Commutative, associative, and distributive over the integers

0:00You already know these three laws. That is exactly the problem. Order does not matter. Grouping does not matter. Multiplication spreads across a sum. All three feel far too obvious to bother proving, and for whole numbers they very nearly are. But something happened recently. Multiplication got stretched. It used to mean repeated adding, defined on counting numbers, and it has just been pushed out past zero onto numbers it was never built for.

0:34So the honest question is not whether these three laws are true. It is whether they came along. A rule that held on one side of a boundary can simply stop at it. Nothing guarantees this one did not. So let us check, and let us be careful about what checking means. Start with the easiest of the three to state. Does swapping the two numbers change the answer? Here are three pairs to try.

1:05Three times minus four, against minus four times three. Minus thirty times twelve, against twelve times minus thirty. And minus fifteen times minus eight, against minus eight times minus fifteen. Work them out and the two columns agree every time. Minus twelve. Minus three hundred and sixty. One hundred and twenty. But three matching pairs is not a reason. It is three matching pairs. So let us take a product apart instead, into the two things it is actually made of.

1:44Every product is really two separate decisions. How big the answer is, and which side of zero it lands on. Take the size first, because it goes quickly. The size of the answer depends on the sizes of the two numbers you started with, and on absolutely nothing else. Minus fifteen times minus eight has size fifteen times eight. And fifteen times eight is whole number arithmetic. Old arithmetic. Nothing was stretched there at all.

2:18That already survives a swap, and for a reason you can see: eight rows of fifteen and fifteen rows of eight are the same rectangle, turned on its side. So half the question is closed before we have really started. The size cannot move. Which leaves the sign, and the sign has very few places to go. Either your two numbers have the same sign, or they have different signs. There is no third case.

2:49Same sign means both positive, or both negative, and either way the answer comes out positive. Now swap them over. Both positive is still both positive. Both negative is still both negative. Different signs means one of each, and the answer comes out negative. Swap those, and you still have one of each. Swapping cannot change how many negatives are in the room, because swapping does not remove anything. So the sign cannot move either. Two halves, and both of them shut.

3:27Which gives us the first law, written in letters. a times b equals b times a, for any two integers there are. And notice what just happened, because it matters more than the law does. We did not try some examples and then hope. We split the claim in two, showed one part was old arithmetic wearing a disguise, and showed the other part had exactly two cases, then walked through both.

3:57That is an argument. It covers numbers nobody is ever going to write down. Hold on to how that felt. Because the second law is not about to get the same treatment. Second law. This time the numbers stay exactly where they are, and the brackets move. Five, times minus three, times four. Bracket the first two together. Five times minus three is minus fifteen, and minus fifteen times four is minus sixty.

4:31Now bracket the last two instead. Minus three times four is minus twelve. Five times minus twelve is minus sixty. Same answer, different route, and nothing was reordered along the way. This is a genuinely different claim from the first one, and it is worth being clear about why. The first law moves numbers. This one moves brackets. Neither of them implies the other. Put the two together, though, and something bigger falls out.

5:04If you may reorder, and you may also regroup, then you may take a long product in absolutely any order you like. Five times four is twenty. Twenty times minus three is minus sixty. Same answer again, and this time the order and the brackets have both changed. Try it with three uglier numbers. Twenty five, minus six, twelve. There are six orders, and two ways to bracket each of them, so twelve routes altogether.

5:36Every single one arrives at minus one thousand eight hundred. That is a very useful freedom. But look carefully at how we just came by it. We did not earn it. We checked it. Twelve routes agreeing is twelve routes agreeing, and nothing more than that. For the swap we had an argument, and it covered every integer in existence. For this one we have some worked instances and a confident statement, and between those two things there is a real gap.

6:10The law is true. That is not in any doubt. But it is worth knowing which of your beliefs you hold for a reason, and which ones you hold because of examples. They are not the same kind of belief, and confusing them is how people end up certain about things that turn out to be false. So here is a case where examples would have walked you into exactly that.

6:37Take minus one, and keep multiplying by minus one. Two of them. Minus one times minus one is one. Three of them. That one, times another minus one, is minus one. Four of them, and we are back to one. Five of them, and it is minus one again. The answer flips every single time you add a factor, because each new minus one carries it across to the other side of zero.

7:07So it alternates forever, and you never have to multiply anything to know where you are on it. You only have to know whether you are holding an even number of them or an odd one. And that goes very much further than minus ones. Take any string of numbers multiplied together, and count how many of them are negative. An even count gives a positive answer, an odd count a negative one, and you never work the product out.

7:39Minus two, minus one, minus five, minus three. Four negatives, so positive, and if you insist, it is thirty. Now here is the trap, and it is the whole reason to be careful with rules that came out of examples. Change that last number to zero. Three negatives now, an odd count, so the rule promises a negative answer. It is not negative. It is zero, which is neither. The rule holds only when none of the factors is zero, and no amount of trying examples without a zero in them was ever going to tell you that.

8:20Third law, and this one comes with a picture. Four rows of tokens. Two columns of green, and three columns of red. Green counts as one each. Red counts as minus one. So the whole rectangle is four rows of two plus minus three. But you can also just count. Eight greens, twelve reds, and the net value is minus four. Now split the rectangle down the middle. The green block is four times two, which is eight.

8:53The red block is four times minus three, which is minus twelve, and eight and minus twelve give minus four. And notice what the picture cures. Both blocks are four rows deep, so the four reaches every part of the sum, and not merely the first piece of it. Three laws, then, and a single claim sitting underneath all of them. Multiplication was extended onto numbers it had never been defined for, and nothing it used to do got broken.

9:28Order survives. Grouping survives. Spreading across a sum survives. Those are statements about multiplying, by the way. They do not come free with every operation you meet. Three minus five is minus two, but five minus three is two. Subtraction is not interested in what you would prefer. And of everything here, the piece that pays off soonest is the counting rule. Because order and grouping stopped mattering, the sign of a long product stopped depending on where you begin, and collapsed into a single count of negatives.

10:08One question, asked once, instead of a whole chain of decisions. That is what a law is for.