Irrational decimals: an expansion with no stop and no repeating block

0:00Divide one by three and the digits never stop. Nought point three three three three, going on for as long as you are willing to keep writing. So here is a question worth getting right the first time. Is that what makes a number irrational? That its decimal never ends? It is not. One third is a fraction. It is about as rational as a number gets, and its decimal runs for ever.

0:29Never ending is half an answer at most. On its own, it is not evidence of anything at all. The real mark of an irrational number is two conditions, and both of them have to hold. The decimal never stops. And no group of digits repeats for ever after. One third fails the second one straight away. A single three, over and over, is exactly a group of digits repeating for ever. So one third lands on the rational side, which is where it belongs.

1:06Both conditions together. Drop either one and the test stops working. Why should two conditions be enough to decide anything? Because there are two arguments here, pointing opposite ways, and between them nothing gets through. The first direction says every fraction stops or repeats. When you divide, the remainder left at each step has to be smaller than the divisor, so there are only so many values it can take.

1:39Keep going and one of them has to come round a second time. And when the remainder comes round, the digits have no choice but to come round with it. So a fraction cannot hand you digits that run for ever without ever settling. It has nowhere to put them. The other direction runs the opposite way. Anything that stops or repeats is a fraction. Shift the decimal along past its block, subtract the original, and the endless tail cancels exactly, leaving a whole number over a whole number.

2:17Put the two arguments together and you have a test that decides in both directions. Every fraction with a divisor below two hundred and sixty was run. Thirty three thousand four hundred and eleven of them. Two thousand four hundred and forty one stopped, thirty thousand nine hundred and seventy repeated, not one did anything else, and every single one converted back into itself. So what does a number look like when it fails both conditions at once?

2:51Root two. One point four one four two one three five, and it keeps going. Nineteen places written out here, twenty significant figures, with no end and no repeating block anywhere in sight. And pi. Three point one four one five nine two six five, and on it goes as well, to exactly the same length. Both of these were computed digit by digit rather than looked up. And neither of them can ever be written out completely. That is not a shortage of space. That is the whole point.

3:30Now the trap that catches almost everybody. A root sign is not a certificate of irrationality. Root eighty one is nine. Run it through the very same machinery that produced those nineteen digits and you get nine point zero zero zero zero zero. It stops immediately. The question was never whether there is a root sign. It is whether the number underneath it is a perfect square. Below two hundred there are only fourteen of those.

4:04Root twelve is a different story. Three squared is nine and four squared is sixteen, so twelve is not a perfect square, and searching every fraction with a divisor under four thousand finds nothing that squares to twelve either. Here is the harder trap, and it is the one this whole idea turns on. One point zero one, zero zero one, zero zero zero one, zero zero zero zero one, and onwards.

4:38You can predict every digit of that number. A one, then a run of noughts, then another one, and each run is one longer than the run before it. There is nothing mysterious about it at all. So it certainly has a pattern. The question is whether it has a repeating block, and those are not the same thing. Look at the gaps. Between the first one and the second there are two noughts. Then three. Then four, then five, then six.

5:10A repeating block is a fixed group of digits that comes back unchanged, for ever. Fixed is the word doing the work. If some block were repeating here, then the gaps between the ones could never grow longer than that block. But these gaps grow without ever stopping. Four thousand digits were searched, against every starting point up to two hundred and every block length up to two hundred. Nothing repeats. And in that stretch the longest run of noughts reaches eighty seven.

5:46Being able to describe every digit and having a repeating block are two different claims. This number has the first and not the second, so it is irrational. Put six decimals side by side and sort them, because the sorting is where the traps live. Root eighty one is nine, so it is rational. Nought point three recurring is one third. Nought point one two three four five, repeating, is four thousand one hundred and fifteen over thirty three thousand three hundred and thirty three.

6:24Root twelve is irrational, and so is the number with the lengthening runs of noughts. And then the last one. Twenty three point five six zero one eight five six, carrying on for twenty one untidy digits, and then stopping. It stops, so it is rational. The orderly looking number is the irrational one. The messy looking number is not. Length and untidiness are not evidence of anything. One more caution, and then the test is safe to use.

7:00A decimal that stops has a second writing. Take two point four seven. Knock one off the last digit and hang endless nines on the end. Two point four six, nine nine nine nine, for ever. That is not a number a little below two point four seven. It is two point four seven. Convert the nines writing back and you land on two hundred and forty seven over a hundred, exactly.

7:31Two thousand four hundred and thirty three terminating fractions were checked this way, and every twin came back to the number it was built from. The plainest case is the one that unsettles people most. Nought point nine nine nine, for ever. Call it x. Ten x is nine point nine nine nine, with exactly the same endless tail. Subtract, and the tails cancel completely. Nine x equals nine, so x is one.

8:04Not nearly one. One. The two writings are two ways of spelling the same number. And notice carefully what this does not do. It does not break the test. Two point four seven stops, and two point four six nine recurring repeats, and stopping and repeating are both on the rational side. One number can carry two signatures. A signature never serves both kinds. So here is what you are holding now.

8:38Hand me any decimal at all. If it stops, it came from a fraction and I can produce that fraction. If it repeats, it came from a fraction and I can produce that one too. And if it does neither, then no fraction made it. Not because nobody has found the right one yet, but because a fraction is incapable of behaving that way. It is not a fingerprint. Two different writings can name one number, and you have just seen that happen.

9:09But it is a signature. And it never lies about which kind of number it came from.