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I found somewhere on the internet that $e$ can be approximated by $$\Large\left(1+9^{-4^{7\times6}}\right)^{3^{2^{85}}} $$

Note that the expression uses each numerical digit exactly once.

I understand that this works because $$e=\lim_n \left(1+\frac{1}{n}\right)^n.$$

Using the same idea I got another approximation to $e$

$$\large\left(1+\left(2^{3}\right)^{(4+5)(6-7)}\right)^{8^9} $$

I know this approximation is not nearly as good as the previous one, but it uses the digits in order. (Is it the best possible?)

Do these types of expressions have a name?

They are called pandigital expressions (Thanks Wojowu)

Are there similar expressions for $ \pi$ or the golden ratio or any other interesting number?

Is there a way of constructing this type of expressions for any given integer, rational or real ?

edit: (Inspired by Arthur's comment) Are there expressions similar to these not using base ten?

King Ghidorah
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    They are called "pandigital expressions". I wouldn't think much research has gone into these, but at least now you have some term to Google. – Wojowu Nov 25 '16 at 10:40
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    Mathematically, pandigital expressions are not that important / interesting, because they depend on using base ten. Recreationally, they might be great fun. – Arthur Nov 25 '16 at 10:46
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    You can extend your idea to higher bases to get as accurate as you like. For example in base $N=2^{3456789}+1$:

    $$\Large{\left({1+(2^{3456789}})^{(10-11)^{(-12+13)\cdots(-(N-2)+(N-1))}}\right)}^{N}$$

     – Ian Miller Nov 25 '16 at 10:58
  • @IanMiller You should make that an answer – Simply Beautiful Art Nov 25 '16 at 12:59
  • While probably not what you are looking for, $e^x\approx1+x+\frac{x^2}2+\dots+\frac{x^k}{k!}$ provides good approximations to $e^x$ and you can calculate them in your head. – Simply Beautiful Art Nov 25 '16 at 13:01
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    See also http://math.stackexchange.com/questions/1945026/an-amazing-approximation-of-e – Gerry Myerson Nov 27 '16 at 10:20
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    And also http://math.stackexchange.com/questions/449877/pandigital-rational-approximations-to-the-golden-ratio-and-the-base-of-the-natur – Gerry Myerson Nov 27 '16 at 10:22
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    Here is link for $\pi$: http://math.stackexchange.com/questions/445277/pi-estimation-using-integers . – Oleg567 Dec 03 '16 at 18:27

2 Answers2

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You can extend your idea to higher bases to get as accurate as you like. For example in base $N=2^{3456789}+1$:

$$\Large{\left({1+(2^{3456789}})^{(10-11)^{(-12+13)\cdots(-(N-2)+(N-1))}}\right)}^{N}$$

Ian Miller
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Yes there is a pandigital expression for π -

π = (71*5)/(98 + 6 + 4 + 3 + 2)

This equals 355/113 which is accurate upto 6 decimal places for π (3.141592...)

amit s
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