## 50,000 YEARS… AND COUNTING

The idea of numbers goes back long before recorded history began, as some archaeological evidence suggests that humans were counting as far back as 50,000 years ago. The earliest sense of numbers that primitive humans had was probably recognising the concepts of more and less, when some objects were added to or taken from a small group. In the very beginning of mathematics, people could only distinguish between one and many. As societies evolved and tribes formed, it became important to be able to know how many members were in the group, and perhaps how many were in the enemy’s camp.

Thus, counting developed. Most likely the original counting method was tallying. Tally sticks first appeared as animal bones carved with notches during the Upper Palaeolithic. In the late 1930s, archaeologist Karl Absolom, sifting through soil in Czechoslovakia, uncovered a 30,000-year-old wolf bone with a series of notches carved into it. This wolf bone had 55 notches in it, arranged into groups of five, and there was a second notch after the first 25 marks.

“Nobody knows whether Gog the caveman used the bone to count the deer he killed, the paintings he drew or the days he had gone without a bath, but it is pretty clear that early humans were counting something. A wolf bone was the Stone Age equivalent of a supercomputer.” (“Zero: The Biography of a Dangerous Idea” by Charles Seife, 2000). Another example of this kind of tool is the Ishango Bone, discovered in 1960 in Ishango in the Democratic Republic of the Congo (the former Belgian Congo). Reported to date from around 18,000 to 20,000 BC, it is a dark brown length of bone, the fibula of a baboon.

## SPEAKING NUMBERS

If numbering preceded the concept of numbers, on the way to more advanced levels of civilisation, enumeration preceded numbering. In this context, enumeration means the correspondence between a given object and an “account”. The Bugilai tribe in Papua New Guinea used a sequence of different parts of the body to count to a very high number without resorting to any words or symbols. But objects other than body parts (seeds, scratches on stones or grooves in sticks) were also used as counters in the first stages of enumeration. As methods for counting developed, and as language progressed as well, spoken words for numbers would appear. How this happened is still impossible to trace. However, after ten, it is possible to detect some patterns. For example, eleven comes from “ein lifon” in early Teutonic, meaning “one left over”.

## THE EVOLUTION OF A SYSTEM

The ancient custom of recording notches on a staff or scratches on stones soon became insufficient for everyday needs. The development of societies made symbols convenient for writing down numbers and the methods for making calculations with those numbers. The symbols for writing numbers are called numerals and the methods for making calculations are called algorithms. When taken together, they form what we call number systems. The most common system in the world is the Hindu-Arabic system, which was probably invented between the 1^{st} and 4^{th} centuries by Indian mathematicians and adopted into Arabic mathematics by the 9^{th} century.

Composed of the ten symbols {0,1,2,3,4,5,6,7,8,9}, this is a base-ten (decimal) system where each digit’s position in a number increases in powers of 10 – for example, a 3 in the second position means 30 (3 x 10), if it is in the third position it means 300 (3 x 100) and so on (possibly because the ten fingers of both hands were used to count in the early days of numbering). Furthermore, this system is positional, which means that the position of a symbol has a bearing on the value of that symbol within the number. Unlike this one, two other important base-ten systems – the Roman and the Egyptian – are non-positional. The numeric system represented by Roman numerals originated in Ancient Rome (753 BC–476 AD) and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages (14^{th} and 15^{th} centuries).

Found in 1799 in a small village in the Nile Delta called Rosetta (Rashid), the Rosetta Stone is one of the most famous objects in the British Museum in London. It has a message carved in 196 BC using three scripts: hieroglyphic, demotic and Greek, written by a group of priests in Egypt to honour the Egyptian pharaoh. The Rosetta Stone was an important clue that helped experts learn to decipher Egyptian hieroglyphs.

Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, are based on seven symbols: I, V, X, L, C, D, M. Despite having been replaced in most contexts by the Hindu-Arabic numerals, the Roman system persists in some minor applications to this day, from names of monarchs and popes to the number of chapters in books or numerals on a watch dial. The earliest known Egyptian numerals are inscribed in hieroglyphs on a royal staff from about 3,400 BC, when Pharaoh Menes united Upper Egypt and Lower Egypt. These symbols were used to denote large numbers associated with the spoils of war: the capture of 120,000 human prisoners, 400,000 head of livestock and 1,422,000 goats.

For the numbers from one to nine, they repeated a small vertical line and then used special symbols for the different powers of 10, from 10 to 107. These symbols were used in combination and repeated as many times as necessary to express any number. Representations of the different symbols have been interpreted in different ways: 10 was the handle of a basket or bow, for instance. This writing system prevailed from about 3,000 BC until the early centuries of the Christian era and, at some point, it began to be used only for formal inscription, or on stone, wood or metal monuments. In addition to hieroglyphic writing, the Egyptians used two other writing systems. Hieratic writing, the cursive form of hieroglyphs, was employed throughout the pharaonic period for administrative and literary purposes as a faster and more convenient method of writing. First encountered about 660 BC, demotic writing was originally developed expressly for official government use, which required a standardised cursive script.