What is something you learned in math that made you loose a sense of innocence-- knowledge that you can never unlearn that changes the world forever?
at some point I was told that the decimal (or any other base) expansion of such irrational numbers as $\Pi$ contained any other number
but that can't be! if it contained all of the digits of $\Pi$, in sequence, in its fractional part, then the sequence of digits of $\Pi$ would be a repeating sequence, therefore it would be rational rather than irrational!
indeed, even rational numbers with a periodic component to their fractional expansion cannot fit in the expansion of $\Pi$
but that can't be! if it contained all of the digits of $\Pi$, in sequence, in its fractional part, then the sequence of digits of $\Pi$ would be a repeating sequence, therefore it would be rational rather than irrational!
indeed, even rational numbers with a periodic component to their fractional expansion cannot fit in the expansion of $\Pi$
This sounds like the claim that Pi is "normal" in a particular base. It's not actually known whether Pi is normal in any particular base, I don't think, but in some sense most real numbers are. It doesn't say that every other number is contained somewhere in pi, just that any finite string of digits is. So no irrational or repeating numbers required, if I have it right.
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This addresses inter alia the claim that every (finite?) string of numbers appears somewhere in Pi. https://en.m.wikipedia.org/wiki/Normal_number
