eolas/zk/Binary_units_of_measurement.md

title	categories	tags
Binary units of measurement	Computer Architecture Mathematics	binary

Binary units of measurement

A single place or symbol in a decimal number is called a digit. For example the number 343 is a number containing three digits. A digit can be any one of the ten numerals through 0-9.

The equivalent entity in the binary number system is the bit. For example the binary number 110 has three bits. A bit can only have one of two values in contrast to a digit which can have one of ten values. These values are 0 and 1.

Sequences of bits

Informational complexity

The informational complexity of a single digit is much larger than a single bit: it can represent one of 10 states whereas a bit can only represent one of two states.

We can think of how much data can be stored in a number in terms of the total number of unique arrangements of bits or digits. With this in mind, compare a two digit digital number to a two bit binary number. For the decimal number each digit can represent one of ten states, hence the total number of unique states is equal to 2^{10} (1024):

With the binary number we have 2^{2} (4), giving us far fewer possible unique states. They are so few we can easily list them:

00
01
10
11

Bytes

In order to express larger binary numbers and greater complexity we work with sequences of bits.

The standard base sequence of bits is called a byte. This is a binary number comprising eight bits. For example the number 11001110 is a byte equivalent to 206 in decimal.

A byte allows for a complexity of up to 256 possible states: 2^{8} = 256

Metric units: kilobytes, megabytes etc

Having established that the core quantity of information is the byte, the convention is to apply the standard metric prefixes to the byte to establish units:

Prefix	Symbol	Expression as base ten exponent	Value	English word
Terra-	T	10^{12}	1,000,000,000,000	trillion
Giga-	G	10^9	1,000,000,000	billion
Mega-	M	10^6	1,000,000	million
Kilo-	k	10^3	1,000	thousand

Hence 2MB is two million bytes, 4kb is four thousand bytes etc.

Whilst the metric prefixes are conventionally used to refer to aggregates of bytes, they are only approximate values due to the fact that metric is base ten whereas bytes are base two. Hence 1MB is not 1,000,000 bytes it is 1,048,576 bytes.

This is usually fine as a shorthand but an accurate representation of byte values is as follows:

Prefix	Symbol	Expression as base two exponent	Value	English word
Terra-	T / Ti	2^{40}	1,099,511,627,776	trillion
Giga-	G / Gi	2^{30}	1,073,741,824	billion
Mega-	M / Mi	2^{20}	1,048,576	million
Kilo-	k / Ki	2^{10}	1,024	thousand

A special IEEE standard was introduced to distinguish base two within the metric nomenclature: kibi, mebi, gibi etc but this was not widely adopted. The symbols for this are included above as they are sometimes used by Linux programs (Gi, Mi, Ki etc)