The laws addition and scaling obey, and what they buy you

0:00Here is an equation. Twice a known arrangement, plus three times an unknown one, equals five times a second known arrangement. You already know how to add two arrangements and how to multiply one by a number. Neither of those tells you whether you are allowed to take twice the first arrangement across to the other side. That is what a law is for. Not the sentence it states, which usually sounds too obvious to bother with. The move it licenses.

0:34There are six of them here. Four are about adding and two are about multiplying by a number, and every one of the six is settled by the same three-step move, done six times. So the plan is: learn the move once. Then watch the six laws pay for every step of that equation, one law per line. Take two arrangements of the same order and add them the two ways round.

1:03On the left, the first plus the second. On the right, the second plus the first. The claim is that these two are the same arrangement. Now here is the move. Do not look at the whole thing. Look at one position, and it does not matter which, so pick the one in row two, column one. In that position, the left side holds one entry plus another. In the same position, the right side holds the second entry plus the first.

1:35Those are two ordinary numbers added the two ways round, and ordinary numbers do not care. So the two sides agree in that position. The position was arbitrary. Nothing in that argument used which position it was. So the two sides agree in every position, and two arrangements of the same order that agree in every position are the same arrangement. That is the whole proof. Three steps: drop to a position, use the fact about numbers there, come back up.

2:12It is worth saying plainly what just happened, because it is going to happen five more times. A statement about whole arrangements was settled by a statement about single entries. The arrangement contributed nothing to the argument except its shape. That is why the law about adding carries the same condition that adding itself carries. If the two orders differ there is no sum, and if there is no sum there is nothing for a law to be true of.

2:45Put a number on that. Build every arrangement you can from the entries minus two, minus one, zero, one and two, at the four smallest orders. That is six hundred and eighty arrangements. Pair each with each and you have four hundred and sixty-two thousand, four hundred ordered pairs. Of those, three hundred and ninety-one thousand nine hundred have matching orders, and the order of adding never mattered in a single one of them.

3:17The other seventy thousand five hundred pairs are not counterexamples. They are pairs the question was never asked about. A law is exactly as wide as the operation it is about, and no wider. Three arrangements now, all the same order. Add the first two, then add the third. Or add the last two first, and put the result on the end of the first. Same move. Drop to one position. On the left that position holds one entry plus a second, and then a third added on. On the right it holds the first entry, with the second and third added together first.

4:02And ordinary addition regroups. That is the only fact used, and it is the fact the whole line rests on, so it is worth saying out loud rather than sliding past it. Come back up, and the two groupings give the same arrangement. Fifty arrangements at the two smallest rectangular orders make one hundred and twenty-five thousand ordered triples. Thirty-one thousand two hundred and fifty of those have all three orders alike, and in every one of them the grouping made no difference.

4:39Here is a test of whether that argument was doing any work, or whether it was decoration on something that was going to be true anyway. Keep the arrangements. Change what the entries are and what adding them means. Let an entry be a word, and let adding two words mean writing the second one after the first. A followed by B is A B. B followed by A is B A. Those are different words.

5:09Now build the same population: every arrangement of words at the same four orders, from a four-word stock. That is two hundred and ninety-two arrangements. Of the sixty-six thousand and sixty-four pairs that have matching orders, ten thousand two hundred and ten still come out the same either way round. Fifty-five thousand eight hundred and fifty-four do not. So the law about order is false here. Not almost false. False in most cases.

5:43Change one thing more. Let an entry be a number again, but let adding two of them mean taking the midpoint between them. The midpoint of A and B is the midpoint of B and A, so the order law survives. But group three of them one way and then the other and you get different answers: of the eight thousand one hundred and ninety-two triples with matching orders, only five hundred and twelve regroup safely. Seven thousand six hundred and eighty do not.

6:17Two carriers, two different failures. Joining reorders badly and regroups fine. The midpoint reorders fine and regroups badly. So the two laws really are two laws, and each one is exactly as true as the fact about entries underneath it. That is what the three-step move was buying. Third law. There is an arrangement that changes nothing. Every entry zero. Add it to something and drop to one position: an entry plus zero is that entry. Come back up, and the arrangement is unchanged.

6:56Two things about it are easy to lose. The first is that it is not one object. There is a zero arrangement at every order, and the one that changes nothing is the one whose order matches. Across the four orders in our population that is four different zero arrangements, and offering the wrong one gets you no answer at all, not an unchanged one. The second is that it has to work from both sides. Added on the right it must leave the arrangement alone, and added on the left it must too.

7:33That sounds like fussiness until you find a rule where it fails. Suppose adding two entries just means keeping the first one and discarding the second. Then anything at all leaves an arrangement unchanged when you add it on the right, because the right-hand entries are thrown away. Add on the left instead and the arrangement is destroyed. Over that rule, of one hundred and two arrangements only four survive the test from both sides, and those four are the ones that were already all zeros. Ninety-eight fail. A one-sided identity is not an identity.

8:18Fourth law. Every arrangement has one that cancels it. Take an arrangement and negate every entry. Add the two and drop to a position: an entry plus its negative is zero. Come back up and you have the zero arrangement of that order. Notice what is different about these last two laws. The first two were proved. These two are really about something existing. Here is the sharpest way to see that. Keep ordinary addition, which reorders and regroups perfectly well, but restrict the entries to positive numbers only.

8:58Both of the first two laws still hold. But now there is no zero entry to build a zero arrangement out of, and no negative entry to cancel with. The question cannot even be put. So the six laws divide two and four, not by difficulty but by kind. Two of them say the operation behaves. Four of them, once the zero and the negative exist, need nothing further said about them at all.

9:30Now the two laws about multiplying by a number. First: take a number, and an arrangement plus another arrangement. Multiply the sum by the number, or multiply each of them and then add. The claim is that these agree. Same move, and this is the fifth time. Drop to one position. The left holds the number times the sum of two entries. The right holds the number times the first, plus the number times the second. Ordinary multiplication spreads across a sum, so they agree. Come back up.

10:09Across six numbers and every pair of the fifty small arrangements, that is seven thousand five hundred cases where both sides exist, and it held in every one. The other seven thousand five hundred are order mismatches again, where there is no sum to multiply. Second: two numbers added together, multiplying one arrangement. Or each number multiplying it separately, and the two results added. Drop to a position, and it is the same ordinary fact read the other way round. Come back up.

10:48Thirty-six pairs of numbers against fifty arrangements is one thousand eight hundred cases, with no order condition anywhere, because multiplying by a number never refuses. It held in all one thousand eight hundred. Are these two laws the same law twice? No, and here is the evidence. Invent a wrong rule. Instead of multiplying an entry by the number, multiply it by the number squared. Run the first law on that rule and it passes everywhere: not one failure in seven thousand five hundred. Run the second law on the same rule and it fails in one thousand two hundred cases out of one thousand eight hundred.

11:35One rule, obeying one law and breaking the other. So they are two laws. And a second wrong rule, adding the number to each entry instead of multiplying, breaks both: six thousand two hundred and fifty failures on the first, one thousand seven hundred and twenty-eight on the second. Back to the equation from the start, and now every step has a name. Twice a three-by-two arrangement, plus three times an unknown, equals five times a second three-by-two arrangement.

12:12The first arrangement holds eight and zero, four and minus two, three and six. The second holds two and minus two, four and two, minus five and one. Step one: add the negative of twice the first arrangement to both sides. That is allowed because that negative exists, which is the fourth law. Step two: on the left you now have twice the first, plus three times the unknown, plus that negative. Regroup it so the two outer terms come together, and reorder so they are adjacent. Grouping and order, the second law and the first.

12:53Step three: twice the first, plus its negative, is the zero arrangement. The fourth law again, this time doing the work. Step four: the zero arrangement plus three times the unknown is three times the unknown. The third law. Four laws, four steps, and nothing else has happened. The right-hand side is arithmetic. Five times the second arrangement is ten and minus ten, twenty and ten, minus twenty-five and five. Twice the first is sixteen and zero, eight and minus four, six and twelve.

13:33Subtract, and three times the unknown is minus six and minus ten, twelve and fourteen, minus thirty-one and minus seven. Multiply by a third. The unknown arrangement is minus two and minus ten thirds, four and fourteen thirds, minus thirty-one thirds and minus seven thirds. Four of its six entries are not whole numbers, which is worth noticing: nothing anywhere said the answer had to be tidy. Two unknown arrangements now, and two equations relating them.

14:12Their sum is five and two above zero and nine. Their difference is three and six above zero and minus one. Add the two equations. On the left, the second unknown and its negative meet and cancel, which is the fourth law visible in one step, and twice the first unknown is left. On the right, eight and eight above zero and eight. So the first unknown is four and four above zero and four.

14:42Subtract instead and twice the second unknown is two and minus four above zero and ten, so the second is one and minus two above zero and five. That is elimination, exactly as you would do it with two ordinary equations in two ordinary unknowns, and the six laws are the reason it is legal here. Which becomes obvious the moment the coefficients are not one. Twice the first unknown plus three times the second is two and three above four and zero. Three times the first plus twice the second is two and minus two above minus one and five.

15:27Nothing cancels by itself now. You have to scale both equations first, and every one of those scalings is licensed by one of the two laws about multiplying by a number. The answers come out in fifths. The first unknown is two fifths and minus twelve fifths above minus eleven fifths and three. The second is two fifths and thirteen fifths above fourteen fifths and minus two. Six of those eight entries are not whole numbers. Substitute them back into both equations and both come out right, which is the only reason to believe any of it.

16:11So far the unknowns have been whole arrangements. Now let them be single numbers sitting inside one. Twice an arrangement holding an unknown, a five, a seven and another unknown less three, plus an arrangement of plain numbers, equals a third arrangement of plain numbers. The laws build the left-hand side: multiply by two, then add. And then equality takes it apart again, position by position. Four positions, so four equations. Two of them mention no unknown at all, and both are already true, which is worth checking rather than assuming.

16:54The other two say twice the first unknown plus three is seven, and twice the second unknown minus four is fourteen. So the unknowns are two and nine. That pattern covers a great many questions. One arrangement scaled, another added, read off position by position and solved. It also covers the harder case where the same unknown appears on both sides of the equals sign. Three times an arrangement of four unknowns equals an arrangement holding one of those unknowns, a six, a minus one and twice another, added to a fourth arrangement holding a four, a sum of two unknowns, another sum, and a three.

17:42Every one of the four positions has an unknown on both sides. Collect them and the four come out two, four, one and three. One case with the numbers meaning something. Two market stalls, three varieties of apple, sales recorded for two months. Rows are the stalls, columns are the varieties, and both months use the same arrangement, which is what makes any of the following possible. The first month: ten thousand, twenty thousand and thirty thousand for the first stall, fifty thousand, thirty thousand and ten thousand for the second.

18:25The second month: five thousand, ten thousand and six thousand, then twenty thousand, ten thousand and ten thousand. Combined sales are the sum. Fifteen thousand, thirty thousand and thirty-six thousand above seventy thousand, forty thousand and twenty thousand. Two hundred and eleven thousand in all. The drop from the first month to the second is the difference. Five thousand, ten thousand and twenty-four thousand above thirty thousand, twenty thousand and zero.

19:02Every entry of that difference is positive or zero, so the second month was never better than the first in any variety at either stall. Now the zero. It is not a gap and it is not missing data. It says the second stall sold exactly ten thousand of the third variety in both months. Zero is a value, and in a difference it is one of the more informative ones.

19:30Two per cent of the second month is multiplying by a number, so it asks nothing about order and always exists: one hundred, two hundred and one hundred and twenty above four hundred, two hundred and two hundred. Six laws, and one move that settled all six: drop to one position, use the fact about entries there, come back up. The move is the part worth keeping. You will use it again. It is how the statements about turning an arrangement on its side get proved, and how the ones about arrangements that equal their own reflection get proved after that. Anyone who skipped it here has to invent it there.

20:17Each of the four laws about adding carries the same order condition that adding carries. Neither of the two about multiplying by a number carries any condition at all, because that operation never refuses. But two of the six divide off from the rest, and not by difficulty. Before you can settle whether the zero arrangement changes nothing, there has to be a zero arrangement; before you can settle whether a negative cancels, there has to be a negative. Restrict the entries to positive numbers and the first two laws survive untouched while the other two cannot even be asked.

21:00And that is what the six were for. Not to be recited. To be the reason you were allowed to move a matrix across an equals sign, cancel it against its negative, and drop the zero that was left.