Definition:Number Base/Real Numbers

Definition

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:

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

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 $16$: