From 0 to Infinity in 26 Centuries (2 page)

When you count on your fingers, or, more importantly, use them to show that Greek barman how many shots of Metaxa you’d like to buy, it does not really matter which fingers you use –
we tend to use what is anatomically more comfortable. For example, it is far easier to show the number four using the fingers of one hand rather than a thumb and three fingers on the same hand (try
it!). It does mean, however, that there are ten different ways of showing one on your fingers, and that the finger system relies on a person’s ability to count up all the fingers that are
shown.

Some cultures have navigated these potential pitfalls by assigning a value to different parts of their body. On the Torres Strait Islands between Australia and Papua New Guinea, thirty-three
different body parts are used for counting, and indeed as words to represent the numbers. For example, to a Torres Strait Islander the ring finger on your left hand means sixteen, your right
shoulder means eight, your left knee twenty-four and your little toe on the right foot thirty-three. This system has evolved to allow for effective communication between islands that are home to
several different languages.

The Stone Age humans persisted in their ways, hunting and gathering without the need for many numbers at all. However, approximately 10,000 years ago, people in fertile areas
around the world’s great rivers decided to settle down and get civilized. This led to the need for much bigger numbers, methods of recording them and every school kid’s favourite
– arithmetic.

Early Civilized Maths

F
ROM
H
UNTER
-G
ATHERERS TO
H
ERDERS

According to historians there are five main stages involved in a society becoming ‘civilized’. The first stage is the ability to make and control fire –
Homo sapiens
and their ancestors have been creating fires for approximately half a million years. The second stage is the cultivation of crops, which really requires the help of domesticated
animals. The Neolithic Revolution, which occurred in independent locations across the globe approximately 10,000 years ago, saw humans begin to stay in one place, grow crops and domesticate and
rear livestock.

Counting sheep

The first shepherds would have needed a method for counting their animals, so it seems obvious that they would have been good at counting. Or would they? Equally, the first
farmers would have needed a method of working out what time of year it was so they knew when to plant their seeds – surely more numbers were involved here?

The Neolithics used a system called pebble counting. In order to count their herds of sheep the shepherds placed one counter – perhaps a pebble or a fruit pip – in a bag for each
member of the flock. To find out if all of the animals in the herd were safe the shepherds then removed a pip for each sheep counted. If by the time they got through all of the sheep they still had
pips remaining they knew then that one or more of the sheep had been lost. Hopefully, if they were half-decent shepherds, this number would have been one of the low numbers (1–10), which we
now know hunter-gatherers were equipped to deal with. The farmers would have used a similar system to count from a key event, such as the rains beginning, or the birds flying south, to let them
know when to plant their crops.

The economy takes shape

Surplus is an inevitable consequence of agriculture and the domestication of animals. Once there was an excess of food, early civilizations could start to trade their surplus,
which in turn promoted the idea of value. This is the third sign of civilization – the idea of an economy; of goods being bartered or
sold. Naturally, they needed a way
to cart around their surplus of goods, so the fourth sign of civilization is the wheel, which seemed to be widespread by
c.
4000
BC
.

T
HE
D
AWN OF THE
A
CCOUNTANTS

With a reliable surplus of food it then became possible for some people to fill their time doing things other than finding or growing food in order to survive. By
c.
3000
BC
towns and cities were filled with such people. These large urban societies needed organizing in order to function properly, and the ability to count large numbers was key
to this. The need also arose for these new, large numbers to be recorded – and hence written – for the first time. Among the myriad flagship new professions that arose –
craftsmen, soldiers, farmers, merchants – a new literate class – the scribe – was to be found; almost certainly many of these scribes were numerate accountants and –
inevitably – tax collectors.

The Bronze Age

The fifth sign of a civilization is the use and working of metals, which started at approximately the same time as the first towns and cities arose. The easiest metal to work is
copper, which is used to make bronze. Hence we call this period in history the Bronze Age.

Most of the earliest civilizations developed around the fertile areas near rivers, where the land was suitable for farming and raising livestock. Three notable
civilizations that arose during this period, and about which we know a great deal, were:

As some cities rose in importance and began to dominate the surrounding area, either economically or by strength of arms, certain regions began to adopt the most effective
numbers system. Unfortunately the early records from many such cultures have not survived, perhaps because these people were sited next to large rivers that flooded often. But if we take a quick
look at some of the highlights of the archaeological findings that
have
survived from each civilization, we gain some idea of an evolution of written numbers and number systems from our
earliest urbanite ancestors.

M
ESOPOTAMIAN
M
ATHEMATICS

Mesopotamia means ‘between rivers’ in Ancient Greek, and refers to the cultures that sprang up between the Tigris and Euphrates rivers in present-day Iraq – a
very fertile area of land often called the Cradle of Civilization. Despite their early beginnings, we know a fair amount about the Mesopotamians because they performed all of their writing on clay
tablets, which are hardy enough to withstand repeated soakings.

Clay tablets, however, are not easy to write on. The Mesopotamians began with a pictographic language, when the written symbol for a word is a stylized picture of the thing being described.
However, drawing decent images in thick wet clay is tricky, so they took to using the end of a wedge-shaped stylus (a rod-like implement with a pointed end) to make marks.

Number system

The Mesopotamian’s number system was base 60 (also referred to as
sexagesimal
), which means they counted in blocks of 60 rather than in blocks of 10, as we do
today. Lots of numbers go into 60 (mathematicians would say 60 has many factors), which makes it a convenient number with which to do arithmetic. We still see a few reminders of it today – 60
seconds in a minute and 60 minutes in an hour hark back to the Babylonians.

Cunningly, their number system contained only one symbol, made using the end of a stylus:

They would use up to nine of these symbols. They would show a 10 by rotating the stylus by 90 degrees to get a slightly different symbol:

So the number 47 would look like this:

The Mesopotamians could write up to the number 59 in this way, using a sub-column for each of the tens and units digits. To write numbers larger than 59, they would write a new
number alongside the tens and units (just as we can go as high as 9 in one column before we have to move on to the next one). Whereas our columns follow the pattern units, tens, hundreds,
thousands, etc., according to our way of thinking the Mesopotamians’ columns went units, sixties, three-thousand six-hundreds, two-hundred and sixteen-thousands. A Mesopotamian would think of
our number 437 as being 7 lots of 60 (7 × 60 = 420) plus 17, which they would write like this: