Ancient Indian mathematicians were also familiar with techniques for working out square roots, as can be found in a text known as the Bakhshali Manuscript, written on birchbark and found near the village of Bakhshali, in what is today Pakistan, in 1881. Dated between the second century BCE and the third century CE, it contains techniques for solving a wide variety of mathematical problems, such as a technique for working out square roots.3
An ancient Chinese method for proving Pythagoras’ theorem. Consider the triangle in the top left of the square with its right angle shown. Now see if you can make a connection between the areas of the squares of its three sides.
Likewise, The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, by Zhou Bi Suan Jing, is an ancient Chinese mathematical text that dates from the period of the Zhou dynasty (1046–256 BCE). It is an anonymous collection of 246 mathematical problems, each with detailed steps and answers, and contains one of the first recorded proofs of the Pythagorean theorem, to which a later Chinese mathematician, Chou Kung, provided an accompanying diagram (above), which is one of the simplest ways of seeing how the area of the square of the hypotenuse equals the sum of the areas of the squares of the other two sides.
Another example of mathematical prowess predating the ancient Greeks shows honours shared between the Babylonians and the Egyptians and concerns the determination of the value of pi (the constant ratio of a circle’s circumference to its diameter). The value of pi is what is known as an ‘irrational number’, one that cannot be written exactly as a ratio of two whole numbers, which was a concept only fully understood by Greek mathematicians. Thus, written in decimal notation the exact value of pi is an infinite sequence of numbers. The extent to which we have been influenced by the Greek contribution to mathematics shows in many people’s assumption that it was the Greeks who first figured out that such a fundamental constant exists. This is mainly because we universally use the Greek letter π as its symbol. It is also sometimes referred to as Archimedes’ constant, after the Greek scientist who first estimated it rigorously by geometric means. But of course it was known long before the Greeks came on the scene. What is interesting is the way ancient mathematicians worked it out. Clearly to a very rough approximation, it has a value of 3. That is, the perimeter of any circle is three times the length of the line straight across the middle. For many purposes this might be good enough. But both the Babylonians and Egyptians needed to do better.
Rather than give this constant a name or even recognize its existence as a mathematical quantity, the ancients had it encoded within arithmetical rules for evaluating, for example, the area of a circle. We learn at school that this is given by π times the square of the radius, just as we learn that the circumference of a circle is π times twice the radius. The Babylonians often wrote that the area of a circle is given by one-twelfth of the square of the circumference. This may sound rather strange, but with a little not too complicated algebraic manipulation we see that this implies that they were using a value for π of exactly 3. One clay tablet dating back to the early second millennium BCE suggests the use of a more accurate value of 3⅛, or 3.125, which is slightly smaller than the correct value of 3.1415… 4
The Egyptians used a different formula, which stated that the area of a circle was given by the square of eight-ninths of the diameter. Again, a little re-jigging tells us that this is equivalent to their use of a value for π of 3.16, a little on the large side but still better than the rough value of 3.
On many other mathematical matters, the Babylonians, particularly during the Hammurabi dynasty (about 1780 BCE), made a number of advances, and we have thousands of cuneiform tablets that confirm this. For instance, their system of multiplication tables is so ingenious that it is far better than that of Ptolemy two thousand years later. Of course, I should not oversell the Babylonians’ prowess and achievements in mathematics; while more advanced than the Egyptians, they were inevitably totally eclipsed by the Greeks, in particular by geniuses like Pythagoras, Archimedes and Euclid. It should be noted, however, that Ptolemy’s Almagest used the Babylonian sexagesimal numbering system, although only for writing fractions. In common with all astronomical texts right up to Islamic times, integers were always written as letters very much like Roman numerals.
The Arabic alphabetic notation, copied from the Greek and Hebrew traditions, dates back to the early years of Islam. It was known as the abjad system since the first four numbers: 1, 2, 3, 4, were denoted by the first four letters of the alphabet: alif (‘a’), bā (‘b’), jīm (‘j’) and dāl (‘d’). For instance, a number like 365, which Ptolemy would have written as τξε, would have been written by Islamic mathematicians as . In each case, the three letters symbolize the numbers 300, 60 and 5. The point to note here is that these are very different from the decimal notation we use today, where just nine symbols, plus zero, are used for all numbers. Instead, the numbers 3, 30 and 300, for instance, would each be represented by a different letter.
Muslim mathematicians, even after inheriting the Indian decimal system, continued to use, and improve upon, Babylonian sexagesimal arithmetic, particularly in astronomical calculations, to such an extent that it was referred to as ‘the astronomer’s arithmetic’.
The prototypes of the number symbols we use today all come from India. They are found in the Ashoka inscriptions from the third century BCE, the Nana Ghat inscriptions about a century later and in the Nasik Caves from the first and second centuries CE – all in forms that have considerable resemblance to today’s symbols.5 For instance, the numbers 2 and 3 are well-recognized cursive derivations from the ancient = and ≡. However, none of these early Indian inscriptions contains any notion of place value or of a zero that would make modern place value possible. Hindu literature provides some evidence that the zero may have been known earlier, but there are no extant inscriptions from India with such a symbol before the ninth century.
A positional, or place-value, notation is a numeral system in which each position is related to the next by a constant multiplier called the base. Our decimal system of course has a base of ten and credit for its development can be traced back to two great medieval Indian mathematicians, Āryabhata (476–550), who developed the place-value notation itself, and Brahmagupta a century later.6 More recent authors have argued that the oldest known authentic document containing the place-value system is the Jaina cosmological text Lokavibhaga, which was completed in 458.7 By about 670 the system had reached northern Syria, where a bishop by the name of Severus Sebokht praised its Hindu inventors as discoverers of things more ingenious than those of the Greeks and spoke of their ‘nine signs’. The zero was, it seems, unknown to him.
It is not clear when the Indian numerals would have become known to the scholars of Abbāsid Baghdad. It may have been as early as the time of al-Mansūr, when Brahmagupta’s Siddhanta was first translated into Arabic, either directly from Sanskrit or from Persian. Two of the most famous Baghdadi scholars, the philosopher al-Kindi and the mathematician al-Khwārizmi, were certainly the most influential in transmitting Hindu numerals to the Muslim world. Both wrote books on the subject during al-Ma’mūn’s reign, and it was their work that was translated into Latin and transmitted to the West,8 thus introducing Europeans to the decimal system, which was known in the Middle Ages only as Arabic numerals. But it would be many centuries before it was widely accepted in Europe. One reason for this was sociological: decimal numbers were considered for a long time as symbols of the evil Muslim foe.
But there was a more important practical reason for the long delay. For most purposes in daily life, Roman numerals proved adequate and it was only with the emerging interest in science during the Renaissance that the importance of mathematics was understood and numbers appreciated as being at the heart of mathematics, and therefore the very foundation of modern science.
The Hindu-Arabic numbering system was finally popularized in Europe by the great mathematician Leonardo of Pisa (Fibonacci) (c. 1170–1250
), who had travelled throughout the Mediterranean world to study under the leading Arabic mathematicians of the time. He returned from his travels around the year 1200 and within a couple of years, aged 32, wrote what he had learnt in the Liber Abaci (Book of Abacus, or Book of Calculation). However, the historian George Sarton makes the following point: ‘A single example will suffice to indicate the slowness of the integration of Hindu numerals in Western usage. As late as the eighteenth century the Cour des Comptes of France (the national Audit Office) was still using Roman numerals.’9
The evolution of Hindu-Arabic numerals.
Considering the awkwardness of Roman numerals when performing arithmetic operations such as multiplication one would think that the Hindu-Arabic decimal system would have been embraced enthusiastically. But what is important about the system is of course not the symbols themselves used for the nine digits, or even that there were just nine of them (plus the zero). After all, just seven letters are needed to denote any integer up to a thousand in Roman numerals. What is crucial is the place-value system itself: that the Hindu-Arabic symbols could define any number up to infinity. They also allow numbers to be manipulated and combined far more efficiently than Roman numerals. Consider multiplying two numbers like 123 and 11. This is straightforward enough on paper for most of us and can even be done in your head if you are that way inclined. The answer is 1,353. But try doing the same multiplication using Roman numerals. You would have to multiply CXXIII by XI to give MCCCLIII. There is a method for doing this but it is somewhat cumbersome.10 The technique was most likely originally discovered by accident in ancient Egypt and gradually refined as its practitioners became more adept.
Let us return though to the mathematicians of Baghdad who first inherited the Hindu decimal system. Al-Khwārizmi’s great work on arithmetic, The Book of Addition and Subtraction According to the Hindu Calculation, written around 825, no longer exists in the original Arabic. Even this title is just a guess. It was probably the first book on the decimal system to be translated into Latin, under the title of Liber Algorismi de Numero Indorum, and begins with the words Dixit algorismi (or ‘Al-Khwārizmi says’). It goes on to describe the procedures for various computational instructions and is the origin of the word ‘algorithm’, derived from the Latinized name of al-Khwārizmi. This and other early translations of his work met with much opposition in Europe and were referred to as dangerous Saracen magic.
But the Muslim world was very reluctant to abandon its old ways too. Despite being introduced to the Hindu decimal notation by al-Kindi and al-Khwārizmi, Arabic mathematicians preferred to stick with what they knew best: either the Babylonian sexagesimal system or the Greek and Roman tradition of using letters of the alphabet to denote numbers.11 This is what was done routinely in astronomical tables, and continued the tradition they had learnt from texts such as Ptolemy’s Almagest. Five centuries after al-Khwārizmi, the decimal system remained no more than a curiosity, and the Babylonian sexagesimal system continued to be used widely. Consider the example of the monograph on geography of the polymath al-Bīrūni: The Determination of the Coordinates of Cities, which was completed in 1025. In it, mathematical formulae are carefully derived, and then worked examples are shown, such as the coordinates of Ghazna, the city in central Afghanistan where al-Bīrūni lived, which are determined relative to both Baghdad and Mecca. Latitude determinations presented no difficulties to a genius like al-Bīrūni and computations were carried out according to the primitive technique of converting all sexagesimals into decimal integers. But for the longitude computations, which were more difficult at the time, all operations were performed with sexagesimals directly in the Babylonian manner.12
This custom was to continue for hundreds of years. The mathematical, astronomical and geographical tables published in Arabic during the Middle Ages contain hardly any decimal numbers. This can be seen as late as the fourteenth century in the tables of longitudes and latitudes compiled by the geographer Abū al-Fidā (1273–1331). So one can hardly criticize the Europeans for rejecting Hindu-Arabic numerals for so long when the Muslim world did not embrace them either.
Amid the considerable historical confusion surrounding the origin of many scientific ideas and, in the context of Arabic science, the broad span of opinions ranging from ‘they did nothing more than pass on the knowledge of the Greeks and Indians’ to ‘we owe everything we know to them’, one of the most problematic yet fascinating has been over the origin of zero.
The reason for this is not so much because of conflicting evidence, or the lack of it, to support a particular claim, but rather because the question: ‘Who first discovered zero?’ can mean several different things, and the answer is different in each case. So, let me be more precise:
Is the question: when was the very first use of a symbol, or mark, to indicate a blank placeholder within a number?
Or, more specifically: when was a symbol for such a placeholder used within our current decimal system, such as to distinguish between, say, the numbers 11 and 101?
Does it allude to the first appreciation of zero as a philosophical concept that symbolizes the absence of anything (emptiness, the void)?
Or does it mean the earliest reference to zero as a proper number in its own right, having the same status as any other and sitting on the boundary between positive and negative numbers?
There are clearly different levels of sophistication in understanding the concept of zero. We are not looking for a mathematician who simply woke up one morning and thought ‘I know what is missing from our number system that would make arithmetic much more versatile and useful: the zero.’
The crudest definition of zero is that of a positional notation within a number. The ancient Babylonians, in the early second millennium BCE, needed to be able to distinguish between numbers in their sexagesimal tabulations. They appreciated from the very beginning the ambiguity in the meaning of their numbers. For example, consider the number (1,20). This might denote any one of the following numbers:
(a) (1,20), which means 60 × 1 + 20 = 80
(b) (1,0,20), which means 3600 × 1 + 60 × 0 + 20 = 3620
(c) (1,20,0), which means 3600 × 1 + 60 × 20 + 0 = 4800.
If there is no zero placed in the appropriate unit box, how can we tell what the number is? The Babylonians got around the problem in two ways. The cuneiform symbols for 1 and 10 were and , respectively. Hence 20 would be written as , and the number 80, or (1,20), would be written as . But to distinguish this from the number 3620, or (1,0,20), they would simply leave enough of a gap between the symbols to denote the zero, or empty slot: . Of course this still leaves the problem of how to write the number 4800 (1,20,0). For this, they would simply write , just as they would for (1,20) and trust the context in which it was written to make clear that it is 4800 and not 80 that is implied.
Much later, the Seleucid Babylonians, who ruled over Mesopotamia as the successors of Alexander the Great, invented a symbol to replace this ambiguous ‘gap’ that the old Babylonians employed. Thus, the earliest known symbol for zero () is found on many Babylonian cuneiform clay tablets from around 300 BCE. But it was only ever used to keep apart other number symbols, no different from our use today of the zero to distinguish between, say, 25, 205 and 2005. Strangely, they continued the custom of the old Babylonians of never placing the symbol at the end of a number, only in between other symbols.
One can argue about the extent to which this first use of a zero symbol constitutes the real invention of zero. Interestingly, the empty positions requiring a zero symbol occur far less frequently in the sexagesimal system than they do in our base-10 decimal system. It is not needed at all for integers less than 60 and only 59 times for integers less than 3600 (compared with 917 occurrences of the nought in the decimal system). So the Babylonians did not feel such an urgent need for it.
A little later than the Babylonians and on the other side of the world, the Mayans of Central America developed their own vigesimal positional (base-20)
number system using very few symbols (a dot for 1 and a bar for 5). This allowed them to have a combination of dots and bars up to 19, then they would move to the next units. And, just like the Babylonians, they used a symbol for zero as a placeholder. The earliest recorded example of this dates to around 36 BCE.
The Greeks, who were very strongly influenced by Babylonian astronomy and the associated sexagesimal system, used their letters for whole numbers but sexagesimal notation for fractions. For this they needed a zero symbol too and picked a Greek letter for this, omicron (like the English letter ‘o’). But in all three cases – the Babylonians, Mayans and Greeks – this zero is not a number or even a concept in its own right. Nevertheless, it would be correct, in answer to the question ‘who first invented the symbol for zero?’ to say: the Babylonians.13
What about the concept of zero as representing nothingness? Of course philosophical references to the ‘void’ can be regarded in some sense as being on a par with the mathematical notion of zero. If so, then the ancient Greeks got there first. One prominent historian of mathematics, Carl Boyer, has argued that Aristotle was thinking and writing about zero as a mathematical concept in the fourth century BCE.14 In his Physica, Aristotle describes the idea of the mathematical zero in relation to a point on a line. He also mentions the impossibility of dividing by zero in the context of the speed of an object being inversely proportional to the resistance (or density) of the medium through which it is moving. Thus the speed in a vacuum (or void) would be infinite as there is no resistance. This, argued Aristotle, proved the impossibility of the existence of the void. We therefore see that, at around the same time, the Babylonians invented a symbol for zero, while the Greeks first described the concept of zero.
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