Commutative and associative: why order and grouping are free

0:00A drone lifts off from the middle of a flat rooftop, and we will measure everything from there. It climbs six metres, and then it comes back down four. How far above the roof is it now? Two metres. And the line you would write for that is six minus four. But we know what to do with a minus by now. We turn it into a plus. Six, plus negative four.

0:26Two terms. The six, and the negative four. One term for each move the drone made. Up six, down four. And the sign inside each term is the direction it went. That is the whole set-up. Now we can ask the interesting question. What happens if we swap those two terms round? Negative four, plus six. In drone terms that is a machine which drops four first, and then climbs six.

0:57It starts by going below the roof, which the first one never did. A genuinely different flight, with a genuinely different shape. But where does it finish? Two metres above the roof. Exactly the same place. So the two terms can be written in either order, and the total does not care. That is worth sitting with, because it is not obvious in the slightest. One drone dipped below the roof and the other never did.

1:27Only the finishing height came out the same. Now the mistake this invites, and it is a big one. If six minus four can be swapped, then surely six minus four is the same as four minus six? It is not. Six minus four is two. Four minus six is negative two. Those are different numbers. They are four apart. So what went wrong there? We swapped the wrong things. We swapped the two numbers where they sat, and left the minus sign sitting exactly where it was.

2:03The swap that works is a swap of the terms, and the second term is negative four. Not four. The sign travels with its term. Move the term, and its sign goes too. Which is why the rewriting comes first, every single time. Write it as a sum, and then shuffle. Never shuffle first. Next question, and it is the one that stops this being a slogan. We have tried it on one pair of numbers. Does it still hold when the terms are negative?

2:37Take negative seven plus negative eleven. Swap them: negative eleven plus negative seven. Negative eighteen, both ways round. Try one of each. Negative twenty plus three, against three plus negative twenty. Negative seventeen, both ways round. And here is why it holds, rather than just that it does. Think of each term as a small pile of counters. Positive counters, or negative ones. Adding the terms means pushing the piles together into one heap.

3:14It does not matter which pile you push in first. The heap is the same heap. So much for two terms. Now let us see what happens with three of them. Negative seven, plus ten, plus negative eleven. You have to start somewhere, so start by adding the first two. Negative seven and ten is three. Then three and negative eleven is negative eight. Now do it the other way. Add the last two to each other first.

3:46Ten and negative eleven is negative one. Then negative seven and negative one is negative eight. Negative eight again. But look at what did change. The number in the middle. One route went through three. The other went through negative one. Different middles, same ending. That is the thing worth noticing. There is a third route as well, and it is the one you would actually pick. Add the two negative terms to each other first.

4:19Negative seven and negative eleven is negative eighteen. Then negative eighteen and ten is negative eight. Through negative eighteen this time. And still negative eight. In fact you can try every order and every grouping of those three terms. There are twelve of them. Six orders, with two groupings each. Twelve arrangements, and one answer. So the terms of a sum can be added in whatever sequence you like. Not only moved along the line, but bracketed up any way you please, which is a second and separate freedom.

4:59Those two freedoms each have a name, and the names are worth keeping well apart from each other. Changing the order of the terms is called the commutative property of addition. Six plus negative four, or negative four plus six. Same total. Changing which terms get added to each other first is called the associative property. Bracket the first pair, or bracket the last pair. Same total. One is about where the terms sit. The other is about which of them join up first.

5:35They are easy to blur together, so here is a one-line test. If something moved along the line, that is the commutative one. If everything stayed where it was and a bracket moved instead, that is the associative one. And both of them are about a sum of terms. Neither is about a line you have not rewritten yet. Here is where this stops being theory. Somebody is adding a column of five numbers by hand.

6:06One thousand three hundred and forty two. Seven hundred and seventy four. Eight thousand six hundred and eleven. Nine thousand and fifty five. One thousand and twenty two. It takes five minutes, and the total comes out as eleven thousand seven hundred and forty nine. Then comes the horrible moment. One of them was skipped. The fourth one, nine thousand and fifty five, never went in. Does the whole thing have to be done again?

6:40No. Add nine thousand and fifty five to the total that is already there. Twenty thousand, eight hundred and four. Done. That is one addition instead of four, and it is allowed precisely because the order is free. The skipped number was sitting in the middle of the column, and it goes in at the end. That is the property, doing a job. One more thing before the end, about expressions that have a multiplication sitting inside them.

7:13Thirty plus five times four. Its terms are thirty, and five times four. Settle each term first. Thirty, and twenty. Then add them, in whichever order you fancy. Fifty either way. A longer one. Five times a bracket, three plus two, then plus seven times eight, then plus three. The terms come to twenty five, fifty six, and three. There are six orders you could add those in, and every one of them gives eighty four.

7:45So the freedom belongs to the terms, once you have settled what each one is worth. Settle, then add, in any order at all. That is the method complete. Last thing, and it is a warning about the word order. Order does not stop mattering in general. It stops mattering for the terms of a sum. Here is a test for telling the difference, and the good thing about it is that it works outside mathematics as well.

8:16Put on a hat, then your shoes. Or your shoes, then a hat. You end up looking exactly the same, because a hat and a pair of shoes go in different places. Now put on socks, then shoes. Then try that the other way round. Shoes first, socks over the top. Same two items, and a very different morning. The difference is whether the two actions touch the same place.

8:43Different places, and the order is free. The same place, and the order decides everything. The terms of a sum are separate piles that all end up in one heap, and that is exactly why they can go in any order you like.