Fast mental multiplication, powered by distribution

0:00Here is a multiplication, and a condition attached to it. Three thousand eight hundred and seventy-four, times eleven. The condition is that you write the answer straight down, in one line, without setting out a column sum underneath. Most people meet this as a trick, learn the trick, and never find out why it works. It is not a trick, and there is nothing here to memorise. The answer, since you will want it, is forty-two thousand six hundred and fourteen.

0:33By the end of this you will be able to write that down as fast as you can say it, and say why it is right. And the same single idea will handle a hundred and one, a thousand and one, ninety-nine and nine hundred and ninety-nine. Start by refusing to treat eleven as one thing. Eleven is ten plus one. So the product is the number, times ten plus one.

1:02Multiplying by a sum means multiplying by each part and adding the results. Times ten gives thirty-eight thousand seven hundred and forty. Times one gives the number straight back. Add those two and you have the answer, and nothing has been skipped or approximated. That is the whole method. Everything that follows is about doing that addition without writing it out. You are always allowed to split the multiplier. Splitting it into ten and one is what earns the speed.

1:38Multiplying by ten does not really multiply anything. It slides the digits one place to the left and puts a zero underneath. Three, eight, seven and four are still there, in the same order, standing one place further along. So the two things you have to add are the same digits, in different places. Split eleven as six plus five instead, and you have two genuine multiplications and a harder problem than you started with.

2:11A power of ten is the only split that turns a multiplication into a slide. Nothing about eleven is special. What is special is being one place away from a power of ten. Write the two copies one above the other, the shifted one on top. Three, eight, seven, four, zero. And underneath, pushed one place to the right, three, eight, seven, four. That zero is not a digit of the answer. It is the units place of the number after it slid.

2:45Now stop reading across the rows and read down the columns instead. The units column holds four, and nothing else, because there is nothing beneath it to pair with. The next column holds four and seven: the units digit and the tens digit of the original, side by side. Every column in the middle holds two digits of the original that were next-door neighbours. Do it once with letters and it stops being about this particular number.

3:20Call the four digits, reading from the left, d, c, b and a. The top row is d, c, b, a, zero. The bottom row, pushed one place right, is d, c, b, a. Read the column totals from the left: d, then c plus d, then b plus c, then a plus b, then a. The two outer ones are copies. The leading digit of the answer is the leading digit of the number, and the units digit is the units digit.

3:51They are copies because those two columns have nothing to pair with. Everything in between is the total of two neighbours. Which gives the rule everybody carries away: the first digit, then the neighbour totals, then the last digit. Try it here. Three and eight make eleven. Eleven is not a digit. It will not fit into one column of the answer. The other two pairs are no better. Eight and seven make fifteen, and seven and four make eleven again.

4:26So that rule is not wrong about where the answer comes from. It is silent about what happens when a column overflows. Sometimes nothing overflows and it works perfectly. Three thousand two hundred and forty-one times eleven is thirty-five thousand six hundred and fifty-one, straight off the neighbours. Across every number from ten to nine hundred and ninety-nine it is right three hundred and seventy-five times out of nine hundred and ninety, and the smallest one it gets wrong is nineteen.

5:03The repair is to do the addition column by column, from the right, and carry. Units column: four. Write four. Next: four and seven make eleven. Write the one and carry the one. Next: seven and eight make fifteen, and the carried one makes sixteen. Write six, carry one. Next: eight and three make eleven, and the carried one makes twelve. Write two, carry one. Last: three, and the carried one, is four. Three of the five columns handed something on.

5:38The running answer goes four, fourteen, six hundred and fourteen, two thousand six hundred and fourteen, forty-two thousand six hundred and fourteen. Now state it for a number of any length, because a rule that mentions four digits is not the rule. Write the units digit of the number as the units digit of the answer. Then work leftwards, writing each neighbour total in turn, and carry whenever a total reaches ten.

6:10When the pairs run out, write the leading digit plus whatever is still being carried. That is all of it, and it does not care how long the number is. Ninety-four times eleven is one thousand and thirty-four. Four hundred and ninety-five times eleven is five thousand four hundred and forty-five. A seven-digit number defeats anyone who memorised a four-digit shape, and this rule does not notice the difference. A hundred and one is one past a hundred, in exactly the way eleven is one past ten.

6:50So the number times a hundred and one is the number slid two places, plus the number. Stack those, and the gap between the copies is two places instead of one. Now the column totals read d, then c, then b plus d, then a plus c, then b, then a. The pairs are no longer neighbours. They stand two apart, because the copies stand two apart. And four digits survive untouched at the ends instead of two.

7:21Eighty-nine times a hundred and one is eight thousand nine hundred and eighty-nine, which is the number written twice. Push the gap out again and nothing changes except the gap. A thousand and one slides the copy three places, so the pairs stand three apart. Ten thousand and one slides it four. Every extra place leaves two more digits of the answer untouched at the ends. One thousand one hundred and eleven times a thousand and one is one million, one hundred and twelve thousand, one hundred and eleven.

8:01Two hundred and sixty-five thousand eight hundred and thirty-one times a thousand and one is two hundred and sixty-six million, ninety-six thousand, eight hundred and thirty-one. Which you can very nearly read straight off: the number, then the number again, with the overlap tidied up in the middle. Now go the other way. Ninety-nine is a hundred, less one. So the second copy is taken away instead of added, and the slide is unchanged.

8:36Nine thousand seven hundred and thirty-four times ninety-nine is nine hundred and seventy-three thousand four hundred, less nine thousand seven hundred and thirty-four. Nine hundred and sixty-three thousand six hundred and sixty-six. Nine hundred and ninety-nine is a thousand less one, and behaves the same way with a wider slide. Twenty-three thousand four hundred and seventy-eight times it is twenty-three million, four hundred and fifty-four thousand, five hundred and twenty-two.

9:15So there are not five methods here. There is one move. Write the multiplier as a power of ten, give or take one. Nine, eleven, ninety-nine, a hundred and one, nine hundred and ninety-nine, a thousand and one: the same move each time, with a different slide and a different sign. Methods built this way were worked on at length by Brahmagupta, in six hundred and twenty-eight. By Sridharacharya, around seven hundred and fifty. And by Bhaskaracharya, around eleven fifty.

9:52Brahmagupta's name for them was ishta-gunana, which means multiplication by a chosen amount. Chosen is the important word. You are picking the split that makes the work easy. Multiplying a sum is the same as multiplying each part and adding. Choosing the parts well is where the speed lives.