A pinch of history: where solving for an unknown came from

0:00The oldest name for this subject is not algebra. In mathematics written in India more than a thousand years ago it was called bijaganita, and bija means seed. That is a strange word to choose for a board full of equations, and it was chosen carefully. A seed does not look like a tree. Everything the tree is going to be is already inside it, folded up small. What a gardener does is not to invent the tree.

0:32It is to let it out, a little at a time. An unknown is a seed in exactly that sense. The answer is already sitting inside the situation before you pick up a pen. Solving is not making it up. It is coaxing it out. So where does the word algebra come from? Not from Europe. Around the year four ninety nine, Aryabhata proposed a systematic method for handling a single unknown.

1:03Around six twenty eight, Brahmagupta set out how to calculate with unknowns properly. Just under two hundred years later those ideas were being read and rewritten in Arabic. Around eight twenty five a mathematician working in what is now Iraq, Al-Khwarizmi, wrote a work whose title means calculating by restoring and balancing. The word for restoring is al-jabr. By the twelfth century that work had been translated into Latin, and al-jabr had become algebra.

1:40The route ran east to west. And notice what both of the old names describe. Not the answer. The method. Here is what an expression looked like in that older writing. Where we write two x plus one, they wrote: ya, two, ru, one. Ya is short for a phrase meaning as much as so much, which was their name for the unknown. Ru is short for a word meaning form, and it marks a plain number.

2:10Now look at the order. The marker comes before its number, not after it. We write the two and then the x. They wrote the ya and then the two. Nothing mathematical has changed. But you cannot read their line at all until somebody tells you that. Now two x minus eight. They wrote ya two ru eight, and put a small dot above the eight. There is no minus sign anywhere on that line.

2:40The dot does the whole job. That is worth stopping on, because a dot is easy to miss and a minus sign is not. Write both expressions out and the only mark separating them is that dot. And at every value of x, the two differ by sixteen. A notation is a set of promises about what you will notice. What about a whole equation? Three x plus four equals two x plus eight was written as two lines, one under the other.

3:14Ya three ru four on top. Ya two ru eight underneath. There is no equals sign anywhere. The stacking is the equals sign. So this is not our notation with odd symbols in it. The marker leads, a dot carries the negative, and the two sides sit one above the other. Read that stack the modern way and it is an equation you can solve. It comes out at four. Sixteen on the top line, sixteen on the bottom.

3:47You might assume they had one letter and we have twenty six. They had five. Ya was the first unknown. The others were named after colours: ka from the word for black, ni from the word for blue, and two more after those. Our x, y and z are no more systematic than that. They are letters we happened to agree on. So when you meet an old notation that looks alien, ask whether the strangeness is in the mathematics or only in the marks.

4:19Here it is only in the marks. Here is a problem written down around eleven fifty. One man has three hundred coins and six horses. A second man has ten horses, and owes a hundred. The two of them are exactly equally rich, and every horse is worth the same. What is one horse worth? Call it x. The first man is worth six x plus three hundred. The second man's horses are worth ten x, and then there is the debt.

4:54A debt is not a smaller pile of coins. It comes off. He is worth ten x minus one hundred. Six x plus three hundred equals ten x minus one hundred. Solve that and a horse is worth one hundred. Check it. Three hundred and six hundred is nine hundred. A thousand take away a hundred is nine hundred. Equally rich, exactly as the problem said. Now the uncomfortable part.

5:25Suppose you had written the debt as a plus. Six x plus three hundred equals ten x plus one hundred. That has an answer too, and it is a clean one. Fifty. And it checks perfectly. Six hundred on one side, six hundred on the other. Substituting back cannot tell you that you modelled it wrong. It only ever tells you that you solved what you wrote down. The sign has to be argued for before you solve, not after.

5:57Take two ordinary equations. Five x plus four equals three x plus eight. And three x minus six equals two x plus four. Solve them the usual way and you get two, and ten. Now stop looking at the letters and look at the numbers. Each equation is really four numbers. A five, a four, a three and an eight. A three, a minus six, a two and a four.

6:25And the question worth asking is not what x is. It is whether some arithmetic on those four numbers could hand you the answer directly, every time, without solving anything. So write the shape itself. A x plus B equals C x plus D. Now do exactly what you would always do. Take C x off both sides. A minus C, times x, plus B, equals D. Take B off both sides. A minus C, times x, equals D minus B.

7:01Divide both sides by A minus C. x equals D minus B, over A minus C. That is Brahmagupta's rule, and it is not a cleverer method. It is the same method, run once in general instead of every time in particular. Watch the order, though. D minus B on top. A minus C underneath. Turn either one of those round and your answer comes out with the wrong sign.

7:32Try it on something already solved. Six hundred and fifty m plus four thousand equals five hundred m plus five thousand and fifty. Top: five thousand and fifty minus four thousand is one thousand and fifty. Bottom: six hundred and fifty minus five hundred is one hundred and fifty. One thousand and fifty over one hundred and fifty is seven. Seven months, and both plans stand at eight thousand five hundred and fifty.

8:06That was one line. Written out the long way it takes four. Your turn. Two x plus three equals four x plus five. The bottom is two minus four, which is minus two. The top is two. So x is minus one. And both sides come to one. A rule you can trust is a rule that tells you when it does not apply. What happens if A and C are the same?

8:34Then A minus C is zero, and there is nothing to divide by. The rule does not give a wrong answer. It gives no answer at all. And that is honest, because the equation has no single answer to give. Two x plus three equals two x plus three is true for every number you try. Two x plus three equals two x plus five is true for none of them.

9:01Same stall, opposite reasons. And the two are told apart by B and D alone. So what was all of it for? Two things. It lets you generalise: one statement covering every case at once, instead of a list you can never finish. And it lets you justify. Take a claim like this one. Two odd numbers always add to an even number. You can test it. Three and five, eight. Seven and nine, sixteen.

9:35You could test pairs all afternoon and still not have shown it. Or write one odd number as two m plus one, and another as two n plus one. Add them: two m plus two n plus two. Twice something. Even. Always. And now you know why. That is what was inside the seed.