Why a debt times a debt is a fortune

0:00Start with the case nobody argues about. You take on four debts, each of three coins. Four slips, three coins owed on each. Add them up and you are twelve coins down. A debt of three, four times over, is a debt of twelve. Nobody needs convincing of that one. Multiplying by four means doing something four times, and what you did four times was lose three coins. Check it on every size of debt and every number of them, from one to twenty, and the running total always matches the product. Four hundred cases, no exceptions. That is not surprising, and it is worth having in hand before the surprising case arrives.

0:50Now change one thing, and watch the story fall apart. A debt of three, minus four times. Read that as repetition and it is nonsense. You can do a thing four times. You cannot do a thing minus four times. There is no such action. So either the expression means nothing at all, or multiplying by a negative means something other than repeating. Those are the only two options, and the first one is not acceptable, because we already decided that an operation which sometimes refuses is not an operation.

1:31Here is the move that rescues it, and it is worth naming. Multiplying by a positive number adds copies. Multiplying by a negative number takes copies away. Repetition, and removal. Same operation, read in two directions, exactly as the two halves of the number line were the same road read in two directions. And removal is not a mathematician's invention. Debts get cancelled all the time. Somebody forgives what you owe them, and the slip is torn up.

2:08Notice this is not a definition anyone is free to reject. It is the only reading left once repetition has been ruled out, and it is a reading that already existed outside mathematics, in every ledger that ever had a debt written off. So run it. You are holding four debt slips of three coins. Somebody cancels all four. Tear up the first: you owe nine instead of twelve. The second: six. The third: three. The fourth: nothing.

2:44You started twelve down and you finished level. You are twelve coins better off. And here is the part worth being careful about. No coins arrived. Your cash in hand is exactly what it was. What changed is what you are worth, and it went up by twelve. Cancel four debts of three, and a debt of three taken minus four times, come to the same number. Twelve, and positive.

3:15That is a good argument. It is also a story, and stories can be argued with. Somebody can always say the debts are a metaphor and metaphors prove nothing. So let us do it again with no story in it. No coins, no slips, nobody forgiving anybody. Just two rules we already have, and the question of what value is left standing when both of them hold. The advantage of this route is that there is nothing in it to disagree with. If you accept the two rules, you are going to accept the conclusion, whether you find debts persuasive or not.

3:56The first rule: anything multiplied by nought is nought. Now write down something that is obviously nothing. Minus four, plus four. Those two cancel; a number and its opposite always do. So take a debt of three and multiply it by that bracket. The bracket is nought, so the whole thing is nought. No argument, no interpretation, just the rule. Whatever else happens, the answer to this is zero. The second rule: multiplying across a bracket. Multiply the sum, or multiply each piece and add. Same answer either way.

4:41So expand the left side. Minus three times minus four, plus minus three times four. The second piece we already know. A debt of three, four times, is a debt of twelve. That was the case nobody argues about. So the whole expression is: the thing we are trying to find, plus minus twelve. And we already established that the whole expression is nought. Look at what that leaves. Some number, take away twelve, is nought.

5:15Do not solve it in your head. Search it. Try every whole number from minus two hundred to two hundred: four hundred and one candidates. Exactly one survives. Twelve. That is the difference between a rule you were told and a rule you were left with. Nobody chose twelve. Four hundred candidates were eliminated and twelve was what remained standing. And notice which part was ever in doubt. Nobody wondered whether the answer might be seven, or a hundred. The size was always going to be three times four. The only question was the sign, and that is the habit worth keeping: separate the size from the sign, settle the size by ordinary multiplication, and spend your attention on the sign alone.

6:10You might suspect three and four were chosen to be convenient. So run the same elimination on every pair from one to fifteen. Two hundred and twenty five pairs, each one searched over six hundred and one candidates. Every single one comes out the same way: one survivor, and it is the two numbers multiplied together, positive. And to be sure the search is doing work rather than agreeing with us, break one of the two rules on purpose. Say that a number times nought gives back the number instead of nought. Now the survivor is nine, not twelve. Flip the sign on the easy case instead, and the survivor becomes minus twelve.

6:58The argument uses both rules, and it uses them load-bearingly. Take either one away and it gives a different answer. Which settles a question people ask and are usually fobbed off on. Is the sign rule a convention? Something mathematicians agreed to, that could have gone the other way? No. There is nothing left to agree about. Once you want multiplying by nought to give nought, and you want multiplying across a bracket to work, the value is fixed. Choosing anything else does not give you a different mathematics; it gives you a broken one.

7:40The rule is not an extra assumption bolted on. It is a consequence you cannot refuse without giving up something you wanted more. Now the mistake this whole topic exists to prevent. Two negatives make a positive. People remember that sentence and then apply it to addition. Look at the same pair, minus five and minus four, under both operations. Added, they give minus nine. Multiplied, they give twenty. It is not a near miss. Take every pair of debts from minus one to minus thirty. Nine hundred pairs. Not one of them adds to a fortune, and all nine hundred multiply to one.

8:27The two operations disagree on every single pair. There is no overlap to be confused by, which is what makes it such an expensive mistake. The sentence was never wrong. It was just never about addition. Two negatives make a positive is a statement about multiplying, and about nothing else. One last connection, and then it is finished. You have already met removal once, in a smaller form. Ten, take away a debt of five, is fifteen. One slip torn up.

9:05A debt of three, times minus four, is four slips torn up. Same move. Once, or four times. Taking away a debt and multiplying by a negative are not two rules that happen to look alike. They are one idea at two scales, and the sign rule is what that idea looks like when you write it in symbols. So when you are asked for minus twelve times five, and you write minus sixty, and for minus eight times minus seven, and you write fifty six, nothing is being remembered. Something is being read.