π is irrational, and what that rules out

0:00Divide one by seven the long way and watch the remainders, not the digits. Seven into ten goes once, remainder three. Three carries down: seven into thirty goes four, remainder two. Then remainder six, then four, then five - and then one, which is where we started. The moment a remainder repeats, everything after it repeats too, because the remainder is the whole state of the division. And a remainder has to repeat, because dividing by seven there are only six of them that are not zero.

0:38So the block is not something one seventh happens to do. It is something the division cannot avoid. The same argument works for any divisor at all. One third repeats a block of one digit. One eleventh repeats a block of two. One seventh repeats a block of six, and it cannot be longer than six, because there are only six remainders to run out of. That ceiling is real and it is sometimes reached exactly: the longest block under a hundred belongs to one ninety-seventh, and it is ninety-six digits.

1:19Every divisor up to three hundred was checked, and not one broke the rule. So here is a test that every fraction in existence passes. Write it as a decimal, and sooner or later it settles into a block and repeats it for ever. Now run the test on our circle constant. Two thousand of its digits were taken, and every block length up to three hundred was tried against them.

1:47Each length was allowed to start anywhere in the first fifty places, so a slow beginning would not hide a block that eventually settles. Nothing. Not one length, at any starting point, describes those digits. And be careful about what is doing the work here, because there is a trap in it. One third never stops either, and one third is a perfectly ordinary fraction. It is not the not-stopping that matters, it is the not-repeating - and this number does neither.

2:23Which brings us to the word. A rational number is one whole number divided by another, with the divisor not zero - a half, seven thirds, six over one. Irrational means: not that. It is a statement about what the number is not. It does not say the number is strange, or unknowable, or beyond arithmetic. It says one thing, precisely: no pair of whole numbers, however enormous, divides to give it.

2:55Everything people believe about the word beyond that is invention. And most of the invention is wrong. First, though, be honest about what that digit search actually established. It is worth something real: a fraction with divisor below three hundred must show a block shorter than three hundred, so finding no such block rules every one of them out. That is a genuine result, and it is finite. It is not irrationality.

3:29No search of any length can be, because a block could always begin one digit past wherever you stopped looking. The digits are the symptom. Aryabhata and Zu Chongzhi both wrote in ways that suggest they suspected the truth, and neither could settle it. Lambert settled it in seventeen sixty-one, without knowing much of anything about the digits at all. So take three beliefs away. Irrational does not mean unknown.

4:01We can produce more correct digits of this number than anybody will ever have a use for, and every one of them is forced. Irrational does not mean it cannot be written exactly. There is a formula that writes it exactly - a quarter of it is one minus a third plus a fifth minus a seventh, running on for ever in the odd numbers. That is an equation, not an approximation.

4:29And irrational does not mean uncomputable. What it forbids is one thing only: being a fraction. Here is the consequence worth the whole video. There is no closest fraction to this number. Not one that is hard to find - one that does not exist. Offer any fraction you like, and a nearer one is waiting. The reason is short enough to give in full. Your fraction is not the number, so the distance between them is some positive quantity, however tiny.

5:04Now cut the number's own decimal expansion off far enough along that the piece you kept lands inside that distance. What you kept is a fraction - it is a whole number over a power of ten - and it is nearer than yours. Watch it work on real candidates. Against three, one decimal place is already enough to beat it. Against twenty-two sevenths, three places do it. Against three point one four, three places again.

5:37Against three hundred and fifty-five over one hundred and thirteen, which is very good indeed, seven places are needed - and seven places are available. And the recipe never runs dry, because you can always keep one more digit. Cut it off at fifty places and the fifty-first is nearer still. There is no last step, and there is no winner. Which is why two symbols have to be kept apart.

6:07This number is approximately twenty-two sevenths, and this number is not equal to twenty-two sevenths. Both of those are true, and they belong on the same line. They agree on three point one four, and part company at the third decimal place, where twenty-two sevenths is high by about thirteen ten-thousandths. Keep the same discipline everywhere. The square root of two is approximately one point four one four, and is not equal to it.

6:41Those two part company at the fourth place - and you can catch it without knowing anything about square roots at all. One point four one four, squared, is one point nine nine nine three nine six. The best working fraction anybody has is that three hundred and fifty-five over one hundred and thirteen. It is high by under three ten-millionths. That is four thousand seven hundred times nearer than twenty-two sevenths.

7:13Every divisor below fifteen thousand was tried, keeping the nearest fraction each one can make, and not a single one of them beats it. A hundred and thirty-two of them tie with it, and those are simply the multiples of a hundred and thirteen writing the same fraction again. And it is still not the number. It is a very good fraction that is not it. By now that should sound inevitable rather than disappointing.

7:45Now the most tempting wrong idea available here. People say a circle cannot be squared with a straight edge and compasses because the constant is irrational. That reason cannot be right, and the square root of two is the counter-example. It is irrational, and it is the exact diagonal of a square of side one, which anyone can draw in a few seconds. The real difference is this: the square root of two answers to a polynomial equation with whole-number coefficients, namely x squared equals two.

8:23Our constant answers to none. Every polynomial of degree four or less with whole-number coefficients between minus six and six was tested - three hundred and seventy-one thousand two hundred and ninety-two of them. The square root of two survives three hundred and fourteen, which are exactly the multiples of x squared minus two; the constant survives none of them, and that is the stronger property that actually defeats the compasses.

8:58So we live with approximations, and we mark them. There is a well-known sentence whose word lengths count out three, one, four, one, five, nine, two - the first seven digits, carried in a form a person can remember. There are two days in the calendar too. The fourteenth of March, written three fourteen, celebrates the first three digits. The twenty-second of July, written twenty-two seven, celebrates the fraction. And the second date is the more honest of the two, because the number you actually calculate with is always an approximation of something you can never write down as a fraction.

9:42Irrational is not a warning label. It is a precise statement of what this number is not, and everything that follows from it follows from that one exclusion.