Definition:Number Base

Definition

Integers

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

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

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

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

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 $b$.

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:

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

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 known as

A number base is also known as a radix.

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 $16$:

• Definition:Binary Notation
• Definition:Hexadecimal Notation

• Results about number bases can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): base: 1. (of a number system)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): base: 1. (of a number system)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): base (for representation of numbers)