eli5, is it true that irrational numbers like pi contains every single number combination in it?
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First thing: I suspect what you mean is: “do the *decimal expansions* of irrational numbers like pi contain every *finite* number combination?”
It’s important to distinguish between the number itself and its decimal representation. Also, the decimal expansion of pi clearly doesn’t contain every infinite number combination (since, for example, an infinite sequence of 0’s never occurs there).
That said: there are clearly irrational numbers that *don’t* have this property. For example, consider this number:
0.01101010001010001…
where the n’th decimal place is a 1 if *n* is prime, and 0 otherwise. This number is irrational, but its decimal representation clearly doesn’t have every finite number combination since it doesn’t contain any digit other than 0 or 1.
Another question: does pi have this property? **Pi is conjectured to have this property, but it hasn’t been proven**. In fact, pi is conjectured to have the stronger property of being “normal.” A normal number’s decimal representation is contains every finite digit sequence of a given length roughly the same number of times (in other words, the digits are approximately uniformly distributed).
Interestingly, there are a large number of irrational numbers that are conjectured to have this property, but it is very difficult to prove, and proofs are only known for a small number of numbers.
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