1.1. Bits#
The current digital computers are mostly binary machines\footnote{We don’t consider q-bit used in quantum computers.} and use a bit \(b\) as the smallest unit of information where \(b=0\) or \(1\) (or we write it equivalently as \(b=\{0,1\}\)). Inside a computer, information is generally encoded in a string of bits such as \(01100101000101100\cdots\). The number of unique expressions depends on the length of the string, which is measured as the number of bits. An \(N\)-bit string
can express \(2^N\) different values. For instance, there are only four possible realizations of a 2-bit string: \(00\), \(01\), \(10\), \(11\). \(N\) can be very large but always finite and limited by the size of hardware.[1] The common lengths of the binary string in the present computers are \(8\), \(16\), \(32\), and \(64\). The string of 8 bit is called a byte. The number of different expressions these strings can have is shown in Table 1.1.
size in bits |
size in bytes |
# of expressions |
|---|---|---|
8 |
1 |
\(2^8=256\) |
16 |
2 |
\(2^{16}=65536\) |
32 |
4 |
\(2^{32}=4,294,967,296\) |
64 |
8 |
\(2^{64}=18,446,744,073,709,551,616\) |
We encode numbers and characters in binary strings and decode binary strings to get human-readable information. Encoding/decoding is not a one-to-one map.
The same one byte of string may correspond to multiple different things, integer, character, and others as shown in the following sections. For example, \(01000001\) can be character ‘A’ or integer \(65\). Some computer languages (dynamical language) such as python choose an appropriate encoding scheme based on the context but in compiler-based languages, programmers must declare the type of each variable before using it or otherwise the computer issues an error message.
Last modified on 02/09/2024 by R. Kawai.