Here we also see the search for a logic in the construction of the symbols. For example, the 60 and the 3,600 are multiplied by 10 when a circle is included inside them. With the arrival of cuneiform writing, these first symbols were gradually transformed into the following:
Because it was so close to Mesopotamia, Egypt did not lag far behind in adopting writing, and developed its own numeration symbols from the start of the third millennium.
The system was then purely decimal: each symbol had a value ten times greater than the one before.
These additive systems, in which one just has to add the values of the written symbols, became very popular throughout the world, and numerous variants were unveiled throughout antiquity and also much of the Middle Ages. They were used, in particular, by the Greeks and the Romans, who simply employed the letters of their respective alphabets as numerical symbols.
Alongside the additive systems, a new form of notation for numbers gradually emerged: numeration by position. In these systems, the value of a symbol was defined to depend on the position it occupies within the number. Once again, the Mesopotamians were the first to come up with this.
In the second millennium BC, it was the town of Babylon that now shone brightly over the Near East. Cuneiform writing was still in vogue, but now, just two symbols were used: the simple nail that had the value ‘1’ and the chevron with the value ‘10’.
These two signs were used to provide an additive notation for all numbers up to 59. For example, the number 32 was written as three chevrons followed by two nails.
Then, from 60 onwards, groups were introduced, where the same symbols were used to denote groups of 60. Thus, in the same way as in our present-day notation, where the figures read from right to left denote the units, then the tens, then the hundreds, in this Babylonian numeration the units are read first, then the sixties, then the three-thousand-six-hundreds (that is, sixty sixties), and so on, where each rank has a value sixty times greater than its predecessor.
For example, the number 145 consists of two sixties (which make 120) to which you have to add 25 units. The Babylonians would therefore have denoted it as follows:
Based on this system, Babylonian scholars developed an extraordinary knowledge. They had a good understanding of the four basic operations of addition, subtraction, multiplication and division, and also of square roots, powers and reciprocals. They produced extremely detailed arithmetic tables and developed very good solution techniques for equations that they set themselves.
However, all this was soon forgotten. The Babylonian civilization was in decline, and a large part of its mathematical advances would be consigned to oblivion. There would be no more numeration by position and no more equations. Indeed, there was a time lag of centuries before these questions became the flavour of the day again, and it was only in the nineteenth century that the decoding of cuneiform tablets reminded us that the Mesopotamians had tackled these things before everyone else.
Following the Babylonians, the Mayans also devised a positional system, in this case with base 20. Then it was the turn of the Indians to invent a system with base 10. This last system was used by Arab scholars before it reached Europe at the end of the Middle Ages. Its symbols became known as Arabic figures, and soon spread all over the world.
0 1 2 3 4 5 6 7 8 9
With numbers, mankind gradually came to understand that it had just invented a tool for describing, analysing and understanding the world around it that surpassed any purpose it might have hoped for.
Sometimes we have been so pleased with numbers that we have even overdone it. The birth of numbers represented at the same time the birth of the practice of various forms of numerology. This involves attributing magical properties to numbers, interpreting them beyond the bounds of reason, and attempting to read into them messages from the gods and about the fate of the world.
In the sixth century BC, the Greek scholar Pythagoras made numbers the fundamental concept of his philosophy when he declared: ‘Everything is a number.’ According to him, it is numbers that produce geometric figures, which in turn give rise to the four elements of matter – fire, water, earth and air – of which all living things are made. Pythagoras thus created a whole system around numbers. The odd numbers were associated with the masculine, while the even numbers were feminine. The number 10, represented as a triangle called the tetractys, became a symbol of harmony and of the perfection of the cosmos. Pythagoreans were also behind the origin of arithmancy, which claimed to read people’s characters by associating numerical values with the letters comprising their names.
In parallel, there began to be discussions about what constitutes a number. Some authors believed that the unit – 1 – is not a number, because a number denotes a plurality and so can only be considered from 2 onwards. It was even asserted that in order for it to be able to generate all the other numbers, 1 must be simultaneously even and odd.
Later, increasingly animated discussions developed concerning the zero, the negative numbers and the imaginary numbers. In each case, the admission of these new ideas to the circle of numbers led to debate and forced mathematicians to broaden their ideas.
In short, numbers have never ceased to raise questions, and human beings still need time to learn to master these strange creatures that are their brainchildren.
LET NO ONE IGNORANT OF GEOMETRY ENTER
Once numbers had been invented, it did not take long for the discipline of mathematics to spread its wings. Various core branches such as arithmetic, logic and algebra gradually sprouted within it, developed to maturity and asserted themselves as disciplines in their own right.
One of these, geometry, rapidly won the popularity stakes and captivated the greatest scholars of antiquity. It was this that singled out the first celebrities of mathematics, such as Thales, Pythagoras and Archimedes, whose names still haunt the pages of our textbooks.
However, before it became a subject for great minds, geometry gained its place on the ground. Its etymology bears witness to this: it is first and foremost the science of the measurement of the Earth, and the first surveyors were hands-on mathematicians. Problems concerning the division of territory were then classics of the craft. How to divide a field into equal parts? How to determine the price of a plot of land from its area? Which of two plots is closer to the river? What route should the future canal follow to make it the shortest possible?
All these questions were paramount in ancient societies where the whole economy still revolved in a vital way around agriculture and hence around the distribution of land. In response to this, geometrical know-how was built up, enriched and transmitted from generation to generation. Anyone equipped with this know-how was certain to hold a central and indisputable place in society.
For these measurement professionals, the rope was often the primary instrument of geometry. In Egypt, ‘ropestretcher’ was a profession in its own right. When the Nile floods led to regular inundations, it was the ropestretchers who were sent for to redefine the boundaries of plots that bordered the river. Using information they recorded about the ground, they planted