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:


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

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