Converting a terminating or repeating decimal back to p/q

0:00Nought point three five. Turning that into a fraction takes no cleverness. It is thirty five hundredths. Thirty five over a hundred, and then you tidy it up. Both numbers divide by five, so it is seven twentieths. That works because the decimal stopped. A decimal that stops has a last place, and that last place tells you which power of ten to put it over. So the easy direction is finished before it starts. The interesting question is the other kind.

0:36Nought point six, six, six, and onwards for ever. Try the same move and it falls apart immediately. Over ten? Over a hundred? There is no last place to point at, so there is no power of ten that is the right one. And you cannot stop somewhere and call it near enough. Six tenths is not this number. Nor is sixty six hundredths. Every truncation is wrong, and stays wrong however far you go.

1:07What is needed is a way to get rid of the tail entirely. Not to approximate it. To remove it. Here is the idea, and it is worth seeing before any arithmetic. Call the number x. Now multiply it by ten. The digits all shift one place to the left, past the point. Look at what that did to the tail. The original had sixes running on for ever after the point. The new one has sixes running on for ever after the point as well. The same digits, in the same columns.

1:46Two different numbers carrying the identical infinite tail. Subtract one from the other and the tails go. Not approximately. Completely, because they matched digit for digit. Do it properly. Let x be nought point six recurring. Ten x is six point six recurring. Write the two underneath each other and take the smaller from the larger. On the left, ten x minus x is nine x. On the right, six point six recurring minus nought point six recurring. The tails cancel and what is left is six. A whole number, out of two numbers that had no end.

2:32So nine x equals six, and x is six ninths. And you are not finished until you have reduced it: six ninths is two thirds. Divide two by three and the sixes come back, which is the check. That step deserves suspicion, so let us be precise about why it is allowed. It is not a general licence to subtract infinite decimals. Take two endless tails that differ and subtract them, and you have not obviously done anything meaningful.

3:07What makes this legitimate is that the second tail was manufactured. It was not found lying around, matching by luck. It was built by multiplying by exactly the right power of ten, so that every digit landed in the same column as its partner. The whole method is one idea wearing different clothes: engineer a second number whose tail is identical, then subtract. Everything after this is bookkeeping about which power of ten does the engineering.

3:44Nought point four five, four five, four five. Now the block is two digits long, and that changes what you multiply by. Try ten anyway and watch it fail. Ten x minus x leaves forty five elevenths, which is four point nought nine, nought nine, and still repeating. The tails did not line up, because shifting one place moved the block only halfway round. Shift by the whole block instead. Multiply by a hundred. Now a hundred x is forty five point four five recurring, the tails match, and subtracting leaves ninety nine x equals forty five.

4:29So x is forty five ninety ninths, which reduces to five elevenths. And five elevenths is exactly the fraction whose long division produced this decimal in the first place. The two directions have met on the same number. Now a harder shape. Nought point one, then sixes for ever. The one does not repeat. Only the sixes do. And that single stray digit breaks the method as it stands, because shifting by the block no longer lines the tails up.

5:06So deal with it first. Multiply by ten, once, purely to move the one across the point. Ten x is one point six recurring, and what is left after the point is now a clean repeating block with nothing in front of it. That is the first shift, and it has a job of its own: park the non-repeating digits on the left where they cannot cause trouble. It is not the shift that cancels anything.

5:37Then the second shift does the cancelling, exactly as before. From ten x equals one point six recurring, multiply by ten again. A hundred x is sixteen point six recurring. Now these two have identical tails, so subtract them and take ninety x equals fifteen. x is fifteen ninetieths, which reduces to one sixth. Two multiplications, and each one had a different purpose. The first cleared the digits that would not repeat. The second advanced by one whole cycle.

6:17Try the general case. Two point three five, then sevens for ever. Two digits do not repeat, so multiply by a hundred. One digit does, so multiply by ten again. Nine hundred x equals two thousand one hundred and twenty two, and x is one thousand and sixty one over four hundred and fifty. Put the numbers that x got multiplied by into a single column, and something obvious appears that was invisible one example at a time.

6:54Nine. Ninety nine. Ninety. Nine hundred. Nine thousand nine hundred. Every one of them is ten to the total number of digits you shifted through, minus ten to the number that did not repeat. Nine is ten minus one. Ninety is a hundred minus ten. Nine thousand nine hundred is ten thousand minus a hundred. So you never have to remember the procedure. Count the digits that do not repeat, count the ones that do, and both powers of ten are decided for you before you write a single line.

7:33There are exactly two ways this goes wrong, and both were measured rather than guessed at. The first is multiplying by ten whatever the block length. Every fraction with a divisor under three hundred was converted this way. Thirty nine thousand five hundred and thirty six of them have a block of two digits or more, and multiplying by ten gets every single one of them wrong. Not most. All of them.

8:03The second is collapsing the two shifts into one. That one is more dangerous, because it works beautifully on the twenty four thousand six hundred and ninety two decimals whose block starts right after the point. Then it fails on all seventeen thousand two hundred and twenty six that have digits in front of the block. Exactly those, and nothing else. A shortcut that is right whenever you first meet it and wrong the moment the problem gets interesting.

8:38Now run the machinery on a decimal that most people feel they already know the answer to. Nought point nine, nine, nine, for ever. It is a repeating decimal with a one digit block, so it gets exactly the treatment nought point six recurring got. Let x be it. Ten x is nine point nine recurring. Subtract. Nine x equals nine. So x equals one. Not close to one. Not a whisker under it. One. The instinct that it must fall short comes from imagining the nines stopping somewhere, and they do not stop anywhere. There is no last nine for the gap to hide behind.

9:26Which has a consequence that reaches back over everything else. If nought point nine recurring is one, then two point four six, then nines for ever, is two point four seven. That is not a special case. Every decimal that stops can be rewritten by dropping one from its last place and letting nines run on for ever. Three thousand six hundred and eighty one of them were rewritten that way and checked, and every rewriting landed on the number it started from.

9:59So a decimal expansion is a signature, but it is not a fingerprint. Numbers that stop have two of them. And here is what has actually been built. Forty four thousand eight hundred and fifty fractions were divided out into decimals, then converted back with nothing but counting and subtraction, and every one landed on the number it came from. Every fraction gives a decimal that stops or repeats. Every decimal that stops or repeats gives a fraction back. The two directions are one thing, and the tail that seemed to make it impossible was the thing that made it work.