Predicting the expansion from the denominator's prime factors

0:00Nought point three seven five. We write it as though it were a different kind of object from a fraction. It is not. Nought point three seven five is three hundred and seventy five thousandths. A fraction, with a thousand underneath. Every terminating decimal is a fraction whose denominator is a power of ten, and nothing else. Which turns a vague question into a sharp one. Asking whether a fraction's decimal stops is asking whether that fraction can be rewritten with a power of ten underneath.

0:39And you can answer that by looking at the denominator. No dividing at all. So look at what a power of ten is actually made of. Ten is two times five. A hundred is two squared times five squared. A thousand is two cubed times five cubed. There is a pattern, and it is a strict one. Every power of ten is some number of twos multiplied by the same number of fives. That is the whole ingredient list. No threes ever appear. No sevens.

1:15That was checked all the way to the fortieth power of ten. Not one of them carries a prime other than two or five, because there is nowhere for one to come from. Now take three twentieths and try to force it into that shape. Twenty is two times two times five. Two twos, one five. Compared with a power of ten, which wants them in equal numbers, this one is short of a five.

1:45So supply it. Multiply top and bottom by five. The bottom becomes a hundred, the top becomes fifteen, and the fraction has not changed at all, because multiplying top and bottom by the same thing never does. Fifteen hundredths. Nought point one five. The decimal stopped, and it stopped because the denominator could be topped up to a power of ten. That is the whole criterion, and it is worth saying slowly.

2:19Put the fraction in lowest terms. Factorise the denominator. If every prime you find is a two or a five, the decimal stops. If anything else turns up, it repeats. This was tested against the actual division. Every fraction with a denominator below four hundred, which is seventy nine thousand four hundred and one of them, predicted by the primes and then divided out by hand to see. Seventy nine thousand four hundred and one agreements. Zero disagreements. Three thousand six hundred and eighty one of them stopped and seventy five thousand seven hundred and twenty repeated, so the rule had both kinds to get wrong, and it got neither wrong.

3:11The half of this that usually gets skipped is why any other prime is fatal. It is the more interesting half. Suppose the reduced denominator has a three in it. To reach a power of ten, you would have to multiply it by something that cancels that three away. But multiplying only ever adds factors. It cannot remove one. So the three would still be sitting there in the answer, and a power of ten has no three in it. There is nowhere for it to go.

3:45Every denominator below five hundred was checked against every power of ten up to the fortieth. The ones carrying an outside prime reach none, not a single one. The ones built only from twos and fives all reach one. Watch that happen to a real fraction. Four fifteenths. Fifteen is three times five. The five is welcome. The three is the problem, and reducing will not help, because four and fifteen share no factor. It is already in lowest terms.

4:23So the prediction is: this one repeats. Now divide, and see whether the prediction was worth anything. Nought point two, then six, six, six, and it never gets away from that six. One digit before the block, and a block one digit long. The three was enough on its own. Here is the part the criterion is usually not asked for, and it comes free. Once you know a fraction terminates, its denominator is some twos and some fives. Topping up the short side takes it to ten to the power of whichever count was larger. And that exponent is exactly the number of decimal places.

5:09Thirteen two hundred and fiftieths. Two hundred and fifty is one two and three fives. The larger count is three, so the prediction is three places, before any dividing. Top up the twos. Multiply by four, and two hundred and fifty becomes a thousand and thirteen becomes fifty two. Nought point nought five two. Three places, exactly as promised. It is easy to misread which number does the work here, so take one more.

5:44Eighteen over a hundred and twenty five. A hundred and twenty five is five cubed. No twos at all, so the short side is the twos, and you top them up by multiplying by eight. A thousand underneath, a hundred and forty four on top. Nought point one four four. Three places, and three was the larger exponent. Notice what it is not. It is not how many prime factors there are, and it is not how many different ones. Forty has four prime factors below the line, two of them different, and gives three places. The exponent is the number that matters.

6:28Now the condition everybody nods past. In lowest terms. Three sixths. Factorise the six as it is written and you find a three, so the rule as stated says: repeats. Divide it, and you get nought point five and it stops dead. Because three sixths was never really about six. It is one half, and the three was cancelling anyway. The rule was not wrong. It was asked about the wrong denominator.

7:00Below two hundred there are six hundred and twenty nine fractions that the unreduced test gets wrong, and the very first one is that three sixths. Reduce first and it gets all six hundred and twenty nine right. Which means something slightly uncomfortable, and it is worth sitting with. Six fifteenths and four fifteenths have the same number underneath. One of them terminates and the other does not. Six fifteenths is really two fifths, and the three vanished in the reducing. Four fifteenths keeps its three, because four brings nothing to cancel it with.

7:43So the decimal is not a property of the denominator you happen to have written down. It is a property of the number. Reducing is not tidying up before the real work. It is the first step of the real work. One last question, and it is the kind you can now answer without dividing anything. Which fractions take exactly four decimal places? Four places means ten to the fourth underneath, so the larger exponent has to be four. The twos reach four, or the fives do.

8:20One sixteenth is nought point nought six two five, and sixteen is two to the fourth. One six hundred and twenty fifth is nought point nought nought one six, and six twenty five is five to the fourth. But three eightieths is also four places, and eighty is neither. Eighty is two to the fourth times five, and it is the twos alone that get it there. The answer is an or, not an and.

8:50Four and a half million fractions were searched for one that takes four places without either condition holding. There is not one. So here is what you can now do to a fraction you have never seen. Reduce it. Factorise the bottom. If anything but twos and fives shows up, it repeats, and you are done. Otherwise it stops, in as many places as the larger of the two counts.

9:20Six of them, side by side. Eight, twenty, a hundred and twenty five, two hundred and fifty and forty all stop. Fifteen does not. Four of the six take three places. Three thousand six hundred and eighty one terminating fractions had their place count predicted this way and then measured against the division. The rule got none of them wrong. And notice what has really happened. The division was never the source of the answer. The answer was sitting in the denominator the whole time, and dividing was only ever a way of asking it out loud.