# Babylonian Numbers

The Babylonian number system is old. It started about 1900 BC to 1800 BC but it was developed from a number system belonging to a much older civilisation called the Sumerians. It is quite a complicated system, but it was used by other cultures, such as the Greeks, as it had advantages over their own systems. Eventually it was replaced by Arabic numbers.

### Count with Babylonian numbers

The Babylonians writing and number system was done using a stylus which they dug into a clay tablet. This explains why the symbol for one was not just a single line, like most systems. I am using a yellow background to represent the clay!

Enter a number from 1 to 99999 to see how the Babylonians would have written it, or enter a number to count with.

Enter number:       every

### How Babylonian numbers worked

Like the Egyptians, the Babylonians used two ones to represent two, three ones for three, and so on, up to nine. However, they tended to arrange the symbols into neat piles. Once they got to ten, there were too many symbols, so they turned the stylus on its side to make a different symbol. Eleven was ten and one, twelve was ten and one and one, twenty was ten and ten, just like the Egyptians. This is a unary system. However, something strange happened at sixty (see below). The symbol for sixty seems to be exactly the same as that for one. Sixty one is sixty and one, which therefore looks like one and one, and so on. Surely this is very confusing! However, the Babylonians were working their way towards a positional system (see below).

   1-10 11-20 21-30 31-40 41-50 51-60 61-70 1 2 3 4 5 6 7 8 9 10

### Positional number system

A positional number system is one where the numbers are arranged in columns. We use a positional system, and our columns represent powers of ten. So the right hand column is units, the next is tens, the next is hundreds, and so on. If you want to add large numbers (and you've lost your calculator!) you line the numbers up so their units are in the same column. Then you can add each column, carrying forward to the next, if necessary. The Babylonians had the same system, but they used powers of sixty rather than ten. So the left-hand column were units, the second, multiples of 60, the third, multiplies of 3,600, and so on.

x 3600x 60UnitsValue
1
1 + 1 = 2
10
10 + 1 = 11
10 + 10 = 20
60
60 + 1 = 61
60 + 1 + 1 = 62
60 + 10 = 70
60 + 10 + 1 = 71
2 x 60 = 120
2 x 60 + 1 = 121
10 x 60 = 600
10 x 60 + 1 = 601
10 x 60 + 10 = 610
3600 (60 x 60)
2 x 3600 = 7200

You can now see why they piled the units up into neat piles! They needed to distinguish one plus one or two, from one times sixty plus one meaning sixty one. Both these have two symbols for one. But the representation of two has the two ones touching, while the representation for sixty one has a gap between them. A careless clerk might make mistakes that way, but if you were careful, it should be all right.

### Lack of zero

A more serious problem was that to start with they had no symbol for zero. We use zero to distinguish between 10 (one ten and no units) and 1 (one unit). The Babylonian symbol for one and sixty are the same. Believe it or not, this didn't worry them. After all, if you were counting things, you would tend to know if you were counting individual things or counting in lots of sixty (or even 3,600!) So the Babylonians didn't bother with a zero at the end of the number. However, it is more serious with gaps in the middle of the number. The number 3601 is not too different from 3660, and they are both written as two ones. You could say that there should be a bigger gap for 3601, since the gap represents nothing in the sixty column, but how easy to make a mistake! So the Babylonians DID have a zero, which they used only in only in the middle of numbers. Let's assume that a Babylonian is counting things. He knows that there are large amounts of them, so a single one represents 3,600. The strange slanting symbol is the zero.

 = 60 x 60 = 3,600 = 3,600 + 60 = 3,660 3,600 + 0 + 1 = 3,601

The great advantage of the positional system is that you need only a limited number of symbols (the Babylonians only had two, plus their symbol for zero) and you can represent any whole number, however big. You can also do arithmetic far easier, although I'm not quite sure about learning multiplication tables up to 60! But you do really need a zero. The Babylonians had a sophisticated number system, but it didn't quite work. It is a base 10 / base 60 system, and quite hard to understand.

### Base 60 in modern times

You many wonder why they seemed to like the number sixty so much. Sixty is a very good number for a base. There are many factors (numbers which divide into it).

 Factors of 10: 2 5 Factors of 60: 2 3 4 5 6 10 12 15 20 30

In fact, the Babylonians have given their base 60 to us. There are 60 minutes in an hour, and 60 seconds in a minute. There are also 360 degrees in a circle (6 x 60), and a single degree can be broken down still further. There are 60 minutes in a degree, and 60 seconds in one of these minutes. (There is no connection between angle minutes & seconds and time minutes & seconds.)

### Counting base 60 on fingers

We are used to base 10 or decimal, because we count on our fingers and there are ten of them. But it is possible to fingers to count to 60 using our fingers, and so use them for a base 60. The British TV programme, QI, said that the Babylonians did this, but I don't know what proof there is that they did. It does give the "base 12 within base 60" effect, which is suggestive.

Take one hand. Turn it palm side down. Ignore the thumb. Now count the knuckles. There are 12 of them, three on each finger. I'm not sure which finger they started counting or in what direction, but the picture gives a suggestion.

That gives a count of 12. Now turn down a finger (or thumb) on the other hand, and start again. Since you have 5 fingers (including the thumb) on the other hand, then you can count to 5 x 12 or 60.

There are other finger counting systems here.