergent immediately before the point from which it repeats is 170/39 and Lagrange‘s theory says that

x = 39,  y = 170

will be the smallest solution to Pell‘s equation

19x2 + 1 = y2.

To find the infinite series of solutions take the powers of 170 + 39√19. For example

(170 + 39√19)2 = 57799 + 13260√19


x = 13260, y = 57799

will give a second solution to the equation. Again

(170 + 39√19)3 = 19651490 + 4508361√19


x = 4508361,  y = 19651490

as the next solution. Here are the first few powers of (170 + 39√19), starting with its square, which gives the first few solutions to the equation 19x2 + 1 = y2

57799 + 13260√1919651490 + 4508361√196681448801 + 1532829480√19

2271672940850 + 521157514839√19

772362118440199 + 177192022215780√19

262600848596726810 + 60244766395850361√19

89283516160768675201 + 20483043382566906960√19

Although the continued fraction approach to solving Pell‘s equation is a very nice one for small values on n, the difficulty of the method has been analysed to see if it is the most efficient for large n. A polynomial time method in the length of the input n would be an algorithm which took time bounded by a fixed power of log n (the length of the input). The continued fraction method is not a polynomial time algorithm, and indeed it is now known that no polynomial time algorithm exists for solving Pell‘s equation.


An article on infinity in a History of Mathematics Archive presents special problems. Does one concentrate purely on the mathematical aspects of the topic or does one consider the philosophical and even religious aspects? In this article we take the view that historically one cannot separate the philosophical and religious aspects from mathematical ones since they play an important role in how ideas developed. This is particularly true in ancient Greek times, as Knorr writes in [26]:-

The interaction of philosophy and mathematics is seldom revealed so clearly as in the study of the infinite among the ancient Greeks. The dialectical puzzles of the fifth-century Eleatics, sharpened by Plato and Aristotle in the fourth century, are complemented by the invention of precise methods of limits, as applied by Eudoxus in the fourth century and Euclid and Archimedes in the third.

Of course from the time people began to think about the world they lived in, questions about infinity arose. There were questions about time. Did the world come into existence at a particular instant or had it always existed? Would the world go on for ever or was there a finite end? Then there were questions about space. What happened if one kept travelling in a particular direction? Would one reach the end of the world or could one travel for ever? Again above the earth one could see stars, planets, the sun and moon, but was this space finite or do it go on for ever?

The questions above are very fundamental and must have troubled thinkers long before recorded history. There were more subtle questions about infinity which were also asked at a stage when people began to think deeply about the world. What happened if one cut a piece of wood into two pieces, then again cut one of the pieces into two and continued to do this. Could one do this for ever?

We should begin our account of infinity with the “fifth-century Eleatic” Zeno. The early Greeks had come across the problem of infinity at an early stage in their development of mathematics and science. In their study of matter they realised the fundamental question: can one continue to divide matter into smaller and smaller pieces or will one reach a tiny piece which cannot be divided further. Pythagoras had argued that “all is number” and his universe was made up of finite natural numbers. Then there were Atomists who believed that matter was composed of an infinite number of indivisibles. Parmenides and the Eleatic School, which included Zeno, argued against the atomists. However Zeno‘s paradoxes show that both the belief that matter is continuously divisible and the belief in an atomic theory both led to apparent contradictions.

Of course these paradoxes arise from the infinite. Aristotle did not seem to have fully appreciated the significance of Zeno‘s arguments but the infinite did worry him nevertheless. He introduced an idea which would dominate thinking for two thousand years and is still a persuasive argument to some people today. Aristotle argued against the actual infinite and, in its place, he considered the potential infinite. His idea was that we can never conceive of the natural numbers as a whole. However they are potentially infinite in the sense that given any finite collection we can always find a larger finite collection.

Of relevance to our discussion is the remarkable advance made by the Babylonians who introduced the idea of a positional number system which, for the first time, allowed a concise representation of numbers without limit to their size. Despite positional number systems, Aristotle‘s argument is quite convincing. Only a finite number of natural numbers has ever been written down or has ever been conceived. If L is the largest number conceived up till now then I will go further and write down L + 1, or L2 but still only finitely many have been conceived. Aristotle discussed this in Chapters 4-8 of Book III of Physics (see [36]) where he claimed that denying that the actual infinite exists and allowing only the potential infinite would be no hardship to mathematicians:-

Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untransversable. In point of fact they do not need the infinite and do not use it. They postulate only that the finite straight line may be produced as far as they wish.

Cantor, over two thousand years later, argued that Aristotle was making a distinction which was only in his use of words:-

… in truth the potentially infinite has only a borrowed reality, insofar as a potentially infinite concept always points towards a logically prior actually infinite concept whose existence it depends on.

We will come to Cantor‘s ideas towards the end of this article but for the moment let us consider the effect Aristotle had on later Greek mathematicians by only allowing the potentially infinite, particularly on Euclid; see for example [36]. How then, one may ask, was Euclid able to prove that the set of prime numbers is infinite in 300 BC? Well the answer is that Euclid did not prove this in the Elements. This is merely a modern phrasing of what Euclid actually stated as his theorem which, according to Heath‘s translation, reads:-

Prime numbers are more than any assigned magnitude of prime numbers.

So in fact what Euclid proved was that the prime numbers are potentially infinite but in practice, of course, this amounts to the same thing. His proof shows that given any finite collection of prime numbers there must be a prime number not in the collection.

We should discuss other aspects of the infinite which play a crucial role in the Elements. There Euclid explains the method of exhaustion due to Eudoxus of Cnidus. Often now this method is thought of as considering the circle as the limit of regular polygons as the number of sides increases to infinity. We should strongly emphasise, however, that this is not the way that the ancient Greeks looked at the method. Rather it was a reductio ad absurdum argument which avoided the use of the infinite. For example, to prove two areas A and B equal, the method would assume that the area A was less than B and then derive a contradiction after a finite number of steps. Again assuming the area B was less than A also led to a contradiction in a finite number of steps.

Recently, however, evidence has come to light which suggests that not all ancient Greek mathematicians felt constrained to deal only with the potentially infinite. The authors of [32] have noticed a remarkable way that Archimedes investigates infinite numbers of objects in The Method in the Archimedes palimpsest:-

… Archimedes takes three pairs of magnitudes infinite in number and asserts that they are, respectively, “equal in number”. … We suspect there may be no other known places in Greek mathematics – or, indeed, in ancient Greek writing – where objects infinite in number are said to be “equal in magnitude”. …The very suggestion that certain objects, infinite in number, are “equal in magnitude” to others implies that not all such objects, infinite in number, are so equal. … We have here infinitely many objects – having definite, and different magnitudes (i.e. they nearly have number); such magnitudes are manipulated in a concrete way, apparently by something rather like a one-one correspondence. … … in this case Archimedes discusses actual infinities almost as if they possessed numbers in the usual sense …

Even if most mathematicians accepted Aristotle‘s potentially infinite arguments, others argued for cases of actual infinity, others argued for cases of actual infinity. In the first century BC Lucretius wrote his poem De Rerum Natura in which he argued against a universe bounded in space. His argument is a simple one. Suppose the universe were finite so there had to be a boundary. Now if one approached that boundary and threw an object at it there could be nothing to stop it since anything which stopped it would lie beyond the boundary and nothing lies outside the universe by definition. We now know, of course, that Lucretius’s argument is false since space could be finite without having a boundary. However for many centuries the boundary argument dominated debate over whether space was finite.

It became largely theologians who argued in favour of the actual infinite. For example St Augustine, the Christian philosopher who built much of Plato‘s philosophy into Christianity in the early years of the 5th century AD, argued in favour of an infinite God and also a God capable of infinite thoughts. He wrote in his most famous work City of God:-

Such as say that things infinite are past God’s knowledge may just as well leap headlong into this pit of impiety, and say that God knows not all numbers. … What madman would say so? … What are we mean wretches that dare presume to limit his knowledge.

Indian mathematicians worked on introducing zero into their number system over a period of 500 years beginning with Brahmagupta in the 7thCentury. The problem they struggled with was how to make zero respect the usual operations of arithmetic. Bhaskara II wrote in Bijaganita:-

A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.

It was an attempt to bring infinity, as well as zero, into the number system. Of course it does not work since if it were introduced as Bhaskara II suggests then 0 times infinity must be equal to every number n, so all numbers are equal.

Thomas Aquinas, the Christian theologian and philosopher, used the fact that there was not a number to represent infinity as an argument against the existence of the actual infinite. In Summa theologia, written in the 13th Century, Thomas Aquinas wrote:-

The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited.

This objection is indeed a reasonable one and in the time of Aquinas had no satisfactory reply. An actual infinite set requires a measure, and no such measure seemed possible to Aquinas. We have to move forward to Cantor near the end of the 19th Century before a satisfactory measure for infinite sets was found. The article [15] examines:-

… mathematical arguments used by two thirteenth-century theologians, Alexander Nequam and Richard Fishacre, to defend the consistency of divine infinity. In connection with their arguments, the following question is raised: Why did theologians judge it appropriate to appeal to mathematical examples in addressing a purely theological issue?

Mathematical induction began to be used hundreds of years before any rigorous formulation of the method was made. It did provide a technique for proving propositions were true for an infinite number of integer values. For example al-Karaji around 1000 AD used a non-rigorous form of mathematical induction in his arguments. Basically what al-Karaji did was to demonstrate an argument for n = 1, then prove the case n = 2 based on his result for n =1, then prove the case n = 3 based on his result for n = 2, and carry on to around n = 5 before remarking that one could continue the process indefinitely. By these methods he gave a beautiful description of generating the binomial coefficients using Pascal‘s triangle.

Pascal did not know about al-Karaji‘s work on Pascal‘s triangle but he did know that Maurolico had used a type of mathematical induction argument in the middle of the 17th Century. Pascal, setting out his version of Pascal‘s triangle writes:-

Even though this proposition may have an infinite number of cases, I shall give a very short proof of it assuming two lemmas. The first, which is self evident, is that the proposition is valid for the second row. The second is that if the proposition is valid for any row then it must necessarily be valid for the following row. From this it can be seen that it is necessarily valid for all rows; for it is valid for the second row by the first lemma; then by the second lemma it must be true for the third row, and hence for the fourth, and so on to infinity.

Having moved forward in time following the progress of induction, let us go back a little to see arguments which were being made about an infinite universe. Aristotle‘s finite universe model with nine celestial spheres centred on the Earth had been the accepted view over a long period. It was not unopposed, however, and we have already seen Lucretius’s argument in favour of an infinite universe. Nicholas of Cusa in the middle of the 15th Century was a brilliant scientist who argued that the universe was infinite and that the stars were distant suns. By the 16th Century, the Catholic Church in Europe began to try to stamp out such heresies. Giordano Bruno was not a mathematician or scientist, but he argued vigorously the case for an infinite universe in On the infinite universe and worlds (1584). Brought before the Inquisition, he was tortured for nine years in an attempt to make him agree that the universe was finite. He refused to change his views and he was burned at the stake in 1600.

Galileo was acutely aware of Bruno‘s fate at the hands of the Inquisition and he became very cautious in putting forward his views. He tackled the topic of infinity in Discorsi e dimostrazioni matematiche intorno a due nuove scienze (1638) where he studied the problem of two concentric circles with centre O, the larger circle A with diameter twice that of the smaller one B. The familiar formula gives the circumference of A to be twice that of B. But taking any point P on the circle A, then OP cuts circle B in one point. Similarly if Q is a point on B then OQ produced cuts circle A in exactly one point. Although the circumference of A is twice the length of the circumference of B they have the same number of points. Galileoproposed adding an infinite number of infinitely small gaps to the smaller length to make it equal to the larger yet allow them to have the same number of points. He wrote:-

These difficulties are real; and they are not the only ones. But let us remember that we are dealing with infinities and indivisibles, both of which transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness. In spite of this, men cannot refrain from discussing them, even though it must be done in a roundabout way.

However, Galileo argued that the difficulties came about because:-

… we attempt, with our finite minds, to discuss the infinite, assigning to it properties which we give to the finite and limited; but I think this is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.

He then gave another paradox similar to the circle paradox yet this time with numbers so no infinite indivisibles could be inserted to correct the situation. He produced the standard one-to-one correspondence between the positive integers and their squares. On the one hand this showed that there were the same number of squares as there were whole numbers. However most numbers were not perfect squares. Galileo says this means only that:-

… the totality of all numbers is infinite, and that the number of squares is infinite.; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and, finally, the attributes “equal”, “greater”, and “less” are not applicable to the infinite, but only to finite quantities.

In [25] Knobloch takes a new look at this work by Galileo. In the same paper Leibniz‘s careful definitions of the infinitesimal and the infinite in terms of limit procedures are examined. Leibniz‘s development of the calculus was built on ideas of the infinitely small which has been studied for a long time.

Cavalieri wrote Geometria indivisibilibus continuorum (1635) in which he thought of lines as comprising of infinitely many points and areas to be composed of infinitely many lines. He gave quite rigorous methods of comparing areas, known as the “Principle of Cavalieri“. If a line is moved parallel to itself across two areas and if the ratio of the lengths of the line within each area is always a : b then the ratio of the areas is a : b.

Roberval went further down the road of thinking of lines as being the sum of an infinite number of small indivisible parts. He introduced methods to compare the sizes of the indivisibles so even if they did not have a magnitude themselves one could define ratios of their magnitudes. It was a real step forward in dealing with infinite processes since for the first time he was able to ignore magnitudes which were small compared to others. However there was a difference between being able to use the method correctly and writing down rigorously precise conditions when it would work. Consequently paradoxes arose which led some to want the method of indivisibles to be rejected.

The Roman College rejected indivisibles and banned their teaching in Jesuit Colleges in 1649. The Church had failed to silence Bruno despite putting him to death, it had failed to silence Galileo despite putting him under house arrest and it would not stop progress towards the differential and integral calculus by banning the teaching of indivisibles. Rather the Church would only force mathematicians to strive for greater rigour in the face of criticism.

The symbol  ∞   which we use for infinity today, was first used by John Wallis who used it in De sectionibus conicis in 1655 and again in Arithmetica infinitorum in 1656. He chose it to represent the fact that one could traverse the curve infinitely often.

Three years later Fermat identified an important property of the positive integers, namely that it did not contain an infinite descending sequence. He did this in introducing the method of infinite descent 1659:-

… in the cases where ordinary methods given in books prove insufficient for handling such difficult propositions, I have at last found an entirely singular way of dealing with them. I call this method of proving infinite descent …

The method was based on showing that if a proposition was true for some positive integer value n, then it was also true for some positive integer value less than n. Since no infinite descending chain existed in the positive integers such a proof would yield a contradiction. Fermat used his method to prove that there were no positive integer solutions to

x4 + y4 = z4.

Newton rejected indivisibles in favour of his fluxion which was a measure of the instantaneous variation of a quantity. Of course, the infinite was not avoided since he still had to consider infinitely small increments. This was, in a sense, Newton‘s answer to Zeno‘s arrow problem:-

If, says Zeno, everything is either at rest or moving when it occupies a space equal to itself, while the object moved is in the instant, the moving arrow is unmoved.

Newton‘s fluxions produced wonderful mathematical results but many were wary of his use of infinitely small increments. George Berkeley‘s famous quote summed up the objections in a succinct way:-

And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?

Newton believed that space is in fact infinite and not merely indefinitely large. He claimed that such an infinity could be understood, particularly using geometrical arguments, but it could not be conceived. This is interesting for, as we shall see below, others argued against actual infinity using the fact that it could not be conceived.

The problem of whether space and time are infinitely divisible continued to trouble people. The philosopher David Hume argued that there was a minimum perceptible size in Treatise of human nature (1739):-

Put a spot of ink upon paper, fix your eye upon the spot, and retire to such a distance that at last you lose sight of it; ’tis plain that the moment before it vanished the image or impression was perfectly indivisible.

Immanuel Kant argued in The critique of pure reason (1781) that the actual infinite cannot exist because it cannot be perceived:-

… in order to conceive the world, which fills all space, as a whole, the successive synthesis of the parts of an infinite world would have to be looked upon as completed; that is, an infinite time would have to be looked upon as elapsed, during the enumeration of all coexisting things.

This comes to the question often asked by philosophers: would the world exist if there were no intelligence capable of conceiving its existence? Kant says no; so we come back to the point made near the beginning of this article namely that the collection of integers is not infinite since we can never enumerate more than a finite number.

Little progress was being made on the question of the actual infinite. The same arguments kept on appearing without any definite progress towards a better understanding. Gauss, in a letter to Schumacher in 1831, argued against the actual infinite:-

I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.

Perhaps one of the most significant events in the development of the concept of infinity was Bernard Bolzano‘s Paradoxes of the infinite which was published in 1840. He argues that the infinite does exist and his argument involves the idea of a set which he defined for the first time:-

I call a set a collection where the order of its parts is irrelevant and where nothing essential is changed if only the order is changed.

Why does defining a set make the actual infinite a reality? The answer is simple. Once one thinks of the integers as a set then there is a single entity which must be actually infinite. Aristotle would look at the integers from the point of view that one can find arbitrarily large finite subsets. But once one has the set concept then these are seen as subsets of the set of integers which must itself be actually infinite. Perhaps surprisingly Bolzano does not use this example of an infinite set but rather looks at all true propositions:-

The class of all true propositions is easily seen to be infinite. For if we fix our attention upon any truth taken at random and label it A, we find that the proposition conveyed by the words “A is true” is distinct from the proposition A itself…

At this stage the mathematical study of infinity moves into set theory and we refer the reader to the article Beginnings of set theory for more information about Bolzano‘s contribution and also the treatment of infinity by Cantor who built a theory of different sizes of infinity with his definitions of cardinal and ordinal numbers.

The problem of infinitesimals was put on a rigorous mathematical basis by Robinson with his famous 1966 text on nonstandard analysis. Kreisel wrote:-

This book, which appeared just 250 years after Leibniz‘ death, presents a rigorous and efficient theory of infinitesimals obeying, as Leibniz wanted, the same laws as the ordinary numbers.

Fenstad, in [17], looks at infinity and nonstandard analysis. He also examines its use in modelling natural phenomena.


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