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:

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

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:

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

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.

Also see

The most common 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 base in computer science, as is $16$.

See binary numbers and hexadecimal numbers.