Why −(−a) = a Is Secretly a Theorem

Everybody knows this one. Turn around, then turn around again, and you are facing the way you started. Flip a coin twice and it shows the same face. Cancel somebody's debt, and you have handed them money. Two minus signs make a plus. So why did one short question about it turn into a twelve hour argument, on a forum full of people learning mathematics? An argument that ended only when one of the two people in it wrote, you are right, my apologies. The two sides went like this. One of them said the fact holds in any system where you can add and cancel, so there is nothing to prove. The other said that the proof on offer assumed the very thing it was trying to prove. You cannot get a theorem by writing a symbol down in a suggestive way. They were both pointing at something real. There is a small, genuine theorem hiding inside this obvious fact, and the argument was about where it lives. To find it we have to work only from the rules. The person who asked was reading the first chapter of Spivak's Calculus, which opens by writing down the rules that numbers obey. From here on we are allowed exactly what the rules say and nothing else. No obviously. No number line. Rule one: grouping does not matter. If you are adding three things, you may bracket the first pair or the last pair, and you get the same answer. Rule two: there is a number called zero, and adding it does nothing at all. And rule three, the one the whole argument is about. For every number a, there is a number, written minus a, that cancels it. a plus minus a is zero, and minus a plus a is zero as well. Now read that third rule again, slowly, in words. Four of those words are going to carry this entire video. There is a number. It says there is a number. It never says there is only one. And notice one last thing before we go on. The fourth rule on that list, that a plus b equals b plus a, we are never going to use.

Three rules and a claim. Before we argue about the claim, we had better be sure what its symbols mean, and the minus sign is where people slip. In this rulebook the minus sign is not a property of a number. It is not negativeness. It is a job title. Minus a means whichever number holds the job of cancelling a. Rule three hands us that number and says what the job is: a plus minus a is zero. Let me draw it. Here is a, here is its canceler, and I will join two numbers with a line whenever they add to zero. If a is five, the canceler is minus five. If a is minus three, the canceler is three, so a symbol with a minus in front can perfectly well be a positive number. And zero cancels itself. Now read the thing we are trying to prove from the inside out. Minus a is the canceler of a. So minus minus a is the canceler of the canceler of a. I am keeping that one grey, with a question mark on it, because at this point in the story we do not know which number it is. Here is where the thread got tangled, and it is a good tangle. Does a cancel minus a? Yes, at once. Rule three says minus a plus a is zero, and that is exactly the job. So a does it, for free. So are we done? Not quite, and the difference is a single word. What we have shown is that a is a canceler of minus a. What the symbol names is the canceler of minus a. Look at rule three once more. It promised that a canceler exists. It never promised there is only one. If minus a had two different cancelers, the symbol would be ambiguous. It might be a. It might be the grey one. Two minuses making a plus would be a coin toss. So here is the theorem underneath the claim. Every number has exactly one canceler. Once you have that, minus minus a equals a is only that theorem, read out at the number minus a. And you might think exactly one canceler is obviously true. It is not.

To see that exactly one canceler is not obvious, let me build a number system where it is false. Four numbers: zero, a, b and c. Its addition is given by a table, the same kind you learned your sums from, only smaller. Check rule two. The zero row and the zero column hand everything straight back, unchanged. Zero does nothing here. Check rule three. Every number here has a canceler. a and b add to zero. c and b add to zero. And zero cancels itself. The table is even symmetric, so this addition commutes too. Now look at the row for b. It has two zeros in it. b is cancelled by a, and b is also cancelled by c. In this world, b has two cancelers. So compute the canceler of the canceler of a. The row for a has exactly one zero, in the b column, so minus a is b. Now the row for b has two zeros, so the canceler of b is a, or it is c. Take c. In this little world, minus minus a is c, and c is not a. Which rule did we break? Not rule two. Not rule three. And commutativity is fine as well. Look at rule one. Add a and b first, then add c: a plus b is zero, and zero plus c is c. Now bracket the other pair: b plus c is zero, and a plus zero is a. And look at the two values we just got. The first was c. The second was a. Those are exactly the two numbers that were fighting over the title canceler of b. That is not a coincidence. It is our theorem, seen from the wrong side.

So what does our number system have that the toy one does not? One thing. The bracket can slide. Here is what that buys us. Suppose two numbers, a and c, both cancel the same number b. So a plus b is zero, and b plus c is zero. I am assuming nothing else about them. Must a and c be the same number? Write all three of them in a row: a plus b plus c. Rule one says this row has one value, no matter how I bracket it. So let us bracket it two different ways and compare. Bracket the left pair first. a and b cancel, so that pair is zero. Zero plus c is c. The whole row is worth c. Now the one motion to remember from this whole video. Watch the bracket leave the left pair, and land on the right pair instead. Now the same thing happens at the other end. b and c cancel, so that pair is zero. a plus zero is a. The row is worth a. Same row. One value. So a equals c, and that is the proof. Any two cancelers of the same number are equal. Cancelers are unique. There is nothing clever in that argument. a and c only had to be in the same room as b, and rule one is what puts them there. In the toy world the bracket cannot slide, so a and c never have to meet, and nothing makes them equal. That is the only difference between the two worlds. Let us write that down as a theorem, because we are about to use it twice. If a cancels b, and c also cancels b, then a and c are the same number. Rule three says a canceler exists. Rule one says there is only ever one.

Before we cash it in, let us fix what the symbol is allowed to mean. Minus minus a is a canceler of minus a, any one of them. We are not assuming there is only one of them, because that is the thing we are proving. Now rule three gives us two promises. First: a has a canceler, and the two of them add to zero. Second: minus a has a canceler too, and minus minus a is the name of one. Write all three in a row, exactly as before: a, then the canceler of a, then the canceler of that. Rule one says this row has one value however I bracket it. Bracket the left pair. a and its canceler make zero, and zero plus the last slot leaves the last slot standing. So the row is worth minus minus a. Now slide it. The bracket leaves the left pair and lands on the right pair instead. And now the other two cancel, by the promise we just wrote down. a plus zero is a. So the row is worth a. Same row, one value. So minus minus a equals a. That is the theorem we came for, and it is now proved. Written as a single chain, the whole proof is five lines. Zero does nothing, so the thing on the left is zero plus itself. That is rule two. That zero is a plus the canceler of a: rule three. And now the only interesting line in the proof. Slide the bracket. Rule one. The middle and the right cancel, by rule three again, leaving a plus zero. And a plus zero is a, by rule two. Done. Now notice what never happened in that chain. We never swapped two terms. The fourth rule, that a plus b equals b plus a, was never used. So this is not really a fact about numbers. It holds anywhere there is a zero, an undo for every move, and a bracket that slides. Rotating a cube. Shuffling a deck. The undo of the undo is the move you started with. So back to the argument. Who was right? Both of them, describing one theorem from opposite ends. The person who asked had the shape of it exactly right. a cancels minus a. Cancelers are unique. Therefore a is the canceler of minus a, and minus minus a is only its name. The objection was right too: you cannot get that middle line by notation. Which is exactly why the small proof earns its keep. It proves the middle line on the spot, with one slide of the bracket. The objector saw that, and said so. That is a good thread.

One more question from the thread, and it is a good one. Does everything we have just done prove that a negative times a negative is a positive? No. And seeing why is worth a minute. Think of the rules as a building. Everything today happened on the ground floor: addition, zero, cancelers, and the bracket that slides. On this floor there is no multiplication at all, and no such thing as a positive or a negative number. Minus a is only the canceler of a, and it might be five, or minus five, or zero. To even say negative times negative you need the next floor up: multiplication, and the rule that ties it to addition. Watch what happens there. Add a to minus one times a, and rewrite the first a as one times a. Pull the a out. One plus minus one is zero, and zero times anything is zero. So minus one times a cancels a. And because cancelers are unique, minus one times a is minus a. That is our engine, borrowed. From there, minus a times minus b works out to a times b, and that step uses today's theorem. But you still have not said the word positive. For that you need a third floor, the rules for which numbers count as positive, and only there does the sentence mean anything. Today's theorem is one brick in that wall. A load bearing one. But it is not the wall. So here is the picture to keep. Draw a line between two numbers whenever they add to zero. Rule three says every number gets at least one such line. Rule one says no number gets two. So the lines make a perfect pairing, and zero is its own partner. Minus minus a equals a says exactly this: go to your partner, then go to your partner's partner, and you are home. Turn around twice: true. Why it is true: one slide of a bracket.