Digital Integers

Why

We want to associate elements of $\Z $ with bitstrings for use on digital computers.¹

Definition

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 $\set{0, 1}^8$.

Correspondence with $\Z $

The bit string $x \in \set{0, 1}^d$ corresponds to the integer $\sum_{i = 1}^{d} x_i 2^i$.

Notation

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.

Signed integers

1. Future editions will discuss digital computers. ↩︎
2. Future editions will include an account of signed integers, or will place this in another sheet. ↩︎