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:


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

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 $\displaystyle n = \sum_{j = 0}^m {r_j b^j}$ 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 $16$.

See binary numbers and hexadecimal numbers.

Real Numbers
Let $x \in \R$ be a real number such that $x \ge 0$.

Let $b \in \N: b \ge 2$.

See the definition of Basis Expansion for how we can express $x$ in the form:


 * $x = \left[{s . d_1 d_2 d_3 \ldots}\right]_b$

Then we express $m$ as above, and arrive at:
 * $x = \left[{r_m r_{m-1} \ldots r_2 r_1 r_0 . d_1 d_2 d_3 \ldots}\right]_b$

or, if the context is clear, $r_m r_{m-1} \ldots r_2 r_1 r_0. d_1 d_2 d_3 \ldots_b$.

Integral Part
In the basis expansion
 * $x = \left[{r_m r_{m-1} \ldots r_2 r_1 r_0 . d_1 d_2 d_3 \ldots}\right]_b$

the part $r_m r_{m-1} \ldots r_2 r_1 r_0$ is known as the integer part, or integral part.

Fractional Part
In the basis expansion
 * $x = \left[{r_m r_{m-1} \ldots r_2 r_1 r_0 . d_1 d_2 d_3 \ldots}\right]_b$

the part $.d_1 d_2 d_3 \ldots$ is known as the fractional part.

Radix Point
The dot that separates the integral part from the fractional part is called the radix point.