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I am having a problem with the final question of this exercise.

Show that $e$ is irrational (I did that). Then find the first $5$ digits in a decimal expansion of $e$ ($2.71828$).

Can you approximate $e$ by a rational number with error $< 10^{-1000}$ ?

Thank you in advance

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    Yes, you can. Do you need to give such a number? Do you know about continued fractions?2012-11-14
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    I don't think I need to find the actual number. But I think I have to just prove its existence. I never actually manipulated with continued fractions2012-11-14
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    This question is very unclear. It could mean one of three things: 1: Can e be approximated by such a rational? Answer: Yes, obviously -- every real number can. 2: Do you know how to approximate it? Answer: Yes, in principle. 3: Find such a rational. Answer: Give me a few moments...2012-11-14
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    I think that a more interesting question is whether there is a good short rational approximation to e, analogous to 355/113 for $\pi$. Approximating e to absurd accuracy is just donkeywork for a computer. What puzzles me here is: if you are a good enough mathematician to prove that e is irrational, why would you ask such a naive question?2012-11-14

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