Zero pairs: why a positive and a negative token cancel

0:00There is an attendant in the lift, and he is bored. So he starts keeping count. Every time he presses the plus button, he drops a green token into his pocket. Every time he presses the minus button, he drops in a red one. An hour later, he empties the pocket out. Five green tokens, and three red ones, all jumbled up together. Now here is the question. Which floor is he standing on?

0:30And notice what he has to work with. The pocket knows how many presses of each kind he made. It has completely forgotten the order he made them in. That turns out to matter more than anything else in this video, so hold on to it. Let us do it the long way first, the way we already know how. Five presses of plus is a movement of plus five. Three presses of minus is a movement of minus three.

1:01So the sum is plus five, plus minus three. And on the shaft, that is up five floors from the entrance, then back down three. Up to the sports floor, then down to the art centre. He is on Floor plus two. That works, and there is nothing wrong with it. But it made us walk the whole journey. There is a faster way to read that pocket, and it is a more interesting one.

1:29Take just one green token and one red token out of the pile. The green one is one press of plus. The red one is one press of minus. Press them one after the other, and where do you end up? Up one floor, then down one floor. Exactly where you started. And it works the other way round too. Down one, then up one. Still exactly where you started.

1:56So one green and one red, taken together, are worth nothing at all. That combination has a name. It is called a zero pair. And the name is a description, not a label somebody attached to it. It is called that because it is worth zero. But be careful about what that means. The two presses still happened. The lift really did go up, and it really did come back. There was a moment when he was on a different floor.

2:28What is zero is not the presses. What is zero is their effect on where he ends up. So we will strike a pair out rather than rub it off. It happened. It just does not count. Now line the pocket up, greens on top and reds underneath, so that each red sits below a green. Five green on top. Three red below the first three. There is a zero pair. And another. And a third.

2:58Each of those three pairs is worth nothing, so each of them can go. Strike them out. You do not have to check anything before you do that. You do not have to work out whether it is safe. A pair is worth nothing, so removing it cannot change what the pile is worth. And look at what is left. Two green tokens, standing on their own, with nothing to cancel them.

3:25Two greens is two presses of plus, which is a movement of plus two. Floor plus two. Which is exactly what the long way gave us. So the pocket can be read in one step. Throw away every pair, and whatever is left over is the answer. The word for what is left is the surplus. Three pairs went, and the surplus was two greens. And that is worth saying carefully, because it sounds like a trick and it is not one. The pairs were worth nothing, so throwing them away changed nothing. What is left has to be the whole value.

4:06Try one where it goes the other way. Plus five, and minus eight. Five green on top. Eight red underneath. Now there are five zero pairs, and out they go. Three red tokens are left standing, with no greens to pair with. Three reds is three presses of minus, so the answer is minus three. Three floors below the entrance. And watch out for one thing here. There are more reds, so the answer is negative. Negative does not mean bigger. On the shaft, minus three is lower than where he started, not higher.

4:46Four to run on tokens. And two of them are strange, so we will do those first. Plus six, plus plus four. That is six green tokens and four more green tokens, and not one red among them. There is nothing to cancel. No pairs at all. So nothing gets thrown away, and the whole pile of ten greens is the answer. Plus ten. Minus three, plus minus two. Five red tokens, no greens. Again nothing cancels, and the whole pile is the answer. Minus five.

5:23Those two are not the model breaking down. They are the model telling you it looked for pairs and found none. Now the other two. Plus five, plus minus seven: five pairs go, two reds left, minus two. And minus two, plus plus six: two pairs go, four greens left, plus four. Now run it backwards. Here are two pockets, already emptied out, and the question is which floor each attendant is on.

5:55The first one. Three green tokens above five red ones. Three green and five red is the sum plus three, plus minus five. Pair them off: three pairs go, and two reds are left standing. So that attendant is on Floor minus two. Two floors below the entrance. The second pocket. Six green above three red. That is plus six, plus minus three. Three pairs go, and three greens are left.

6:26Floor plus three. And notice that a drawn pocket does not show you the answer. It shows you the question. The answer is what is left after the pairs are gone. Now the thing this was all for. Go back to five green and three red, and empty that pocket out again, in a completely different order. Red first this time. Then two greens, then a red, then a green, then the last red, then two more greens.

6:57Pair each token with the first one of the other colour that comes after it. Different tokens are pairing up now. Genuinely different pairs. Three pairs go. Two greens are left. Shuffle it again. Different order, different pairs. Three pairs go, two greens left. There are fifty six different orders that pocket could have come out in. Every single one of them gives three pairs and two greens standing. And that is not luck. It is the pocket itself.

7:32A pocket holds counts. Five of one, three of the other. It has no memory of which press came first, so nothing you read off it can possibly depend on that. Which is a statement about addition, and a strong one. It says that plus five, plus minus three, and minus three, plus plus five, must come to the same number. Not because someone told you the order does not matter, but because the pocket cannot tell the difference between them.

8:05You have probably been told that adding in either order gives the same answer. This is the first time you can see why. One last thing, and it is about how old this idea is. More than two thousand years ago, mathematicians were laying out small counting rods on a board to do arithmetic. They used two colours. Red rods for the quantities we would write with a plus, and black rods for the ones we would write with a minus.

8:37A red rod and a black rod, laid together, were worth nothing, and could be lifted off the board. The colours are the other way round from ours, which tells you something worth knowing: the colours were never the point. What matters is that there are two kinds, that one of each is worth nothing, and that what survives is the answer.