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Take $e$ and $\pi$. Scale $e$ to $a$, $\pi$ to $b$ somehow such that $b-a$ is almost zero.

Then there must be a rational $q$ between $a$ and $b$: $a<q<b$. How would you find this $q$?

If $a$ and $b$ were rationals, I would just say: take $\frac{a-b}{2}$ and define that as $q$. But, if $\frac{a-b}{2}$ is irrational, how to find a rational $q$ between $a$ and $b$?

jupiter_jazz
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  • Use Archimedes' axiom. See e.g. https://math.stackexchange.com/questions/421580/is-there-a-rational-number-between-any-two-irrationals – Kal S. Oct 31 '17 at 15:47
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    What do you mean by "scale" to? Why not take any rational $q$ in $[e,\pi]$, and then scale to some nice $a$,$b$ with $a<q<b$. – Dietrich Burde Oct 31 '17 at 15:47
  • @DietrichBurde scale means to multiply with a scalar. Well, how would you scale $a,b,q$ simultanously? – jupiter_jazz Oct 31 '17 at 16:33

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