# Definition:Number Base

## Definition

### 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:

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

where:

$m$ is such that $b^m \le n < b^{m + 1}$
all the $r_j$ are such that $0 \le r_j < b$.

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 $\displaystyle n = \sum_{j \mathop = 0}^m {r_j b^j}$ as:

$\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$

or, if the context is clear:

${r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$

### 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:

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

Then we express $m$ as for integers, and arrive at:

$x = \left[{r_m r_{m-1} \ldots r_2 r_1 r_0 \cdotp d_1 d_2 d_3 \ldots}\right]_b$

or, if the context is clear:

$r_m r_{m-1} \ldots r_2 r_1 r_0 \cdotp d_1 d_2 d_3 \ldots_b$

### Integer 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.

### 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.

In the basis expansion:

$x = \sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0 \cdotp d_1 d_2 d_3 \ldots}_b$

the dot that separates the integer part from the fractional part is called the radix point.

## Examples

### Base $5$

#### $25$ in Base $5$

The integer expressed in decimal as $25$ is expressed in base $5$ as $100_5$.

#### $32$ in Base $5$

The integer expressed in decimal as $32$ is expressed in base $5$ as $112_5$.

#### $56$ in Base $5$

The integer expressed in decimal as $56$ is expressed in base $5$ as $211_5$.

## Also see

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

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 number base in computer science, as is $16$: