**PROPOSITION 1: ** **For a real number x there exists a sequence $ x_1, x_2, x_3,…$ of integers such that**

$ \hspace{4cm} x=x_1 +\frac{x_2}{2!}+\frac{x_3}{3!} + \cdots + \frac{x_n}{n!} + \cdots, \hspace{2cm} (*) $

**where $x_1$ can be any integer, but for $ n \geq 2$, $x_n \in \{ 0,1,…,n-1 \}.$ Furthermore, if we require that the partial sums be strictly smaller than x, then such a representation is unique.**

**Remark:** One cannot help recalling decimal or binary expansion of numbers. Notice that $\frac{n}{n!}=\frac{1}{(n-1)!}$ (drops back to previous digit), so the bound on $x_n$ is logical. Continue reading “Real Numbers Base…Factorials! And A By-product” »