There is a lovely claim that goes round the internet every Pi Day: because π goes on for ever without repeating, it must contain every number there is. Your phone number, your birthday, your bank PIN, the complete works of Shakespeare written in code. Everything.
It is a wonderful idea. It might even be true. But the reason usually given for it is wrong, and the truth is more interesting.
"It never repeats" is not enough
Here is what is definitely known. In 1761 Johann Lambert proved that π is irrational: it can't be written as one whole number divided by another, so its decimals never end and never fall into a repeating loop. In 1882 Ferdinand von Lindemann went further and proved it is transcendental, which is why the ancient puzzle of "squaring the circle" with a ruler and compass can never be solved.
Neither of those facts means every number is inside π. Look at this number:
0.101001000100001000001…
It never ends and it never repeats (each run of zeros is one longer than the last), so it is irrational too. But it will never contain a 2. Or a 7. Or your phone number. Never repeating is not the same as containing everything.
The idea that would settle it: normal numbers
Mathematicians have a name for the kind of number that would contain everything. A number is normal (in base 10) if every digit turns up one time in ten in the long run, every pair of digits (like 00, 01, … 99) one time in a hundred, every triple one time in a thousand, and so on for strings of any length.
A normal number contains every finite string of digits. Not just once: infinitely many times. So if π is normal, your phone number really is in there, over and over again.
The strange thing is that normal numbers are everywhere. In 1909 Émile Borel proved that almost all numbers are normal: if you could pick a number completely at random, it would be normal with certainty. And yet it is fiendishly hard to prove that any particular number is normal. One of the few famous examples is a number someone built on purpose. In 1933 David Champernowne, then a student, proved that
0.123456789101112131415…
(just the counting numbers written one after another) is normal in base 10. Clever, but it is a number made to order.
What nobody has proved
Nobody knows whether π is normal. Nobody knows whether √2 or e are, either. In fact, nobody has even proved that every digit appears in π infinitely often. As far as a proof goes, π could stop using the digit 7 after some point. Almost every mathematician would bet heavily against it, but a bet is not a proof.
So the honest answer to "is every number in π?" is: probably, but nobody can prove it yet.
What the digits say
While the proof waits, we can look. And every test anyone has run says π's digits behave just like the rolls of a fair ten-sided dice. You can check a small piece of this yourself. In the first 2,000 decimals, every digit appears between 182 and 212 times, close to the 200 you would expect from a fair dice. Researchers have run the same kind of test on trillions of digits, for single digits, pairs and longer strings, and π passes every time.
If the digits really do behave like dice, we can work out how likely your number is to be in the part we searched. Every ten-digit window is one of 10 billion possible numbers. In the first 100 billion decimals, that works out at 99.995% of all ten-digit numbers turning up, with about 454,000 still missing. That is what our table shows, too: the count of missing numbers matches the prediction, which is one more small piece of evidence that π's digits are as random as they look.
Why "we checked" is not "we proved"
This is a good example of the difference between evidence and proof. Checking 100 billion digits tells us a lot about those digits. It tells us nothing for certain about the next ones, and π has infinitely many next ones. A proof would have to explain why the digits are evenly spread, for ever. Nobody has found that explanation yet. Maybe you will.
What about Shakespeare?
Even if π is normal, there is a catch: patience. Each extra digit makes a string about ten times harder to find. A typical ten-digit number first appears around 7 billion digits in. A sentence from Shakespeare, written as digits, is far longer, and would sit so deep in π that no computer that could ever be built would reach it. So when a long number isn't in our digits, we show you its longest piece instead.
Go and look
- Find your number and see whether it is one of the lucky ones.
- Read why your phone number is (probably) in π for the maths of those odds.
- The learn page is every number really in π? has the short version.