We want to associate elements of $\Z $ with bitstrings for use on digital computers.1
A digital integer is a bit-string. The set of $d$-bit integers is the set of length-$d$ bit strings $\set{0, 1}^d$. For example, the set of 8-bit digital integers is the set $\set{0, 1}^8$.
The bit string $x \in \set{0, 1}^d$ corresponds to the integer $\sum_{i = 1}^{d} x_i 2^i$.
We denote the set of 8-bit (16-bit, 32-bit, 64-bit) integers by int64 (int8, int16, int32).
It is easy to embed $x$ in
int8 by considering
$x'$ in int16
defined by
\[
x' = (x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, 0, 0, 0, 0,
0, 0, 0, 0)
\]
Naturally, we associate the integers with bit strings.