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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 (x)_{y}. For example, (10)_{{10}}(in the decimal system) represents the number ten(in English and most natural languages, decimal is assumed); whilst (10)_{2}(in the binary system) represents the number two.[1]


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

In numeral systems[edit]

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:

10decimal systemthe most used system of numbers in the world, is used in arithmetic. Its ten digits are "0–9". Used in most mechanical counters.
12duodecimal (dozenal) systemis often used due to divisibility by 2, 3, 4 and 6. It was traditionally used as part of quantities expressed in dozens and grosses.
2binary numeral systemused 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.
16hexadecimal systemis often used in computing. The sixteen digits are "0–9" followed by "A–F".
8octal systemis occasionally used in computing. The eight digits are "0–7".
60sexagesimal systemoriginated 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 (hours, minutes, and seconds).
64Base 64is also used in computing, using as digits "A–Z", "a–z", "0–9", plus two more characters, often "+" and "/".[citation needed]
256byteis 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]

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 -- radix economy.

See also[edit]

External links[edit]

  1. ^ http://www.wolframalpha.com/input/?i=10%E2%82%82+in+decimal. Retrieved 24 January 2014.  Missing or empty |title= (help)