80.180.112.41 at 08:16, 8 August 2011

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{{Table Numeral Systems}}

In [[numeral system|mathematical numeral systems]], the '''radix''' or '''base''' is the number of unique [[numerical digit|digits]], including zero, that a [[positional notation|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 numeral system|unary]], where radix is 1), the base will always be written as <math>(x)_y</math>. For example, <math>(10)_{10} </math>(in the decimal system) represents the number ten(in English and most natural languages, decimal is assumed); whilst <math>(10)_2 </math>(in the [[binary system (numeral)|binary system]]) represents the number two.<ref group===External links==>{{cite web|url=http://www.wolframalpha.com/input/?i=10%E2%82%82+in+decimal|accessdate=24 January 2014}}</ref>

== 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 <math>3 \times 13^2 + 9 \times 13^1 + 8 \times 13^0</math>.

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

Commonly used numeral systems include:
{| class="wikitable sortable"
! Base
! Name
! Description
|-
| 10
| [[decimal|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 [[counter]]s.
|-
| 12
| [[duodecimal|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 [[dozen]]s and [[Gross (unit)|grosses]].
|-
| 2
| [[binary numeral system]]
| used internally by nearly all [[computer]]s, is [[base two]]. The two digits are "0" and "1", expressed from switches displaying OFF and ON respectively. Used in most electric [[counter]]s.
|-
| 16
| [[hexadecimal|hexadecimal system]]
| is often used in computing. The sixteen digits are "0–9" followed by "A–F".
|-
| 8
| [[octal|octal system]]
| is occasionally used in computing. The eight digits are "0–7".
|-
| 60
| [[sexagesimal|sexagesimal system]]
| originated in ancient [[Sumer]]ia and passed to the [[Babylonia]]ns. It is still used as the basis of our modern [[polar coordinate system|circular coordinate system]] (degrees, minutes, and seconds) and [[time]] measuring (hours, minutes, and seconds).
|-
| 64
| [[base64|Base 64]]
| is also used in computing, using as digits "A–Z", "a–z", "0–9", plus two more characters, often "+" and "/".{{citationneeded|date=December 2013}}
|-
| 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.{{citationneeded|date=December 2013}}
|}

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 number]]s. 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==
*[[Base (exponentiation)]]
*[[List of numeral systems]]
*[[Non-standard positional numeral systems]]
*[[Radix economy]]

==External links==
{{reflist}}
{{wiktionary|radix}}
*[http://baseconvert.com Base Convert, a floating-point base calculator]
*[http://mathworld.wolfram.com/Base.html MathWorld entry on base]

[[Category:Elementary mathematics]]
[[Category:Numeral systems]]

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80.180.112.41 at 08:16, 8 August 2011