Bit Strings

Why

We name sequences in $\set{0, 1}$ to easily discuss codes in this set.¹

Definition

A bit string (or binary string) is a finite sequence in the set $\set{0, 1}$.

If a bit string has length one, we refer to it as a bit. Using this terminology, it is natural to call the sequence terms bits. Other terminology for bit strings includes binary sring, bit sequence and digital datum.

If the bit string has length eight, we refer to it as a byte. Using this terminology, a kilobyte is a length $8 \cdot 2^{10}$ bit string. In other words, a kilobyte is $2^{10} = 1024$ bytes, or roughly one thousand bytes. Likewise a megabyte is a length $8 \cdot 2^{20}$ bit string. A megabyte is $2^{20} = 1048576$ bytes, or roughly one million bytes. Similarly a gigabyte is $2^{30}$ bytes and a terabyte is $2^{40}$ bytes.²

Notation

We often denote the set $\set{0,1}$ by $\B$. Using this notation, we denote the length $n$ bit strings by $\B^n$.

We occasionally use \false to denote the length-1 bit string $(0,)$ and \true to denote the length-1 bit string $(1,)$. In this context, \bool is another name for the set $\set{\true, \false}$³ .

1. Why we want to name these requires more explanation, to be included in future editions. ↩︎
2. Warning: some authors use these monikers with powers of ten. For example, a kilobyte is exactly one thousand bytes. etc. ↩︎
3. Future editions might break this out into its own sheet. ↩︎