Why We Count to 10?
As soon as you start counting, you bring into existence an infinite amount of numbers. And it is all very well to start giving special names to the first few of them, but you can't come up with new names forever, and even if you did, you wouldn't be able to remember them all. It's a bit like the Romans with their sons- after a while you give up trying to be original and just call the next one Sextus - ‘Number Six’.
So the system that most people use to cope with all these numbers is called the decimal, or base ten system. The best people to talk to about this are the Tibetans, because they stick to it the most completely. The Tibetans have come up with words for the numbers zero to nine (as we call them). They have also come up with words for every power of ten (as have we for the most part: ten, hundred, thousand and million). Then they can express in words any number they like by combining their words for zero to nine, with their words for the powers of ten. So, they would call the number 324: 'three hundreds, two tens and four', or actually 'gsum-bryga gnyis-bcu rtsa bzhi'.
Now you might be thinking, with a nationalist rush of anger, that the English language is every bit as logical as the Tibetan. But it isn't quite. For starters, in English, names for numbers get shortened to make them easier to say. 'Two-tens' becomes twenty. 'Five and ten' becomes 'fifteen'. Also, there is the mystery of 'eleven' and 'twelve'. They don't appear to have anything to do with 'two-and ten' or 'one-and-ten', although one theory is that they are different - that is, don't contain any reference to ten- because they are so near to it in sequence. So 'eleven' derives from 'one left' (after ten) and 'twelve' from 'two left' (after ten). When we get to thirteen apparently we are getting too far away from ten to cope without being reminded of where we are.
And then there is the fact that we just got lazy when it came to making up names for powers of ten. While the words 'ten', 'hundred', 'thousand', 'million', 'billion' and even 'trillion' are all commonly used in English, the Tibetans went further. They also have a special name for 'ten thousand' and 'one hundred thousand'. We couldn't be bothered, which is a shame, because it would make writing cheques much easier. So we can't claim to be quite as logical as the Tibetans, but we can claim to be a lot more sensible than the Welsh. They came up with 'two-nines' instead of 'ten and eight'. Where is the sense in that?
You might have wondered why we count like this. And in doing so, you once again proved your potential for genius. That is exactly the same question that a fully-grown Aristotle asked, and he ranks as one of the greatest philosophers of all time: 'Why do all men, whether barbarians or Greeks, count up to ten and not to some other number?'
The short answer to this question is: FINGERS. Your fingers are the most natural tool for counting that you have. At some point, people stopped using them as a tally and started connecting them with numbers.
The long answer to the question is that in fact not everyone has counted like this. Although the vast majority of counting civilizations have used base ten, there are plenty of examples of people who used different bases. This may seem surprising. Our numbers and the way that we use them seem so natural that it is hard to believe that they do not just exist in the world that way. But the base ten system is just one of an infinite number of ways that we could have chosen to put numbers into a system. If you had eight fingers, rather than ten, for example, you would be using base eight, and be just as happy, except you would not be so good at playing the piano.
This is not just a hypothetical situation. Besides base ten, the most common number system used is the vigesimal system, or base twenty. Both the Maya and the Eskimos used base twenty, presumably because they were counting on their fingers and their toes - although what an Eskimo was doing without any shoes on, I don't know.
There are still elements of base twenty thinking around today. If you ask a mysterious stranger in the middle of a windswept English moor how far it is to the nearest pub, he might answer: 'Two score miles and ten'. He actually means 'Two lots of twenty miles and ten more' which to you and me is fifty. (The word 'score' for the number twenty has been used since Biblical times, when the average human lifespan was said to be 'three score years and ten' -i.e. seventy years. The word comes from the old method of keeping tally. When you got to the number twenty, you made an extra large cut, or score, in your counting stick.) Similarly if you ask a Frenchman for eighty onions, and in his surprise at the strength of your need for his national vegetable, he raises his eyebrows, and says: 'Quatre-vingt?', what he means is: 'Four twenties?' Both of these people are using a base twenty system.
Just as base ten developed from
counting the fingers on both hands, and base twenty from fingers and toes, a
base five system developed in several civilisations through people counting on
just one hand. To give you an idea of what you are missing, this is how a
member of the Fulah tribe in
Firstly, he had special names for the numbers from one to four. To make life easier, let's say that these names were, in fact, 'one', 'two', 'three', and 'four'. He also had special names for the powers of five (5, 25, 125 etc.). Let's say they were as follows: 'five' (5), 'high-five' (25) and jackson-five' (125). He could then use this system to name any number he liked.
For example, take the number we call 'three-hundred-and thirty-nine'. We have given this number its name because we think of it as being made up of three hundreds, three tens, and nine units. But the Fulah tribesman did not think of it as being made up in this way at all. He looked at it, and saw it as being made up of two jackson-fives, three high-fives, two fives and four units, and so he named it precisely that. You can check his thinking. He hasn't made any mistakes. It all adds up to 339. It is just a different way of looking at the same number.
(I should add here that I shouldn't really talk about the number 339 as if that is the only way of representing this number in symbols. It isn't, and the Fulah tribesman, if he had got around to writing numbers using symbols, would not have written it like this. But that is something that I will come to later. For now, when I write 339, I simply mean the number that we are referring to when we write down these symbols.)
It is possible that your mathematics teacher never told. You about all of this. It is possible that he kept it to himself, tucking away his knowledge in his tattered leather briefcase next to his Tupperware box containing corned beef sandwiches and an overripe tangerine. But it is all true. The way we count is the result of the design of our bodies. It is a system that we have made up to deal with the consequences of inventing number. And it is by no means an easy one to understand.
1. What would we call this Fulah number: 'four jackson-fives, three high-fives, two fives and one'?
2. What would a Fulah call our number: 'four-hundred and seventy-three?
from ‘Mathematics Minus Fear’ by
Published in 2006 by Marion Boyars Publishing Ltd