What happens to a product when you nudge one factor

0:00Two numbers. Twenty-three and twenty-seven. Their product is six hundred and twenty-one. Now three questions. Answer all three before reading on. If the twenty-three becomes twenty-four, how much does the product gain? If instead the twenty-seven becomes twenty-eight, how much does it gain? And if both of them go up by one, how much then? The first answer most people give to all three is one. All three of those are wrong, and wrong in different ways.

0:34And you can see every real answer without multiplying anything a second time. Draw the product instead of computing it. Twenty-three rows of dots, twenty-seven in each row. Now add one more column to the right. How many dots did you just add? One for every row. Twenty-three of them. That is the whole idea. Multiplying a sum is the same as multiplying each part and adding the results. In letters, a times the quantity b plus c is a times b, plus a times c.

1:12The picture is the proof. The rows never change. The columns split in two, and each group keeps its own rectangle. Everything here comes out of that one sentence. Take c to be one. Then a times the quantity b plus one is a times b, plus a. The increase is a. Not one. The entire extra column. So twenty-three twenty-eights is six hundred and forty-four, which is six hundred and twenty-one plus twenty-three.

1:46And twenty-four twenty-sevens is six hundred and forty-eight, which is six hundred and twenty-one plus twenty-seven. Look at what just happened. Raising one factor by one adds the other factor. That is why the two answers differ, although the two nudges look identical. Neither line needed a second multiplication. Both factors up by one. This is where guessing goes wrong, because the two increases do not simply add. Treat the quantity a plus one as a single thing, and split the other bracket.

2:26That gives a plus one, times b, plus a plus one, times one. Multiply out and it is a times b, plus b, plus a, plus one. Three extra pieces. A strip of b along one edge, a strip of a along the other, and one loose corner where they would meet. That corner is a single dot, and it is the piece everybody forgets. Here the gain is twenty-seven plus twenty-three plus one, which is fifty-one.

2:58Twenty-four twenty-eights is six hundred and seventy-two. Six hundred and twenty-one plus fifty-one. Exactly. Now one factor up by one and the other down by one. Surely those cancel. Split it the same way. A plus one, times b minus one, is a times b, plus b, minus a, minus one. So the change is b minus a minus one. Here that is three, and the product goes up, to six hundred and twenty-four.

3:32Up. Not unchanged. And the sign of that expression depends on which factor is which. Sweeping every pair between minus six and six, one hundred and sixty-nine of them, the product falls in ninety-one cases. It falls exactly when the second factor is not past the first, and it stays put only when the second is exactly one more than the first. A fair objection. All of this was drawn with dots, and you cannot draw minus five rows.

4:07So does any of it survive negative numbers? Take a as minus five and b as eight. Both up by one gives a plus b plus one, which is four. Check it. Minus five eights is minus forty. Minus four nines is minus thirty-six. The gain is four. Take a as minus four and b as minus five. The formula says minus eight, and multiplying out gives minus eight. Nothing new was needed, and nothing was extended.

4:41The letters stood for numbers all along, and splitting holds for whole numbers of either sign. It is worth being exact about what has been established. An identity is an equality between two expressions that holds whenever you replace the letters by numbers. Any numbers. Every time. That is far stronger than an equation true for one value. And here is the part that matters. Substituting three values into an identity does not establish it.

5:16Checking values can only catch a mistake. It never finishes the argument, because there is always another value you did not try. What finishes it is deriving it, and every line so far was derived from the splitting rule. The checks are how you notice you slipped, not why you believe it. So far there have been four separate cases, and that is three too many. Let one factor move by m and the other by n. Both are just numbers, so either can be negative.

5:52Split it twice. A plus m, times b plus n, is a times b, plus a times n, plus b times m, plus m times n. Four pieces, four blocks. The original rectangle, a strip a tall and n wide, a strip m tall and b wide, and a small block m by n in the corner. The increase is everything but the first block. A times n, plus b times m, plus m times n.

6:24Swept across two thousand and twenty-five combinations of factors and shifts, that matched the multiplied-out answer every time. And drop any one of those four pieces and it stops matching, which is how you know none of them is decoration. Now watch the earlier cases fall out of that one line. A plus one, times b minus one, is really a plus one, times b plus minus one. So m is one and n is minus one. The increase is a times minus one, plus b times one, plus one times minus one.

7:05Which is minus a, plus b, minus one. The same three terms as before. There is no separate rule for a decrease. A decrease is a negative shift, and the ordinary sign rules do the rest. Try all four sign combinations here, shifting by one and by two. Up and up gives seventy-five. Up and down gives minus twenty-one. Down and up gives seventeen. Down and down gives minus seventy-one. One statement, four answers.

7:41Here is a question that one statement now makes easy. Push one factor up by two and pull the other down by four. When does the product come back unchanged? Set the increase to zero. A times minus four, plus b times two, plus two times minus four, equals zero. Tidy it and b is two a plus four. That is not one answer but a family, and in the range swept here there are seven.

8:11One and six give six, and three twos give six. Two and eight give sixteen, and four fours give sixteen. The same machinery answers any such question. Down two and up three moves our product by nine. Down three and down four moves it by minus one hundred and sixty-one. One last place this turns up, and it is not where you would expect. Take any calendar month and outline any two by two square of dates.

8:44Say four, five, eleven and twelve. Multiply one pair of corners, four twelves, and you get forty-eight. Multiply the other pair, five elevens, and you get fifty-five. A gap of seven. Try another square anywhere in the month. Across twenty-one squares, every single gap is seven. The reason is the block itself. Its corners are a, a plus one, a plus seven and a plus eight, and multiplying those out cancels the a completely.

9:19What is left is the seven. Change the row width and the gap changes with it. Across nine widths, the gap was the width every time. Count what has been replaced. One factor up. The other up. Both up. One up and one down. Both down. Up by two. Down by four. Every one was a separate rule to remember. They are now one line, with the shifts allowed to be negative.

9:49And each of the four pieces has a picture attached. Two strips and a corner is not a formula to memorise. It is a shape. The corner is the piece worth watching. It vanishes when a shift is zero, and it carries the sign when both shifts are negative. So when someone asks what a nudge does to a product, the honest answer is not a number. It is a question back. By how much, and which factor is the other one?

10:20Because the answer was never about the factor you moved. It was always about the one you left alone.