View source for Radix

{{Short description|Number of digits of a numeral system}}
{{other uses}}
{{Numeral systems|expand=Place-value notation|expand2=By radix/base}}

In a [[positional numeral system]], the '''radix''' ({{plural form}}:{{nbs}}'''radices''') or '''base''' is the number of unique [[numerical digit|digits]], including the digit zero, used to represent numbers. For example, for the [[decimal|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 standard positional numeral system, a number is conventionally written as {{nowrap|(''x'')<sub>''y''</sub>}} with ''x'' as the [[String (computer science)|string]] of digits and ''y'' as its base.  For base ten, the subscript is usually assumed and omitted (together with the enclosing [[parentheses]]), as it is the most common way to express [[value (mathematics)|value]]. For example, <span class="nowrap">(100)<sub>10</sub> is equivalent to  100</span> (the decimal system is implied in the latter) and represents the number one hundred, while (100)<sub>2</sub> (in the [[binary system (numeral)|binary system]] with base 2) represents the number four.<ref name="morris_mano_p13_14"/>

== Etymology ==
[[:wikt:radix|''Radix'']] is a Latin word for "root". ''Root'' can be considered a synonym for ''base,'' in the arithmetical sense.

== In numeral systems ==
Generally, in a system with radix ''b'' ({{nowrap|''b'' > 1}}), a string of digits {{nowrap|''d''<sub>1</sub> ... ''d<sub>n</sub>''}} denotes the number {{nowrap|''d''<sub>1</sub>''b''<sup>''n''−1</sup> + ''d''<sub>2</sub>''b''<sup>''n''−2</sup> + … + ''d<sub>n</sub>b''<sup>0</sup>}}, where {{nowrap|0 ≤ ''d<sub>i</sub>'' < ''b''}}.<ref name="morris_mano_p13_14">
{{cite book
 | first1=M. Morris | last1=Mano
 | first2=Charles | last2=Kime
 | title=Logic and Computer Design Fundamentals
 | date=2014
 | publisher=Pearson
 | location=Harlow
 | isbn=978-1-292-02468-4
 | pages=13–14 | edition=4th
}}</ref> In contrast to decimal, or radix 10, which has a ones' place, tens' place, hundreds' place, and so on, radix ''b'' would have a ones' place, then a ''b''<sup>1</sup>s' place, a ''b''<sup>2</sup>s' place, etc.<ref>{{Cite web|url=https://experimonkey.com/facts/computer-science/binary|title=Binary|website=experimonkey.com|access-date=2023-05-14}}</ref>

For example, if ''b'' = 12, a string of digits such as 59A (where the letter "A" represents the value of ten) would represent the value {{nowrap|''5'' × ''12''<sup>''2''</sup> + ''9'' × ''12''<sup>''1''</sup> + ''10'' × ''12''<sup>''0''</sup>}} = 838 in base 10.

Commonly used numeral systems include:
{| class="wikitable sortable"
! Base/radix
! Name
! Description
|-
| 2
| [[Binary numeral system]]
| Used internally by nearly all [[computer]]s. The two digits are "0" and "1", expressed from switches displaying OFF and ON, respectively. Used in most electric [[counter (digital)|counter]]s.
|-
| 8
| [[Octal|Octal system]]
| Used occasionally in computing. The eight digits are "0"–"7" and represent 3 bits (2<sup>3</sup>).
|-
| 10 
| [[Decimal|Decimal system]]
| Used by humans in the wide majority of cultures. Its ten digits are "0"–"9". Used in most [[mechanical counter]]s.
|-
| 12
| [[Duodecimal|Duodecimal (dozenal) system]]
| Sometimes advocated 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]].
|-
| 16
| [[Hexadecimal|Hexadecimal system]]
| Often used in computing as a more compact representation of binary (1 hex digit per 4 bits). The sixteen digits are "0"–"9" followed by "A"–"F" or "a"–"f".
|-
| 20
| [[Vigesimal|Vigesimal system]]
| Traditional numeral system in several cultures, still used by some for counting. Historically also known as the ''[[score (number)|score]] system'' in English, now most famous in the phrase "four score and seven years ago" in the [[Gettysburg Address]].
|-
|36
|[[Base36]]
|'''Base36''' is a [[binary-to-text encoding]] scheme that represents [[binary data]] in an [[ASCII]] string format by translating it into a [[radix]]-36 representation. The choice of 36 is convenient in that the digits can be represented using the [[Arabic numerals]] 0–9 and the [[Latin alphabet|Latin letters]] A–Z (the [[ISO basic Latin alphabet]]). Each base36 digit needs less than 6 bits of information to be represented.
|-
| 60
| [[Sexagesimal|Sexagesimal system]]
| Originally used in modified form in ancient [[Sumer]] and passed to the [[Babylonia]]ns.<ref>
 {{cite book
  | last1=Bertman | first1=Stephen
  | title=Handbook to Life in Ancient Mesopotamia
  | date=2005|publisher=Oxford Univ. Press
  | location=Oxford [u.a.]
  | isbn=978-019-518364-1
  | page=257
  | edition=Paperback
  | url=https://books.google.com/books?id=1C4NKp4zgIQC&pg=PA257
 }}</ref> Used today as the basis of modern [[minute of arc#Symbols and abbreviations|circular coordinate system]] (degrees, minutes, and seconds) and [[time]] measuring (minutes, and seconds) by analogy to the rotation of the Earth.
|}

{{for|a larger list|List of numeral systems}}

The octal and hexadecimal systems are often used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, since sixteen is the fourth power of two; for example, hexadecimal 78<sub>16</sub> is binary {{gaps|111|1000}}<sub>2</sub>. Similarly, every octal digit corresponds to a unique sequence of three binary digits, since eight is the cube of two.

This representation is unique.  Let ''b'' be a positive integer greater than 1.  Then every positive integer ''a'' can be expressed uniquely in the form

:<math>a = r_m b^m + r_{m-1} b^{m-1} + \dotsb + r_1 b + r_0,</math>

where ''m'' is a nonnegative integer and the ''r'''s are integers such that

:0 < ''r''<sub>''m''</sub> < ''b'' and 0 ≤ ''r''<sub>''i''</sub> < ''b'' for ''i'' = 0, 1, ... , ''m'' − 1.<ref>{{harvtxt|McCoy|1968|p=75}}</ref>

Radices are usually [[natural number]]s. However, other positional systems are possible, for example, [[golden ratio base]] (whose radix is a non-integer [[algebraic number]]),<ref>
{{cite journal
 | doi=10.2307/3029218
 | last=Bergman | first=George
 | title=A Number System with an Irrational Base
 | journal=Mathematics Magazine
 | volume=31 | issue=2 | pages=98–110 | year=1957
 | jstor=3029218
 }}</ref> and [[negative base]] (whose radix is negative).<ref>
{{cite journal
 | author1=William J. Gilbert
 | title=Negative Based Number Systems
 | journal=Mathematics Magazine
 | date=September 1979 | volume=52 | issue=4 | pages=240–244
 | url=https://www.math.uwaterloo.ca/~wgilbert/Research/GilbertNegBases.pdf|access-date=7 February 2015
| doi=10.1080/0025570X.1979.11976792
 }}</ref>
A negative base allows the representation of negative numbers without the use of a minus sign.  For example, let ''b'' = −10. Then a string of digits such as 19 denotes the (decimal) number {{nowrap|1 × (−10)<sup>1</sup> + 9 × (−10)<sup>0</sup>}} = −1.

==See also==
* [[Floating-point arithmetic]]
* [[Mixed radix]]
*[[Polynomial]]
*[[Radix economy]]
*[[Radix sort]]
*[[Non-standard positional numeral systems]]
*[[List of numeral systems]]

== Notes ==
{{reflist|30em}}

== References ==
* {{ citation | last1 = McCoy | first1 = Neal H. | title = Introduction To Modern Algebra, Revised Edition | location = Boston | publisher = [[Allyn and Bacon]] | year = 1968 | lccn = 68015225 }}

==External links==
{{wiktionary|radix}}
*[http://mathworld.wolfram.com/Base.html MathWorld entry on base]

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