Mathematics Minus Fear


I know that you are probably a little defensive about your ability to 'do' numbers. However, first of all, I want to get your achievements into context. Forget about the pressure of the school mathematics test. Wipe from your mind that little destructive silence that your teacher left just after you announced that you had only scored 'five out of ten'. If you are able to do any sum - either in your head or on paper - that is a little miracle in itself. It means that you have already come a long way.

Let's see just how far. When you were born, you knew nothing. This is not an insult. No one knows much just after they have been born, except that it is a blessed relief to get out in the fresh air. As a result of interacting with the world, you gradually began to figure some things out. You knew that there is a difference between one aunt, two aunts, three aunts and four aunts, even if they were trying to distract you from figuring this out by making funny noises and invading your personal space. But any more than this number of aunts, and it was all too much. There could have been ten of them or fifteen of them - you wouldn't have known the difference.

And without any further help, that is as far as the human mind will get in Arithmetic. Occasionally, a child grows up without contact with other humans. Such a person is called a 'wild child'. If they are not discovered early in their life, this is the limit of their understanding of number. Once they have hit puberty, they are rarely capable of improving on this vague sense of the difference between one, two, three, and four objects.

This is exactly the same stage as the wiliest of animals can reach. Some of the best natural mathematicians are certain species of birds, like the crow and the magpie. If you are one of those people who collect birds' eggs and put them in glass cases with neat labels, then please bear in mind this fact to avoid unnecessary cruelty to the poor mother. Don't raid a nest where there are four or less eggs, because the lady magpie will know that one of her future children has gone missing.

Leaving behind the animal kingdom, there have existed whole human civilisations that have not got further than this. Unlike the wild child, they all developed speech, but, in general, they only had numbers for 'one' and 'two'. They could deal with 'three' and 'four' by talking about 'two-one' and 'two-two', but beyond that the average Botocudan from the Brazilian rain forest would just point at his head, and look a bit sorry for himself. This is no comment on the intelligence of Botocudans. They were perfectly capable individuals. It's just that they had no need for numbers greater than this.

There is evidence of our inability to get beyond a concept of 'four' all over the place. For example, the Romans only gave normal names to the first four of their sons. Then the fifth son was always called Quintus ('the fifth'), the sixth son Sextus ('the sixth'), the seventh Septimus ('the seventh') and so on. Similarly, in the original Roman calendar (which only had ten months), the first four months had names unconnected to their position (Martius, Aprilis, Maius, Junius), but the rest of them were named from their order: Quintilis, Sextilis, September, October, November and December. Later, January and February were included, when it was realised that the months were falling out of step with the seasons, and Quintilis and Sextilis were renamed July and August after the emperors Julius Caesar and Augustus.

Now, this is not strictly relevant, but you are probably wondering how a member of the Botocudan tribe managed to keep track of things if she had no concept of a number greater than four. What happened if she found nine identical eggs in the nest of a macaw, and decided to carry them off home for breakfast? How would she know that she hadn't dropped one on the way back, if she didn't know the difference between nine and eight?

Well, the truth is that the Botocudans were perfectly capable of dealing with this sort of thing. They would take a tally. One way of doing this was that for each egg, they would pick up a pebble, or tie a knot in a piece of string, or cut a notch on a stick. And when they got home, as they took out each egg, they would throw away a pebble, or untie a knot, or cross out a notch. This way they could keep track of their belongings, and only ever really deal with the number 'one'. Each egg was a 'one' that they would record in whatever way was most handy.

In fact, one of the most common ways of keeping a tally was to use their own body. Each tribe would come up with a particular order for the different parts of the body. So the Botocudan woman would touch the little finger on her left hand when she put the first egg into her pouch, the second finger on her left hand for the second egg, and so on until she had used all her fingers for the first five eggs. For the sixth egg, she would touch her left wrist, for the seventh egg, her left elbow, for the eighth egg her left shoulder, and for the ninth egg her left breast. When she got home, she just had to go through this sequence again as she took out her eggs. If she ended up pointing at her left breast, then she hadn't dropped any.

This might all sound very primitive, but just to stop you from feeling smug, different methods of tallying have stuck with us through the ages. Up until 1828, the British exchequer sent out tax demands on tally sticks, and kept them as receipts in the basement of the Houses of Parliament. When the system was abolished in 1834, the politicians decided to burn all the sticks. Unfortunately, they lost control of the fire, and burnt down the Houses of Parliament by mistake.

So there you go. At the age of eighteen months, you are already at the same level as many civilisations ever reached altogether. As soon as your parents encourage you to count with your fingers (just like the Botocudans), and to give each new number a name (unlike the Botocudans), you have moved into a place that many inhabitants of this planet have never been. You are a little genius. And it won't be long before you can tell all nine of your irritating aunts to kindly leave you alone.


Getting beyond the number four is only the beginning. 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.

So the next challenge that you have to deal with is to understand the system that we use to cope with all these numbers. And the system that most people use 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 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 at such a tender age, 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 (like the Botocudans), 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 civilisations 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 West Africa would have dealt with numbers. It is a perfectly sensible way of going about things.

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.


For most of my time at primary school, doing mathematics was the same as doing sums. You had two choices when faced with a sum. You could either do it in your head, or you could do it on paper. Doing it in your head was harder, and you got more respect. If you could do a sum very quickly, the chances were that some of your classmates would gasp, and mutter: 'He's CLEVER'. Of course, this only lasted until the beginning of your first year at secondary school. After that, if you did a sum in your head quickly, you were more likely to be beaten up.

Still, there is no denying that doing mathematics in your head is a useful skill, if only to make sure that you never get short-changed. I once had a very short-lived stint as a barman. I was expected to add up the price of a round of drinks in my head. I found this very hard, especially on a Friday night, when the pub was packed, and someone had already ridiculed me for being the only barman not wearing a white shirt (mine was blue - I still don't know what the problem was). I began to make more and more mistakes, and my confidence ebbed away. It began to affect my other duties as a barman. I lost my ability to pour a decent pint of lager. My hands started to tremble. The punters became more aggressive. My boss had a go at me for taking too long to deal with an order, and for failing to add up the prices correctly.

At last, one customer made the simple request of a pint of soda and lime. Just one pint. No problems totting up the price of this order (I thought), and no problems in pouring it. It was the moment of calm that I needed to collect my thoughts. I poured in the lime, and then pressed the button to squirt the soda into the glass from the siphon. As I turned to the customer with an air of nonchalance to tell him the price of his drink, I let go of the button. It was stuck. The siphon continued to shoot out soda water in an impressive jet. I tried to stop it by covering its nozzle with my finger, but this only had the effect of increasing the power of the stream of soda, and sending it fizzing all over the bar and several of the customers. As the whole pub turned to stare, I found myself grappling with the siphon and its metallic cord as if with a futuristic serpent. I had finally wrestled it to the ground, and was about to bite off its head, when my boss calmly turned it off at the main pump. The siphon went limp, and I was taken off bar duty. I spent the rest of the evening arranging crisps in boxes so that the customers could see what flavour they were.

You might wonder what this story has to do with anything. But remember that the root cause of this disaster was my inability to do sums in my head. This flaw in my mental make-up gradually undermined my confidence, and I hold it totally responsible for leaving me lying in a fizzy puddle on the dirty floor of a back-street pub.

In order to improve my mental arithmetic, I went out into the world to find out the different techniques that people use to deal with numbers in real life. Or rather, I stood outside a supermarket in South London, and asked people mental arithmetic questions. It was only one afternoon - just a few short hours - but it taught me many things.

Firstly, I learnt that the world is a hostile place, and people on their way to supermarkets are mean. Many people just pretended that I didn't exist. Others marched past me with a cruel smile playing on their lips. Secondly, I learnt that women are much nicer than men, although it is best to avoid mothers with more than three children in tow. Thirdly, and most importantly, nothing has changed since school. No one has grown up. Everything is exactly the same.

I wonder if you recognise any of these people from your days in a classroom. Do you remember a boy who, after scoring ten out of ten in a mental arithmetic test, pumped his fist and yelled 'I am the King'? In fact, so confident was he in is own ability that he demanded that the teacher give him an extra-hard multiplication sum, just to underline his talent. Or do you recall the girl who just got redder and redder and redder as each question was asked, until she burst into a fit of hysterical laughter, and had to leave the room? What about the boy with an opinion of himself that was slightly too high, who called out the wrong answer to a question that he was not being asked? And the girl who was being asked, and who was trying her level best, who then flew into a rage at him, and questioned his manhood? Ring any bells? Well, they were all there, outside the supermarket.

And not only were the characters the same; their attitudes towards doing sums were the same as well. They were obsessed with knowing whether they had got the answer right. Some of them even asked me to give them a mark at the end of the survey. One man came back half-an-hour later to tell me that he had worked out the answer to question nine. He claimed it was seven - he was wrong.

Many of them were nervous about having to do mathematics in the open. Several people told me that they were rubbish at arithmetic at school. They looked fearfully around them as they answered the questions. And one woman really did run away when I asked her to do a division in her head. Presumably, I had reopened an old and painful wound, and I sincerely apologise to her.