---
title: Binary units of measurement
categories:
  - Hardware
  - Mathematics
tags: [bits, 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 numeral through 0-9.

The equivalent entity in the [binary number system](/Hardware/Binary/The_binary_number_system.md) 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: 0 or 1.

## Sequences of bits

### Informational complexity

The informational complexity of digit is much larger than a 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 arrangemnets 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](/Electronics/Physics_of_electricity/Prefixes_for_units_of_electrical_measurement.md) to the byte to establish units:

| Prefix | Symbol | Expression as exponent | Expression as decimal 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.