Definition:Number Base

Integers
Let $$n \in \Z$$ be an integer.

Let $$b$$ be any integer such that $$b > 1$$.

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


 * $$n = \sum_{j=0}^m r_j b^j$$

where:
 * $$m$$ is such that $$b^m \le n < b^{m+1}$$;
 * all the $$r_i$$ are such that $$0 \le r_i < b$$.

The number $$b$$ is known as the number base to which $$n$$ is represented.

$$n$$ is thus described as being "(written) in base $$b$$".

Thus we can write $$n = \sum_{j = 0}^m {r_j b^j}$$ as:
 * $$\left[{r_m r_{m-1} \ldots r_2 r_1 r_0}\right]_b$$

or, if the context is clear:
 * $${r_m r_{m-1} \ldots r_2 r_1 r_0}_b$$

The most common base is of course $$10$$.

So common is it, that numbers written in base 10 are written merely by concatenating the digits:
 * $$r_m r_{m-1} \ldots r_2 r_1 r_0$$

$$2$$ is a fundamentally important base in computer science, as is $$16$$.