Multiply as whole numbers, then count the decimal digits

0:00A pen costs nine point five. You buy five of them. You can get there by adding, which is what multiplying means. Nine point five. Nineteen. Twenty-eight point five. Thirty-eight. Forty-seven point five. Correct. And almost useless. Because the moment the second number stops being whole, there is nothing to stack. Ask for nine point five times seven point five, and adding stops. Seven and a half copies of something is not a pile you can build.

0:35So we need a method that does not care what it is handed. The trouble is plainly the point, so the first move is to remove it. Nine point five is nine ones and five tenths. Which is ninety-five tenths. Ninety-five over ten. That is the move, and it works on every decimal — a whole number sitting over ten, or a hundred, or a thousand. Nothing has been rounded and nothing lost. Ninety-five over ten is not close to nine point five. It is nine point five.

1:12The other factor is easier. Five is five over one. So the question has changed shape entirely. Ninety-five over ten, times five over one. And multiplying two fractions is something we can already do. Tops together, bottoms together. There is no point anywhere now, and no new arithmetic to learn. Tops together. Ninety-five times five is four hundred and seventy-five. Bottoms together. Ten times one is ten. Four hundred and seventy-five over ten.

1:51And four hundred and seventy-five tenths is forty-seven point five. The same answer the adding gave. But look at where the point went. It never travelled with the digits. It vanished at the start, came back at the end, and what put it back was the ten underneath. That is worth pausing on, because one habit fights it. When you add decimals, you line the points up first. Here we did the opposite. Both points were thrown away before a single digit was multiplied.

2:27Now try it with a point in both numbers. A car goes twelve point five kilometres on a litre, and you put in seven point five litres. Twelve point five is a hundred and twenty-five over ten. Seven point five is seventy-five over ten. Tops. A hundred and twenty-five times seventy-five is nine thousand three hundred and seventy-five. Bottoms. Ten times ten is a hundred. Nine thousand three hundred and seventy-five over a hundred, which is ninety-three point seven five.

3:05One decimal place in the first number. One in the second. Two in the answer. Hold on to that, because it is about to happen again. So line four of these up and watch only the counts. Nine point five times five gave forty-seven point five. One place, none, one place. Twelve point five times seven point five gave ninety-three point seven five. One, one, two. One point six four times six is nine point eight four. Two places, none, two places.

3:43And one more, chosen to be awkward. Five point seven times thirteen point three five is seventy-six point zero nine five. One place. Two places. Three places. Stare at that last row, because it kills the guess everybody makes — that the answer copies whichever factor has more places. It does not. One and two did not give two. They gave three. The counts add rather than compete. Adding is a strange thing for counts to do. Where does it come from?

4:19From underneath. Five point seven was fifty-seven over ten. Thirteen point three five was one thousand three hundred and thirty-five over a hundred. And the bottoms multiply. Ten times a hundred. One zero, then two zeros. They do not compete, they stand side by side. Three zeros. A thousand. And a thousand underneath means three digits after the point above. So the count of decimal places was never a rule about decimals. It counts the zeros in a denominator you never wrote down.

4:59Which makes this a receipt, not a rule. Here is the whole idea in one picture. Five hundred and ninety-six times two hundred and forty-eight is a hundred and forty-seven thousand, eight hundred and eight. That is a plain whole-number sum, with no decimals in it. Underneath it, put five point nine six times twenty-four point eight. Two places. One place. And the answer is a hundred and forty-seven point eight zero eight.

5:33Three places, which is two plus one. The digits never changed. One, four, seven, eight, zero, eight, on both lines. Only the point moved, and the count told it where to stop. So one whole-number product is worth an awful lot of decimal ones. Suppose the only fact you have is that eighteen times twelve makes two hundred and sixteen. That is the only multiplying anybody does. Eighteen times one point two. One place. Twenty-one point six.

6:10One point eight times twelve. Also one place, so also twenty-one point six. Eighteen times zero point one two. Two places. Two point one six. One point eight times one point two. Two places again, and again two point one six. Zero point one eight times zero point one two. Four places. Zero point zero two one six. And zero point zero one eight times zero point zero one two. Six places, but only three digits to fill them, so zeros get written in front. Zero point zero zero zero two one six.

6:56Now drive it once, with nothing else to lean on. Five point eight times one point two four. Strip both points. Fifty-eight times a hundred and twenty-four is seven thousand one hundred and ninety-two. Count the places. One, and two. Three. So the answer is seven point one nine two. And now check it, because a count you trust and a count you have tested are different things. Five point eight is fifty-eight over ten. One point two four is a hundred and twenty-four over a hundred. Seven thousand one hundred and ninety-two over a thousand.

7:38Seven point one nine two. The shortcut and the reason agree, which is the only reason to trust one. Now a case that looks like it should break all this. Fruit at fifty-six point five zero for a kilogram, and you buy two point two five zero kilograms. Two places and three places. The count says five. But fifty-six point five zero is the same price as fifty-six point five, and two point two five zero the same weight as two point two five.

8:15Written that way it is one and two places, and the count says three. Both counts are right. The first gives a hundred and twenty-seven point one two five zero zero, the second a hundred and twenty-seven point one two five. Identical. The spare zeros land at the tail, where nobody writes them. So here is the honest version. The count tells you how many places the digits produce, and sometimes the last of those are zeros.

8:49Two point five times zero point four counts as two places and gives one point zero zero, which everyone writes as one. The count is never too small. Just occasionally generous. One last question, and it is the one that makes all this quick. What does multiplying by ten actually do to a decimal? Five point seven times ten. That is fifty-seven over ten, times ten over one. The tens cancel, and what is left is fifty-seven.

9:26The digits did not move. The point did, one step to the right. Times a hundred moves it two steps. Times a thousand, three. So zero point three zero six times a thousand is three hundred and six. It has run out of places to give, and it is whole. And twenty-three point zero two becomes two hundred and thirty point two, then two thousand three hundred and two, then twenty-three thousand and twenty.

9:58So, the whole thing in two sentences. Multiply the digits as though the points were not there. Then count the places in both factors, add them, and set the point that many steps in from the right. And if you ever doubt it, you need not have remembered it. Write each decimal as a whole number over ten or a hundred. Multiply the tops, multiply the bottoms, read the answer off. Three lines, and the rule is back.

10:30The one place this still goes wrong is not the arithmetic. It is the units. A coin one point four five millimetres thick, thirty-six of them stacked, is fifty-two point two millimetres. Which, if centimetres were asked for, is five point two two. Everything hard was already done, and that is where answers get lost. A rule you can rebuild is a rule you cannot lose.