# Definition:Number Base

## Contents

## 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:

- $x = \sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$

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

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

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**.

### Radix Point

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 $16$:

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**base**:**1.** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**base**(for representation of numbers)