The Story of Zero

The Sumerian

The Sumerian cuneiforms was the first positional system. However, it is sexigesimal and it does not have a symbol for zero. They used a stylus to carve the numbers on clay tablets and the result is wedge-shaped symbols. In their symbols, = 1 and = 10. We will write sexigesimal numbers in this notation: 1,16,10 = 1�60² + 16�60 + 10. In cuneiforms, this is . The main problem with this system is that there is no symbol for zero, so there is no way to tell the difference between 1,16,10 and 1,0,16,10 or 1,16,0,10, etc. They did leave a space between digits to represent zero, but this will cause ambiguity because if the space is too small you can miss a zero and how much space must be left if there are two consecutive zeroes? It was only at 400BC that they developed symbols for zero (there was no standard symbol for zero), but there is only evidence to show that they use them between digits and not at the end of the number. This goes to show that they did not treat zero as a number, but rather as a punctuation. So how do they distinguish between 120 and 2, both of which are denoted by ? They could only do so by context. We still use this method today. E.g. when the bus driver tells you that the bus fare is one-fifty, you know that he means $1.50. But when a flight ticket is one-fifty, we know it is $150.

The Greek

Like the Egyptian before them, the Greek used an additive number system, hence there was no need for a symbol for zero. It was only in 331BC when Alexander invaded Babylonia, that the Greeks discovered the importance of zero. From their astronomical papyri of the 3rd century BC, the symbol 'O' for zero was found; it looks exactly like a hollow circle. How did the Babylonian wedge-shaped symbol for zero become O. The most common explanation is O is the first alphabet for ouden meaning nothing. The Greek numerals are all the first alphabet of the word they stand for. To prevent mixing up with number 70, which also starts with 'o', zero is always written with a bar over it e.g. o or o. In fact, all numbers have a simple bar over them to distinguish them from words, e.g. h = 8. However, the simple bar was never put over the various symbol for zero, signifying that they did not treat zero as a number per se. In any case, the use of zero was mainly confined to astronomical usage. This begs the question why the Greeks with their advance mathematics would miss out the development of zero as a number and in general a positional number system. The answer is believed to lie in their heavy reliance on geometry. Geometers did their arithmetic geometrically, while merchants did theirs via counting boards. Both treatment does not need a symbol for zero. A counting board would look like this (in modern notation):

Figure 1: A counting board.

In this example, the number represented is 4057. Although the Greeks used counting boards since 700BC, the Babylonians are believed to use them much earlier. Even without the ease of the present decimal positional system, Archimedes (287-212BC) could conjure up extremely large numbers using words. But to say a myriad-myriad units of the myriad-myriadth order of the myriad-myriadth period instead of 10^{80,000,000,000,000,000} is hardly efficient. And it is not easy for calculation.

The Indian

In 326BC, Alexander's quest for domination reached India. Along with war came the influence of the Greeks. The Indian of course have developed their own Mathematics before this, but with the war there definitely was some Greek influence. Aryabhata was an Indian astronomer. He developed a unique way of forming large numbers based on position or place value. There are 9 vowels in Sanskrit, the first 3 being a, i, u. The vowels are the place holders, while the consonants are the digits. E.g. 863 is written gaciju, where g=3, c=6, j=8 and a=one's, i=ten's, u=hundreds. He then extended this to 18 vowels by simply doubling the original vowels (a, a, i, i, u, u, ....) and forming a second group of consonants, so that jasagi is 388, where s=8. He called the places kha. kha is later to become to most common Indian word for zero. This is a positional system in which the places give weight to the consonants. An empty place signifies there is no value in that place, hence there is no need for a symbol for zero. 100 years later, Brahmagupta still has no symbol for zero, but he treated zero like a number, doing arithmetic with it. In the following arithmetic, A is a positive number and B is any non-zero number.

Notice the second last arithmetic is not telling us anything and the last one is wrong. We now know these two are undefined. His work and later Indian mathematicians' put zero firmly amongst the rank of the other nine Indian numerals. al-Khwarizmi (780-850AD) was amongst the first to write a treatise on the Hindu-Arabic system with zero. Bhāskara showed that √0 = 0 in 1150.

The European

However, when Fibonacci brought the Indian number system back to Europe, he speaks of  the 9 Indian figures and the sign zero. His 1202 work Liber Abaci (The Book of the Abacus) was not about abacus or counting boards, but rather about the Arabic numerals and counting system. In 1484, Nicolas Chuquet solved the quadratic equation 3x² + 12 = 12x and obtained the solution x = 2 � √(4-4). He argued that 4 - 4 = 0 and √0 added to 2 gives 2. This hinges on the relation that we saw earlier, namely √0 = 0. Nicole Oresme (1323-1382) talked about fractional non-zero exponents around 1360; Nicolas Chuquet talked about x⁰ but not fractional exponents; Michael Stifel (1487-1567) talked about zero and negative exponents. Further development came from John Napier (1550-1617), who is famous for introducing Logarithm to the world. Prior to him, Omar Khayyam (1048-1131) needed 3 methods to tackle quadratic equations. This is because without zero, there are 3 families of quadratic equations: x² + ax = b, x² + b = ax and x² = ax + b, where a and b are positive numbers. Any quadratic equation can be reduced to one of these forms. Each of these had a different treatment. It was John Napier who moved all the terms to one side of the equation leaving a zero on the other side, namely ax² + bx + c = 0, where a, b and c can be negative. In so doing, it allowed us to factorize the left hand side into (x-d)(x-e) = 0 and hence x = d or e.

The Mayan

Let's side track to the ancient Mayan civilization (300BC-900AD) of the Yucatan Peninsula. It was an isolated civilization from the rest of the world and yet it came up with the concept of zero independently. Theirs is a base-20 or vigesimal positional system (as opposed to our base-10 or decimal number system). They use a dot and a line to represent numbers from 1 to 19, but use various figures (face, flower, shell, etc) to represent zero. They knew how to write numbers ending with zero. The fact that the concept of zero has evolved in different cultures is evidence for the universality of Mathematics.

Short notes on some of the mathematicians mentioned in this page.
1) Aryabhata (Indian) lived around 500AD and there is said to be more than one Aryabhata, maybe three of them.

2) Bhāskara (Indian) published his work Vija-Ganita in 1150.

3) Nicolas Chuquet (French) was a physician. His work Le triparty en la science des nombres (which contained the above problem) was published after his death.

4) Nicole d'Oresme (French) was the Bishop of Normandy. He first proved that the harmonic series is divergent.

5) Michael Stifel (German) was a Lutheran minister and is remembered for his work on algebra Arithmetica Integra.

6) John Napier (Scottish) was a baron. He invented the Logarithm and had a concept of binary system.

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