Professor Stewart's Hoard of Mathematical Treasures (5 page)

Fold and cut this . . .

. . . to make this.

Ancient cultures wrote numbers in many different ways. The ancient Romans, for instance, used letters: I for 1, V for 5, X for 10, C for 100, and so on. In this kind of system, the bigger the numbers become, the more letters you need. And arithmetic can be tricky: try multiplying MCCXIV by CCCIX, using only pencil and paper.

Our familiar decimal notation is more versatile and better suited to calculation. Instead of inventing new symbols for ever-bigger
numbers, it uses a fixed set of symbols, which in Western cultures are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Larger numbers are taken care of by using the same symbols in different positions. For instance, 525 means

The symbol ‘5’ at the right-hand end stands for ‘five’; the same symbol at the left-hand end stands for ‘five hundred’. A positional number system like this needs a symbol for zero, otherwise it can’t distinguish between numbers like 12, 102, and 1,020.

Our number system is said to be base 10 or decimal, because the value of a digit is multiplied by 10 every time it moves one place to the left. There’s no particular mathematical reason for using 10: base 7 or base 42 will work just as well. In fact, any whole number (greater than 1) can be used as a base, though bases greater than 10 require new symbols for the extra digits.

The Mayan civilisation, which goes back to 2000 BC, flourished in Central America from about AD 250 to 900, and then declined, used base 20. So to them, the symbols 5-2-14 meant
which is 2,054 in our notation. They wrote a dot for 1, a horizontal line for 5, and combined these to get all numbers from 1 to 19. From 36 BC onwards they used a strange oval shape for 0. Then they stacked these 20 ‘digits’ vertically to show successive base-20 digits.

Left: the numbers 0-29 in Mayan; right: Mayan for 5 × 20
²
+ 2 × 20 + 14×1

It is often suggested that the Mayans employed base 20 because they counted on their toes as well as their fingers. An alternative explanation occurred to me while I was writing this item. Maybe they counted on fingers and thumbs, with a thumb representing 5. Then each dot is a finger, each bar a thumb, and it can all be done on two hands. Admittedly, we don’t have three thumbs, but there are easy ways round this with hands and it’s not an issue for symbols. As for the oval shape for zero: don’t you agree that it looks a bit like a clenched fist? Meaning no fingers and no thumbs.

This is wild speculation, but I quite like it.

Much earlier, around 3100 BC, the Babylonians had been even more ambitious, using base 60. Babylon is almost a fabled land, with biblical stories of the Tower of Babel and Shadrach in Nebuchadnezzar’s furnace, and romantic legends of the Hanging Gardens. But Babylon was a real place, and many of its archaeological remains still survive in Iraq. The word ‘Babylonian’ is often used interchangeably for several different social groupings that came and went in the area between the Tigris and Euphrates rivers, who shared many aspects of their cultures.

We know a lot about the Babylonians because they wrote on clay tablets, and more than a million of these have survived, often because they were in a building that caught fire and baked the clay rock-hard. The Babylonian scribes used short sticks with shaped ends to make triangular marks, known as cuneiform, in the clay. The surviving clay tablets include everything from household accounts to astronomical tables, and some date back to 3000 BC or earlier.

The Babylonian symbols for numerals were introduced around 3000 BC, and employ two distinct signs for 1 and 10, which were combined in groups to obtain all integers up to 59.

Babylonian numerals from 1 to 59.

The 59 groups act as individual digits in base-60 notation, otherwise known as the sexagesimal system. To save my printer having kittens, I’ll do what archaeologists do and write Babylonian numerals like this:

The Babylonians didn’t (until the late period) have a symbol to play the role of our zero, so there was a degree of ambiguity in their system, usually sorted out by the context in which the number showed up. For high precision, they also had a symbol equivalent to our decimal point, a ‘sexagesimal point’, indicating that the numbers to its right are multiples of
,
×
=
, and so on. Archaeologists represent this symbol by a semicolon (;). For example,
in decimal (to a close approximation).

About 2,000 astronomical tablets have been found, mainly routine tables, eclipse predictions, and so on. Of these, 300 are more ambitious - observations of the motion of Mercury, Mars, Jupiter, and Saturn, for instance. The Babylonians were excellent
observers, and their figure for the orbital period of Mars was 12,59;57,17 days - roughly 779.955 days, as we’ve just seen. The modern figure is 779.936 days.

Traces of sexagesimal arithmetic still linger in our culture. We divide an hour into 60 minutes and a minute into 60 seconds. In angular measure, we divide a degree into 60 minutes and a minute into 60 seconds, too - same words, different context. We use 360 degrees for a full circle, and 360 = 6×60. In astronomical work, the Babylonians often interpreted the numeral that would usually be multiplied by 60×60 as being multiplied by 6×60 instead. The number 360 may have been a convenient approximation to the number of days in a year, but the Babylonians knew that 365 and a bit was much closer, and they knew how big that bit was.

Nobody really knows why the Babylonians used base 60. The standard explanation is that 60 is the smallest number divisible by 1, 2, 3, 4, 5 and 6. There is no shortage of alternative theories, but little hard evidence. We do know that base-60 originated with the Sumerians, who lived in the same region and sometimes controlled it, but that doesn’t help a lot. To find out more, good sites to start from are: