An overview of Egyptian mathematics
Civilisation reached a high level in Egypt at an early period. The country was well suited for the people, with a fertile land thanks to the river Nile yet with a pleasing climate. It was also a country which was easily defended having few natural neighbours to attack it for the surrounding deserts provided a natural barrier to invading forces. As a consequence Egypt enjoyed long periods of peace when society advanced rapidly.
By 3000 BC two earlier nations had joined to form a single Egyptian nation under a single ruler. Agriculture had been developed making heavy use of the regular wet and dry periods of the year. The Nile flooded during the rainy season providing fertile land which complex irrigation systems made fertile for growing crops. Knowing when the rainy season was about to arrive was vital and the study of astronomy developed to provide calendar information. The large area covered by the Egyptian nation required complex administration, a system of taxes, and armies had to be supported. As the society became more complex, records required to be kept, and computations done as the people bartered their goods. A need for counting arose, then writing and numerals were needed to record transactions.
By 3000 BC the Egyptians had already developed their hieroglyphic writing (see our article Egyptian numerals for some more details). This marks the beginning of the Old Kingdom period during which the pyramids were built. For example the Great Pyramid at Giza was built around 2650 BC and it is a remarkable feat of engineering. This provides the clearest of indications that the society of that period had reached a high level of achievement.
Hieroglyphs for writing and counting gave way to a hieratic script for both writing and numerals. Details of the numerals themselves are given in our article Egyptian numerals. Here we are concerned with the arithmetical methods which they devised to work with these numerals
The Egyptian number systems were not well suited for arithmetical calculations. We are still today familiar with Roman numerals and so it is easy to understand that although addition of Roman numerals is quite satisfactory, multiplication and division are essentially impossible. The Egyptian system had similar drawbacks to that of Roman numerals. However, the Egyptians were very practical in their approach to mathematics and their trade required that they could deal in fractions. Trade also required multiplication and division to be possible so they devised remarkable methods to overcome the deficiencies in the number systems with which they had to work. Basically they had to devise methods of multiplication and division which only involved addition.
Early hieroglyphic numerals can be found on temples, stone monuments and vases. They give little knowledge about any mathematical calculations which might have been done with the number systems. While these hieroglyphs were being carved in stone there was no need to develop symbols which could be written more quickly. However, once the Egyptians began to use flattened sheets of the dried papyrus reed as “paper” and the tip of a reed as a “pen” there was reason to develop more rapid means of writing. This prompted the development of hieratic writing and numerals.
There must have been a large number of papyri, many dealing with mathematics in one form or another, but sadly since the material is rather fragile almost all have perished. It is remarkable that any have survived at all, and that they have is a consequence of the dry climatic conditions in Egypt. Two major mathematical documents survive.
You can see an example of Egyptian mathematics written on the Rhind papyrus and another papyrus, the Moscow papyrus, with a translation into hieratic script. It is from these two documents that most of our knowledge of Egyptian mathematics comes and most of the mathematical information in this article is taken from these two ancient documents.
Here is the Rhind papyrus
The Rhind papyrus is named after the Scottish Egyptologist A Henry Rhind, who purchased it in Luxor in 1858. The papyrus, a scroll about 6 metres long and 1/3 of a metre wide, was written around 1650 BC by the scribe Ahmes who states that he is copying a document which is 200 years older. The original papyrus on which the Rhind papyrus is based therefore dates from about 1850 BC.
Here is the Moscow papyrus
The Moscow papyrus also dates from this time. It is now becoming more common to call the Rhind papyrus after Ahmes rather than Rhind since it seems much fairer to name it after the scribe than after the man who purchased it comparatively recently. The same is not possible for the Moscow papyrus however, since sadly the scribe who wrote this document has not recorded his name. It is often called the Golenischev papyrus after the man who purchased it. The Moscow papyrus is now in the Museum of Fine Arts in Moscow, while the Rhind papyrus is in the British Museum in London.
The Rhind papyrus contains eighty-seven problems while the Moscow papyrus contains twenty-five. The problems are mostly practical but a few are posed to teach manipulation of the number system itself without a practical application in mind. For example the first six problems of the Rhind papyrus ask how to divide n loaves between 10 men where n =1 for Problem 1, n = 2 for Problem 2, n = 6 for Problem 3, n = 7 for Problem 4, n = 8 for Problem 5, and n = 9 for Problem 6. Clearly fractions are involved here and, in fact, 81 of the 87 problems given involve operating with fractions. Rising, in [37], discusses these problems of fair division of loaves which were particularly important in the development of Egyptian mathematics.
Some problems ask for the solution of an equation. For example Problem 26: a quantity added to a quarter of that quantity become 15. What is the quantity? Other problems involve geometric series such as Problem 64: divide 10 hekats of barley among 10 men so that each gets 1/8 of a hekat more than the one before. Some problems involve geometry. For example Problem 50: a round field has diameter 9 khet. What is its area? The Moscow papyrus also contains geometrical problems.
Unlike the Greeks who thought abstractly about mathematical ideas, the Egyptians were only concerned with practical arithmetic. Most historians believe that the Egyptians did not think of numbers as abstract quantities but always thought of a specific collection of 8 objects when 8 was mentioned. To overcome the deficiencies of their system of numerals the Egyptians devised cunning ways round the fact that their numbers were poorly suited for multiplication as is shown in the Rhind papyrus.
We examine in detail the mathematics contained in the Egyptian papyri in a separate article Mathematics in Egyptian Papyri. In this article we next examine some claims regarding mathematical constants used in the construction of the pyramids, in particular the Great Pyramid at Giza which, as we noted above, was built around 2650 BC.
Joseph [8] and many other authors gives some of the measurements of the Great Pyramid which make some people believe that it was built with certain mathematical constants in mind. The angle between the base and one of the faces is 51° 50′ 35″. The secant of this angle is 1.61806 which is remarkably close to the golden ratio 1.618034. Not that anyone believes that the Egyptians knew of the secant function, but it is of course just the ratio of the height of the sloping face to half the length of the side of the square base. On the other hand the cotangent of the slope angle of 51° 50′ 35″ is very close to π/4. Again of course nobody believes that the Egyptians had invented the cotangent, but again it is the ratio of the sides which it is believed was made to fit this number. Now the observant reader will have realised that there must be some sort of relationship between the golden ratio and π for these two claims to both be at least numerically accurate. In fact there is a numerical coincidence: the square root of the golden ratio times π is close to 4, in fact this product is 3.996168.
In [38] Robins argues against both the golden ratio or π being deliberately involved in the construction of the pyramid. He claims that the ratio of the vertical rise to the horizontal distance was chosen to be 51/2 to 7 and the fact that (11/14) × 4 = 3.1428 and is close to π is nothing more than a coincidence. Similarly Robins claims the way that the golden ratio comes in is also simply a coincidence. Robins claims that certain constructions were made so that the triangle which was formed by the base, height and slope height of the pyramid was a 3, 4, 5 triangle. Certainly it would seem more likely that the engineers would use mathematical knowledge to construct right angles than that they would build in ratios connected with the golden ratio and π.
Finally we examine some details of the ancient Egyptian calendar. As we mentioned above, it was important for the Egyptians to know when the Nile would flood and so this required calendar calculations. The beginning of the year was chosen as the heliacal rising of Sirius, the brightest star in the sky. The heliacal rising is the first appearance of the star after the period when it is too close to the sun to be seen. For Sirius this occurs in July and this was taken to be the start of the year. The Nile flooded shortly after this so it was a natural beginning for the year. The heliacal rising of Sirius would tell people to prepare for the floods. The year was computed to be 365 days long and this was certainly known by 2776 BC and this value was used for a civil calendar for recording dates. Later a more accurate value of 3651/4 days was worked out for the length of the year but the civil calendar was never changed to take this into account. In fact two calendars ran in parallel, the one which was used for practical purposes of sowing of crops, harvesting crops etc. being based on the lunar month. Eventually the civil year was divided into 12 months, with a 5 day extra period at the end of the year. The Egyptian calendar, although changed much over time, was the basis for the Julian and Gregorian calendars.
Mathematics in Egyptian Papyri
In the article An overview of Egyptian mathematics we looked at some details of the major Egyptian papyri which have survived. In this article we take a detailed look at the mathematics contained in them.
The Rhind papyrus
Ahmes, in the Rhind papyrus, illustrates the Egyptian method of multiplication in the following way. Assume that we want to multiply 41 by 59. Take 59 and add it to itself, then add the answer to itself and continue:-
41 59 _________________ 1 59 2 118 4 236 8 472 16 944 32 1888 _________________
Since 64 > 41, there is no need to go beyond the 32 entry. Now go through a number of subtractions
41 – 32 = 9, 9 – 8 = 1, 1 – 1 = 0
to see that 41 = 32 + 8 + 1. Next check the numbers in the right hand column corresponding to 32, 8, 1 and add them.
41 59 _________________ 1 592 118 4 236 8 472
Notice that the multiplication is achieved with only additions, notice also that this is a very early use of binary arithmetic (see below). Reversing the factors we have:
59 41
_________________
1 41
2 82
4 164
8 328
16 656
32 1312
_________________
2419
Notice that for this method to work we need to know that ever number is the sum of powers of 2. The ancient Egyptians would not have had a proof of this, nor would have appreciated that a proof was necessary. They would just know from practical experience that it could always be done. Basically we can think of the method as writing one of the numbers to base 2. In the examples above we have written
41 = 1.20 + 0.21 + 0.22 + 1.23 + 0.24 + 1.25
and
59 = 1.20 + 1.21 + 0.22 + 1.23 + 1.24 + 1.25.
Division works also using doubling. For example to divide 1495 by 65 we proceed as follows:
1 65
2 130
4 260
8 520
16 1040
We stop at this point because the next doubling will take us beyond 1495. Now we look for numbers in the right hand column which add up to 1495. We see that 1040 + 260 + 130 + 65 = 1495 and we tick the rows in which these numbers occur:
1 65
2 130
4 260
8 520
16 1040
Now add the numbers in the left hand column which are in ticked rows:
16 + 4 + 2 + 1 = 23,
so 1495 divided by 65 is 23.
What happens if the numbers do not divide exactly? Then the Egyptian method will yield fractions as the following example shows.
To divide 1500 by 65 proceed as before:
1 65
2 130
4 260
8 520
16 1040
Again we stop since the next doubling takes us beyond 1500. Now look for the numbers in the right hand column which add to a number n with 1500-65 < n ≤ 1500. [The Egyptians knew that this was always possible: can you prove that this is so?] In this case we have
1040 + 260 + 130 + 65 = 1495
and we are 5 short of our sum. Again tick the rows with these entries:
1 65
2 130
4 260
8 520
16 1040
Now add the numbers in the left hand column which are in ticked rows:
16 + 4 + 2 + 1 = 23,
so 1500 divided by 65 is 23 and 5/65 = 1/13 remaining. Hence the answer is 23 1/13.
We have cheated a little here for the fraction obtained is a unit fraction, that is a number of the form 1/n for n an integer. In fact the Egyptians only had fractions of this type and if the answer had not involved a unit fraction then the Egyptians would have written the fractional part as the sum of unit fractions. We see below how this was done but we examine a more general case.
The next problem is how to multiply and divide numbers involving fractions. The first important point is that the Egyptians only used unit fractions, and to be able to calculate a table was needed to convert twice a unit fraction into a sum of unit fractions. Now it might be supposed that doubling the unit fraction 1/5 would be easy and yield the sum of the unit fractions 1/5 + 1/5. However, for reasons which we do fully understand, this was not their approach. They wrote twice a unit fraction as the sum of distinct unit fractions. For example twice 1/5 would be written as 1/3 + 1/15.
The Rhind papyrus gives a table for doubling unit fractions 1/n for n odd, n between 5 and 101. Note that Ahmes did not need to give the double of 1/n for n even since it is just 1/m where m = 2n. The doubling table for unit fractions begins
Unit fraction Double unit fraction
____________________________________
1/5 1/3 + 1/15
1/7 1/4 + 1/28
1/9 1/6 + 1/18
1/11
1/13
1/15 1/10 + 1/30
1/17 1/12 + 1/51 + 1/68
.... ..................
It is remarkable that there are no errors in the table. Certainly Ahmes would have been expert at calculating and this would not have been simply a copying exercise for him. There are few errors in the Rhind papyrus but those which there are appear to be errors of calculation, not of copying, since the incorrect result is carried forward rather than a return to the correct path which would happen from an error in copying.
There is the fascinating question of how these decompositions were found, and why some decompositions were chosen in preference to others. This is discussed in [6] and further ideas, adding and correcting information from [6], is given in [17], [18], [29] and [35]. The favourite rules which many historians such as Gillings believe guided the scribes in their choice of decomposition of 2/n into unit fractions are (1) prefer small numbers (2) the fewer terms the better, and never more than four (3) prefer even to odd numbers. However other historians such as Bruins argues against such rules. His argument is essentially that before applying these rules one would need to work out all unit decompositions of 2/n and there is no evidence that the Egyptians had any methods to do this.
As an example of how to use the table, let us examine Problem 21 of the Rhind papyrus. Note that 2/3 was an allowable Egyptian fraction despite not being a unit fraction.
Problem 21: Complete 2/3 and 1/15 to 1.
In modern terms, this asks for a fraction x such that
The method of solution was to “get rid of” the fractions by multiplying through. In this case multiply each fraction by 15 to obtain
This is called the “red auxiliary” equation since the scribe wrote this equation in red ink. [Of course it would not appear in this form but rather “complete 10 and 1 to 15”.]
Now the answer to the red auxiliary equation is 4 so the original equation had solution twice × (twice × 1/15). From the doubling table we see that double 1/15 is 1/10 + 1/30. Doubling this gives 1/5 + 1/15 which is the required solution to Problem 21.
Another example of solving an equation is Problem 24 which asks:
Problem 24: A quantity added to a quarter of that quantity become 15. What is the quantity?
Ahmes uses the “method of false position” which was still a standard method three thousand years later. In modern notation the problem is to solve
Ahmes guesses the answer x = 4. This is to remove the fraction in the x/4 term. Now with x = 4 the expression x + x/4 becomes 5. This is not the correct answer, for the expression is required to equal 15. However, 15 is 3 times 5 so taking 3 times his guess of x = 4, namely x = 12, gives Ahmes the correct result. Another interpretation, favoured by some historians, is that Ahmes thought of the method as dividing x into 4 equal pieces of a size to be determined. Now Ahmes computes x + x/4 getting 5 of these equal pieces. Each piece must now be three so that 5 pieces equals 15. Not very different to our previous way of thinking, but one which is likely to come closer to Ahmes‘ way of thinking than our former description. Finally Ahmes checks his solution, or proves his answer is correct. He takes x = 4 × 3 = 12. Then x/4 = 3, so x + x/4 = 15 as required.
The methods of false position is used in Problems 24 to 29 of the Rhind Papyrus. However, in Problem 31 of the Papyrus Ahmes uses the simpler method of pure division. This is discussed in detail in [31].
Let us now see how to multiply, using Egyptian methods, 1 + 1/3 + 1/5 by 30 + 1/3.
1 1 + 1/3 + 1/5 2 2 + 2/3 + 1/3 + 1/15 = 3 + 1/15 4 6 + 1/10 + 1/30 8 12 + 1/5 + 1/15 16 24 + 1/3 + 1/15 + 1/10 + 1/30 2/3 2/3 + 1/6 + 1/18 + 1/10 + 1/30 1/3 1/3 + 1/12 + 1/36 + 1/20 + 1/60
Now here the row beginning 2/3 has been computed from 2/3 of 1 is 2/3, 2/3 of 1/3 is double 1/9 which is 1/6+1/18, 2/3 of 1/5 is double 1/15 which is 1/10 + 1/30.
Next find the numbers in the left hand column which add to 30+1/3. These are the rows marked with a tick:
1 1 + 1/3 + 1/5 2 3 + 1/15
Add the entries in the right hand column of the rows which are ticked to get the result of the multiplication
46 + 1/5 + 1/10 + 1/12 + 1/15 + 1/30 + 1/36.
As a final look at the Rhind papyrus let us give the solution to Problem 50. A round field has diameter 9 khet. What is its area? Here is the solution as given by Ahmes.
Take away 1/9 of the diameter, namely 1; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore it contains 64 setat of land.
Do it thus:
1 9
1/9 1
this taken away leaves 8
1 8
2 16
4 32
8 64
Its area is 64 setat.
Notice that the solution is equivalent to taking π = 4(8/9)2 = 3.1605. This is a remarkable result if one considers the date at which this approximation must have been discovered. The intriguing question is raised as to how such a discovery might have been made. Although we have no way of ever knowing this with certainty, several interesting conjectures have been suggested. In [25] Gerdes gives three ideas which might have led the Egyptians to this result. Two such conjectures suggested in [25] concern African crafts where a snake curve and a set of equidistant concentric rings are often seen. These two geometric designs are widespread in Africa and Gerdes shows how these could have led to a formula for the area of a circle. The third conjecture in [25] relates to a board game “mancala” which was popular throughout Africa and ancient Egypt. The game involves comparing small circles with larger circles and may have provided the motivation for the area formula.
The Moscow papyrus
Although the mathematical methods we have described are found in various Egyptian documents, all the actual examples we have given so far have come from Rhind papyrus. Let us finish this article by looking at an example from the Moscow papyrus which many historians argue is the most impressive achievement of Egyptian mathematics. The problem is number 14 from the papyrus and it concerns the geometrical figure visible in the portion of Moscow papyrus seen in this image.
Example 14. Example of calculating a truncated pyramid. The base is a square of side 4 cubits, the top is a square of side 2 cubits and the height of the truncated pyramid is 6 cubits.
First we remark that by “calculate a pyramid” the author means “calculate the volume of the pyramid”. Also not how appropriate this calculation is for the civilisation which today is universally known for the remarkable construction of pyramids.
The calculation begins by working out the area of the base: 4 × 4 = 16. Then the area of the top is worked out: 2 × 2 = 4. Next the product of the side of the base with the side of the top is computed: 4 × 2 = 8. These three are then added: 16 + 4 + 8 = 28. Now 1/3 of the height is taken, namely 2. Finally the product of 1/3 of the height with the previous sum of 28 is taken and the scribe writes:-
Behold it is 56.
This example means that the Egyptian knew the formula for the volume (although of course not in the algebraic sense which we now think of formulas). If the base square has side a, the top square has side b, and the height is h then
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:
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
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.
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 in [6] 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-Khwarizmi wrote 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-Samawal was 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. As the authors of [12] write:-
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 21st century begin on 1 January 2001. Zero is still causing problems!
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