Representation of Number Base in that Base

Theorem

Let $b \in \Z$ be an integer such that $b > 1$.

Then $b$ is expressed in base $b$ as $10$.

By the Basis Representation Theorem, $b$ can be expressed uniquely in the form:

$\ds b = \sum_{j \mathop = 0}^m r_j b^j$

where:

$m$ is such that $b^m \le n < b^{m + 1}$

all the $r_j$ are such that $0 \le r_j < b$.

As $b = b^1$, we have that:

$b = 1 \times b^1 + 0 \times b^0$

That is, by definition of base $b$:

$b = \sqbrk {1 0}_b$

Hence the result.

$\blacksquare$

The number ten is expressed in decimal notation as $10$.

The number $2$ (two) is expressed in binary notation as $10$.