An identity holds for every value; an equation need not

0:00Pick three square numbers that follow one another. One, four, nine. Add the smallest and the largest: one plus nine is ten. Subtract twice the middle one: twice four is eight. Ten minus eight leaves two. Try it further up. Nine, sixteen, twenty five. Thirty four, minus thirty two. Two. Once more. Twenty five, thirty six, forty nine. Seventy four minus seventy two. Two again. Three different starting points, and the same answer every time.

0:37So the pattern is true, then. That sentence is what this video is about, and it is worth slowing down on. What you have is three results. What a rule needs is something covering every starting point, including the ones nobody will try. Three runs report on three runs. They are silent about the fourth. The honest position, after three successes, is not that it is true. It is that it seems to be.

1:08Here is a second pattern, and this one is written in letters. The square of a plus b equals a squared, plus two a b, plus b squared. There is a picture that makes it look obvious. Draw a square whose side is a plus b, and cut it across and down. You get a square of side a, a square of side b, and two identical rectangles, each a by b. Their areas add to the whole, and that looks like a proof.

1:39But look at what the picture assumed. It drew a and b as lengths, and a length cannot be negative. So it settles the rule for the numbers it is able to draw, and says nothing about the rest. Fair enough. Go looking for the rest. Take a as minus two, and b as minus three. The left side first. a plus b is minus five, and minus five squared is twenty five.

2:08Now the right. a squared is four. b squared is nine. And two a b is two, times minus two, times minus three, which comes to plus twelve. Four plus nine plus twelve is twenty five. The two sides agree. Notice what the middle term did. Two negatives multiplied to something positive, so the term with a plus sign in front really was an addition. It does not always behave.

2:40Try harder, then. Take a as minus two thirds, and b as three quarters. a plus b is one twelfth, so the left side is one over a hundred and forty four. On the right, a squared is four ninths, b squared is nine sixteenths, and two a b is exactly minus one. Over a hundred and forty four, those are sixty four, minus a hundred and forty four, and eighty one. Which add to one. They agree again.

3:11And there is the thing to keep. The middle term came out negative. The plus sign in the written rule belongs to the notation, not to the value. So: a picture, a pair of negatives, a pair of fractions. Three checks, three agreements, and still not proved. Here is why that is not stubbornness. Take a rule that is plainly false. Say the square of a plus b equals a squared plus b squared, with no middle term at all.

3:45Now check it, but only on pairs where one of the two numbers is nought. There are a hundred and sixty one of those on the grid I swept, and the false rule survives every one. Now check the same false rule on the whole grid. Six thousand five hundred and sixty one pairs, and it fails on six thousand four hundred of them. Same rule. Different trials. Opposite verdicts.

4:13A check reports on what you checked, and nothing else. Which makes the arithmetic lopsided: no number of successes makes a rule certain, and one failure ends it outright. So what would settle it? Write the square of a plus b as what it actually means. a plus b, times a plus b. Open the first bracket against the second, every term on the left meeting every term on the right. Four products: a times a, a times b, b times a, and b times b.

4:51Two of them are the same thing, because multiplication does not care about order. That step deserves saying out loud, since it is the one everybody skips. Collect them, and four become three. a squared, two a b, b squared. Now notice what is missing from that line. There is no number anywhere in it. It never chose a value for a, so there is no value it failed to cover.

5:19That is the difference between a check and a proof. A check works on numbers. A proof works because it does not. Which leaves two kinds of statement that look identical written down. Here is one. x squared, minus one, equals twenty four. On a grid of eighty one values, exactly two satisfy it: five, and minus five. It is a question, and it has answers. Here is the other. x plus y, all squared, equals x squared plus two x y plus y squared.

5:57On the same grid, all six thousand five hundred and sixty one pairs satisfy it. Every single one. Both are written with an equals sign. Only the second is an identity. An equation asks which values fit. An identity says the question has no bite, because they all do. And an identity you have actually proved is a tool. Square forty three without multiplying it out: split it as forty plus three.

6:30Forty squared is sixteen hundred. Twice forty times three is two hundred and forty. Three squared is nine. Sixteen hundred, plus two hundred and forty, plus nine, is eighteen forty nine. Two hundred and five is as easy. Forty thousand, then two thousand, then twenty five. Forty two thousand and twenty five. And a and b need not be plain numbers. Five x plus two y, squared, is twenty five x squared, plus twenty x y, plus four y squared. The same three slots, whatever you put into them.

7:12Now a question the identity answers at once. Which is bigger: the square of a sum, or the sum of the two squares? Subtract one from the other, and everything cancels except the middle. The difference is exactly two a b. So the question was never about squares. It is about the sign of a times b. If a and b share a sign, two a b is positive, and the square of the sum is bigger. If their signs differ, it is smaller. And if either of them is nought, the two are equal.

7:48On that grid: three thousand two hundred pairs on the greater side, three thousand two hundred on the smaller, and a hundred and sixty one tied. Not one disagreed with the sign of a b. The sharpest case is one and minus one. The square of their sum is nought. The sum of their squares is two. Which is worth dwelling on, for what it says about the pairs from earlier.

8:16Suppose you had guessed, from the first of them, that the square of a sum is always bigger. Minus two and minus three gave twenty five against thirteen. The guess held. Then came the fractions. Minus two thirds and three quarters. The square of their sum is one over a hundred and forty four. The sum of their squares is a hundred and forty five over a hundred and forty four.

8:45So the guess was already dead. The pair that killed it was on the table before anybody had thought to ask the question. That is what a counterexample looks like, and it is why one of them is enough. Which leaves the trick we started with. Write three consecutive whole numbers as n minus one, n, and n plus one. Expand the outer two squares and add them. The squared terms give two n squared. The minus two n and the plus two n cancel exactly. What is left is two n squared, plus two.

9:25Subtract twice the middle square, which is two n squared, and every trace of n is gone. Two. And that two is not a leftover from n. It is the two isolated ones, one from each outer square. n never had a chance to appear. Two thousand runs all came out two, and that was never the reason. This is the reason. Try four consecutive squares instead, outer pair less inner pair, and the same machinery hands you four: nine, minus one, minus four.

10:02Once you can see where a number comes from, you stop needing to check that it is there.