# Floating point arithmetic

How can I prove that : a real number has a finite representation in the binary system if and only if it is of the form $$\pm \frac{m}{2^n}$$ where n and m are positive integers.

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A representation of a positive number $\alpha$ in the binary system is a series of the form $$\alpha=\sum_{k=d}^\infty e_k 2^{-k}$$ with $e_k\in \{0,1\}$ and is finite if there is an $n$ such taht $e_k=0$ for all $k>n$, i.e. we simply have $$\alpha=\sum_{k=d}^n e_k 2^{-k}.$$ Then multiplication with $2^n$ produces $$2^n\alpha = 2^n\sum_{k=d}^n e_k 2^{-k}= \sum_{k=d}^n e_k 2^{n-k}\in\mathbb Z$$