# $e$ and $\pi$

Theoretically, can $\pi$ contain the entire sequence of digits of $e$ ? Also, can $e$ contain the entire sequence of digits of $\pi$?

Note by Hobart Pao
3 years, 11 months ago

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This reminds me of Banach–Tarski paradox.

Reshared!

- 3 years, 11 months ago

Not necesary, I think, is like saying that the secuence of digits of $e$ is a subset of the secuence of digits of $\pi$

- 3 years, 11 months ago

I mean I've seen these articles where they say "Any/all of your numerical passwords, everyone's birthday, blah blah blah are contained within pi" or something like that. This may be an obvious question, but where's the limit on length of a sequence of digits that can fit into pi?

- 3 years, 11 months ago

Yes! I'll give a constructive argument in a few minutes

- 3 years, 11 months ago

...

• Digits of pi in e = 2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274274663919320030599218174136

• Digits of e in pi = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081056239

- 3 years, 11 months ago

I meant seeing 314159265358979323846...somewhere in $e$ eventually. Without skipping digits. Because the way you did it, of course you could fit all the digits eventually lol

- 3 years, 11 months ago

If it did, then the following would hold:

$e = x + 10^{-i}\pi$

where x is a rational number and i is an integer. I do not think this is true. Although, I cannot prove it.

- 3 years, 11 months ago

ah! interesting...why does x have to be rational? It could be the sum of two irrationals, right? And since $\pi$ has infinite digits, will your rational number have decimals? Because you have to then avoid modifying the $\pi$ portion and maybe have a.0000000000000000000000000000000000000000000000...11111111111111111111111111111111111111111111111111...0000000000000000000000000 just as an example. I could be wrong because I haven't slept much...lol

- 3 years, 11 months ago

Let's say that e was 2.72831415...

In this case, x= 2.728 is the rational part

- 3 years, 11 months ago

so you're talking about putting $\pi$ at the end of $e$? That doesn't make much sense

- 3 years, 11 months ago

because it could just be somewhere in the "middle" of $e$ because $e$ doesn't end

- 3 years, 11 months ago

$\pi$ is irrational and hence doesn't have a terminating decimal expansion; how can it be in the middle of $e$?

- 3 years, 11 months ago

but nonetheless, that is interesting! Would it be okay if $\pi$ were at the end of $e$? Because both are irrational. Would it matter?

- 3 years, 11 months ago

A note: Only one of them can be true. I shall prove it, by Contradiction.

Let $e$ have $a$ digits and $\pi$ have $b$ digits.

Then, $a \leq b \leq a$.

Since a=a, a=b.

a = b implies that $\pi = e$.

- 3 years, 11 months ago

No, both numbers are irrational and they don't terminate, so you can't say that $e$ have $a$ digits and $\pi$ have $b$ digits.

- 3 years, 11 months ago