Brahmagupta's rules, and how long it took the world to accept them

0:00Negative numbers were not invented by mathematicians. They were written down by people keeping accounts, and they were written down a very long time ago. If you owe somebody money, you have to record it somehow, and you have to be able to tell it apart from money you have. That is a practical problem, not a theoretical one, and traders across Asia had solved it centuries before anybody wrote down a rule about it.

0:31So this story does not begin with a great idea. It begins with a ledger. What comes later is the difficult part, and it is not the part you would expect. Here is one of the oldest examples we have, from a Chinese work finished around the first or second century of the common era. It is called The Nine Chapters on Mathematical Art, and in it, numbers are laid out as counting rods.

1:02Rods of one colour for one kind of quantity, and rods of another colour for the opposite kind. Red and black. Now look at the counters underneath. Green ones worth one each, red ones worth minus one each. Same idea. Same picture. Two thousand years apart. The model you would use in a classroom today is not a modern teaching aid. It is very nearly the original one. Older still, and from India, there is a treatise on statecraft and economics written around three hundred years before the common era.

1:43It deals with credit and debit at length, because a state that collects taxes and pays wages has no choice about it. And it does something worth noticing. It lets a balance fall below zero, and it treats that as a real position rather than as a mistake. That is nine hundred years before anybody wrote down a rule for adding such a thing. Nine hundred years of using a number that has no rules yet.

2:13Which tells you something about the order in which mathematics actually gets made. Around three hundred years into the common era, a manuscript from the north of the subcontinent does something that looks strange today. It marks a negative number by putting a small symbol after it, instead of in front. Not minus five. Five, and then the mark. Which is a good moment to notice that the minus sign in front is a convention, and nothing more than a convention.

2:48Somebody chose it. Somebody else chose differently, and their arithmetic worked exactly as well as ours. The notation is not the number. And then, in the year six hundred and twenty eight, Brahmagupta. By this point negative quantities have been in use for the better part of a thousand years. So what he did was not a discovery. Everybody already had these numbers. What he did was decide that positive numbers, negative numbers and zero are one kind of thing.

3:25Not three kinds, with three sets of rules and a list of exceptions between them. One kind, one set of rules, every operation defined for all of them. That is a decision rather than a discovery, and it is the reason his rules are still the rules. He gives five rules for adding, and you can read each one straight off an example. Two positives. Two plus three is five. Add the sizes, keep the sign.

3:58Two negatives. Minus two plus minus three is minus five. Add the sizes, and keep the sign again. One of each. Minus five plus three is minus two. Take the smaller size from the larger, and keep the sign of the larger. A number with its opposite. Two plus minus two is zero. And a number with zero. Minus two plus zero is minus two, and nothing at all happens.

4:29Then five more, for subtracting. Smaller from larger, three take away two is one. Larger from smaller, two take away three is minus one. Taking away a negative. Two take away minus three is five. If you have met that one as a trick, this is where it stops being a trick. A number from itself. Two take away two is zero, and minus two take away minus two is also zero.

4:59And the cases with zero. Minus two take away nothing is minus two, and nothing take away minus two is two. Ten rules altogether, which sounds like a great deal to memorise. It is about to stop being ten. Because before memorising any of them, you can check every single one on a number line. Take the awkward one. Two take away minus three. Subtracting means moving to the left. So subtracting a negative means moving to the left by a negative amount, and that is moving to the right.

5:38Start at two. Move three to the right. Land on five. No new picture was needed. It is the same line as always, and the rule falls out of it. That is the difference between a rule that is true and a rule that has merely been agreed. Now count the rules again. Every subtraction is an addition. Take away three is add minus three. Take away minus three is add three.

6:08So five of the ten were never separate rules at all. They were the first five, wearing a different coat. And the first five collapse as well. Same sign, add the sizes and keep the sign. Different signs, take the smaller size from the larger and keep the sign of the larger. That is the whole of it. The opposite case and the zero case are not exceptions to that sentence. Put them into it and the right answer comes out on its own.

6:42Which is exactly what Brahmagupta was claiming. Not three sets of numbers with a treaty between them. One set, one rule, and nothing held aside as a special case. There is a modern name for a system that behaves like this, and it is a word you will meet again in a few years. It is called a ring. That is all that will be said about it here, because the definition needs more mathematics than we have so far, and closure is not it.

7:14But the thing itself you already have. You have been working inside one all along. So how long did the rest of the world take to agree? The ideas reach the Arab world within about two centuries, by the ninth. They reach Europe by the thirteenth. That is six hundred years, and fair enough, travel was slow. But here is the number that matters. In the seventeen hundreds, five hundred years after these numbers had arrived, a French mathematician named Carnot was still calling them absurd.

7:50They were not missing. They were not unknown. They had been in Europe for centuries, and they were being refused. The obstacle was never information. So what was the obstacle? A negative number is not a thing you can put on a table. Three apples you can point at. Minus three apples you cannot. So the objection was that these are not really numbers at all, only a way of talking about what is owed.

8:22And the answer to that objection is the one Brahmagupta had already given a thousand years earlier. They behave like numbers, under one consistent set of rules, without a single exception. That is the only test there is. Nothing has to be pointed at. And once you accept that, you can begin writing letters in place of numbers, which is the subject called algebra, and it is where all of this goes next.