For other uses, see Radix (disambiguation).

In mathematical numeral systems, the radix or base is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.

In any positional numeral system (except unary, where the radix is 1), the number x and its base y are conventionally written as $(x)_y,$ although for base ten the subscript is usually assumed and not written. For example, $(100)_{10}$ (in the decimal system) represents the number one hundred, whilst $(100)_2$ (in the binary system with base 2) represents the number four.[1]

## Etymology

Radix is a Latin word for "root". Root can be considered a synonym for base in the arithmetical sense.

## In numeral systems

In the system with radix 13, for example, a string of digits such as 398 denotes the decimal number $3 \times 13^2 + 9 \times 13^1 + 8 \times 13^0$.

More generally, in a system with radix b (b > 1), a string of digits $d_1 \ldots d_n$ denotes the decimal number $d_1 b^{n-1} + d_2 b^{n-2} + \cdots + d_n b^0$, where $0\leq d_i < b$.

Commonly used numeral systems include:

Base Name Description
10 decimal system the most used system of numbers in the world, is used in arithmetic. Its ten digits are "0–9". Used in most mechanical counters.
12 duodecimal (dozenal) system is often used due to divisibility by 2, 3, 4 and 6. It was traditionally used as part of quantities expressed in dozens and grosses.
2 binary numeral system used internally by nearly all computers, is base two. The two digits are "0" and "1", expressed from switches displaying OFF and ON respectively. Used in most electric counters.
16 hexadecimal system is often used in computing. The sixteen digits are "0–9" followed by "A–F".
8 octal system is occasionally used in computing. The eight digits are "0–7".
60 sexagesimal system originated in ancient Sumeria and passed to the Babylonians. Used nowadays as the basis of our modern circular coordinate system (degrees, minutes, and seconds) and time measuring (hours, minutes, and seconds).
64 MIME Base64 is also used in computing, using as digits "A–Z", "a–z", "0–9", plus two more characters, often "+" and "/".[citation needed]
85 PostScript Ascii85 used in computing to encode sequences of bits as base 85 numbers
256 byte is used internally by computers, actually grouping eight binary digits together. For reading by humans, a byte is usually shown as a pair of hexadecimal digits.[citation needed]

For a complete list, see List of numeral systems.

The octal, hexadecimal and base-64 systems are often used in computing because of their ease as shorthand for binary. For example, every hexadecimal digit has an equivalent 4 digit binary number.

Radices are usually natural numbers. However, other positional systems are possible, e.g. golden ratio base (whose radix is a non-integer algebraic number), and negative base (whose radix is negative).

Many devices are built to accept numbers in decimal representation and display results in decimal. Often such devices convert from decimal to some internal radix on input, do all internal operations in that radix, and then convert the results from the internal radix to decimal on output. Such devices could in principle use any radix internally. The people who design such computing devices sometimes wonder what would be the "best" radix to use internally -- the question of radix economy.