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 numeral system (except unary, where radix is 1), the base will always be written as "10". For example, "10" represents the number ten in the decimal system; "10" represents the number two in a base two system.

## 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".
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.
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. It is still used as the basis of our modern circular coordinate system (degrees, minutes, and seconds) and time measuring (minutes and hours).
64 Base 64 is also widely used in computing, using as digits "A–Z", "a–z", "0–9", plus two more characters, often "+" and "/".
256 bytes is used internally by computers, actually grouping eight binary digits together. For reading by humans, bytes are usually shown in hexadecimal.

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).