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I read here that any real number can be used as a base in a numerical system. This does seem to me to raise a number of problems.

If you take $\pi$, for example, how would you write the number $2$ in base $\pi$ ? I suppose $ 2=\sum_{i=0}^{\infty}{x_i} \pi^i$

Which digit-symbols $x_i$ would you use ? Integers smaller than $\pi$,( ie $ 1; 2; 3$ ), like you would with an integer-base system ? Or something different ?

And how would you find the numerical expansion of any given number ? Just some normal Horner method ?

Any explanation would be apreciated, thanks.

Eli
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  • "how would you find the numerical expansion of any given number ?" Numbers don't have unique representations. For example, given a representation of $1-3/\pi$, if you replace the initial $.0$ with $.3$, you have a representation of $1$. Will this be a problem? – Rosie F Aug 07 '22 at 12:03
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    Knowing that $\sin\left(\frac{\pi}{2}\right)=1$, one can use the infinite series expansion of $\sin$ to write any number as a power series of $\pi$ by multiplying both sides by that number. $$n=\sum_{k\geq 0}\frac{(-1)^kn}{(2k+1)!}\left(\frac{\pi}{2}\right)^{2k+1}$$

    It seems like a quite peculiar way of writing numbers, but that's because $\pi$ is transcendental; we can't write it as a finite sum involving $+,-,\times,\div$ and powers and square roots of integers, and the converse is true.

     –  Aug 07 '22 at 13:35
  • @mohamedshawky I completely disagree. Doing that you would have negative rational "figures" ; moreover, what about the fundamental issue which is the uniqueness of such a decomposition ? – Jean Marie Aug 07 '22 at 14:13
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    You are looking for Non-integer base of numeration also known as beta expansion of real numbers. – Somos Aug 07 '22 at 17:13

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