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.

== Table of bases ==
Different bases are especially used in connection with computers.
The commonly used bases are 10 ([[decimal]]), 2 ([[Binary number|binary]]), 8 ([[octal]]), and 16 ([[hexadecimal]]).
A [[byte]] with 8 [[bit]]s can represent values from 0 to 255, often expressed with [[leading zero]]s in base 2, 8 or 16 to give the same length.<ref>{{cite web |title=Conversion Table – Decimal, Hexidecimal, Octol, Binary |url=https://www.securitywizardry.com/packets/pdf/Conversion_Table.pdf |website=SecurityWizardry.com |accessdate=7 April 2025}}</ref>

The first row in the tables is the base written in decimal.
<div style=display:inline-grid>
{| class="wikitable"
|+ 0–15
! 10 !! 2 !! 8 !! 16
|-
!0
|00000000||000||00
|-
!1
|00000001||001||01
|-
!2
|00000010||002||02
|-
!3
|00000011||003||03
|-
!4
|00000100||004||04
|-
!5
|00000101||005||05
|-
!6
|00000110||006||06
|-
!7
|00000111||007||07
|-
!8
|00001000||010||08
|-
!9
|00001001||011||09
|-
!10
|00001010||012||0a
|-
!11
|00001011||013||0b
|-
!12
|00001100||014||0c
|-
!13
|00001101||015||0d
|-
!14
|00001110||016||0e
|-
!15
|00001111||017||0f
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 16–31
! 10 !! 2 !! 8 !! 16
|-
!16
|00010000||020||10
|-
!17
|00010001||021||11
|-
!18
|00010010||022||12
|-
!19
|00010011||023||13
|-
!20
|00010100||024||14
|-
!21
|00010101||025||15
|-
!22
|00010110||026||16
|-
!23
|00010111||027||17
|-
!24
|00011000||030||18
|-
!25
|00011001||031||19
|-
!26
|00011010||032||1a
|-
!27
|00011011||033||1b
|-
!28
|00011100||034||1c
|-
!29
|00011101||035||1d
|-
!30
|00011110||036||1e
|-
!31
|00011111||037||1f
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 32–47
! 10 !! 2 !! 8 !! 16
|-
!32
|00100000||040||20
|-
!33
|00100001||041||21
|-
!34
|00100010||042||22
|-
!35
|00100011||043||23
|-
!36
|00100100||044||24
|-
!37
|00100101||045||25
|-
!38
|00100110||046||26
|-
!39
|00100111||047||27
|-
!40
|00101000||050||28
|-
!41
|00101001||051||29
|-
!42
|00101010||052||2a
|-
!43
|00101011||053||2b
|-
!44
|00101100||054||2c
|-
!45
|00101101||055||2d
|-
!46
|00101110||056||2e
|-
!47
|00101111||057||2f
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 48–63
! 10 !! 2 !! 8 !! 16
|-
!48
|00110000||060||30
|-
!49
|00110001||061||31
|-
!50
|00110010||062||32
|-
!51
|00110011||063||33
|-
!52
|00110100||064||34
|-
!53
|00110101||065||35
|-
!54
|00110110||066||36
|-
!55
|00110111||067||37
|-
!56
|00111000||070||38
|-
!57
|00111001||071||39
|-
!58
|00111010||072||3a
|-
!59
|00111011||073||3b
|-
!60
|00111100||074||3c
|-
!61
|00111101||075||3d
|-
!62
|00111110||076||3e
|-
!63
|00111111||077||3f
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 64–79
! 10 !! 2 !! 8 !! 16
|-
!64
|01000000||100||40
|-
!65
|01000001||101||41
|-
!66
|01000010||102||42
|-
!67
|01000011||103||43
|-
!68
|01000100||104||44
|-
!69
|01000101||105||45
|-
!70
|01000110||106||46
|-
!71
|01000111||107||47
|-
!72
|01001000||110||48
|-
!73
|01001001||111||49
|-
!74
|01001010||112||4a
|-
!75
|01001011||113||4b
|-
!76
|01001100||114||4c
|-
!77
|01001101||115||4d
|-
!78
|01001110||116||4e
|-
!79
|01001111||117||4f
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 80–95
! 10 !! 2 !! 8 !! 16
|-
!80
|01010000||120||50
|-
!81
|01010001||121||51
|-
!82
|01010010||122||52
|-
!83
|01010011||123||53
|-
!84
|01010100||124||54
|-
!85
|01010101||125||55
|-
!86
|01010110||126||56
|-
!87
|01010111||127||57
|-
!88
|01011000||130||58
|-
!89
|01011001||131||59
|-
!90
|01011010||132||5a
|-
!91
|01011011||133||5b
|-
!92
|01011100||134||5c
|-
!93
|01011101||135||5d
|-
!94
|01011110||136||5e
|-
!95
|01011111||137||5f
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 96–111
! 10 !! 2 !! 8 !! 16
|-
!96
|01100000||140||60
|-
!97
|01100001||141||61
|-
!98
|01100010||142||62
|-
!99
|01100011||143||63
|-
!100
|01100100||144||64
|-
!101
|01100101||145||65
|-
!102
|01100110||146||66
|-
!103
|01100111||147||67
|-
!104
|01101000||150||68
|-
!105
|01101001||151||69
|-
!106
|01101010||152||6a
|-
!107
|01101011||153||6b
|-
!108
|01101100||154||6c
|-
!109
|01101101||155||6d
|-
!110
|01101110||156||6e
|-
!111
|01101111||157||6f
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 112–127
! 10 !! 2 !! 8 !! 16
|-
!112
|01110000||160||70
|-
!113
|01110001||161||71
|-
!114
|01110010||162||72
|-
!115
|01110011||163||73
|-
!116
|01110100||164||74
|-
!117
|01110101||165||75
|-
!118
|01110110||166||76
|-
!119
|01110111||167||77
|-
!120
|01111000||170||78
|-
!121
|01111001||171||79
|-
!122
|01111010||172||7a
|-
!123
|01111011||173||7b
|-
!124
|01111100||174||7c
|-
!125
|01111101||175||7d
|-
!126
|01111110||176||7e
|-
!127
|01111111||177||7f
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 128–143
! 10 !! 2 !! 8 !! 16
|-
!128
|10000000||200||80
|-
!129
|10000001||201||81
|-
!130
|10000010||202||82
|-
!131
|10000011||203||83
|-
!132
|10000100||204||84
|-
!133
|10000101||205||85
|-
!134
|10000110||206||86
|-
!135
|10000111||207||87
|-
!136
|10001000||210||88
|-
!137
|10001001||211||89
|-
!138
|10001010||212||8a
|-
!139
|10001011||213||8b
|-
!140
|10001100||214||8c
|-
!141
|10001101||215||8d
|-
!142
|10001110||216||8e
|-
!143
|10001111||217||8f
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 144–159
! 10 !! 2 !! 8 !! 16
|-
!144
|10010000||220||90
|-
!145
|10010001||221||91
|-
!146
|10010010||222||92
|-
!147
|10010011||223||93
|-
!148
|10010100||224||94
|-
!149
|10010101||225||95
|-
!150
|10010110||226||96
|-
!151
|10010111||227||97
|-
!152
|10011000||230||98
|-
!153
|10011001||231||99
|-
!154
|10011010||232||9a
|-
!155
|10011011||233||9b
|-
!156
|10011100||234||9c
|-
!157
|10011101||235||9d
|-
!158
|10011110||236||9e
|-
!159
|10011111||237||9f
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 160–175
! 10 !! 2 !! 8 !! 16
|-
!160
|10100000||240||a0
|-
!161
|10100001||241||a1
|-
!162
|10100010||242||a2
|-
!163
|10100011||243||a3
|-
!164
|10100100||244||a4
|-
!165
|10100101||245||a5
|-
!166
|10100110||246||a6
|-
!167
|10100111||247||a7
|-
!168
|10101000||250||a8
|-
!169
|10101001||251||a9
|-
!170
|10101010||252||aa
|-
!171
|10101011||253||ab
|-
!172
|10101100||254||ac
|-
!173
|10101101||255||ad
|-
!174
|10101110||256||ae
|-
!175
|10101111||257||af
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 176–191
! 10 !! 2 !! 8 !! 16
|-
!176
|10110000||260||b0
|-
!177
|10110001||261||b1
|-
!178
|10110010||262||b2
|-
!179
|10110011||263||b3
|-
!180
|10110100||264||b4
|-
!181
|10110101||265||b5
|-
!182
|10110110||266||b6
|-
!183
|10110111||267||b7
|-
!184
|10111000||270||b8
|-
!185
|10111001||271||b9
|-
!186
|10111010||272||ba
|-
!187
|10111011||273||bb
|-
!188
|10111100||274||bc
|-
!189
|10111101||275||bd
|-
!190
|10111110||276||be
|-
!191
|10111111||277||bf
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 192–207
! 10 !! 2 !! 8 !! 16
|-
!192
|11000000||300||c0
|-
!193
|11000001||301||c1
|-
!194
|11000010||302||c2
|-
!195
|11000011||303||c3
|-
!196
|11000100||304||c4
|-
!197
|11000101||305||c5
|-
!198
|11000110||306||c6
|-
!199
|11000111||307||c7
|-
!200
|11001000||310||c8
|-
!201
|11001001||311||c9
|-
!202
|11001010||312||ca
|-
!203
|11001011||313||cb
|-
!204
|11001100||314||cc
|-
!205
|11001101||315||cd
|-
!206
|11001110||316||ce
|-
!207
|11001111||317||cf
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 208–223
! 10 !! 2 !! 8 !! 16
|-
!208
|11010000||320||d0
|-
!209
|11010001||321||d1
|-
!210
|11010010||322||d2
|-
!211
|11010011||323||d3
|-
!212
|11010100||324||d4
|-
!213
|11010101||325||d5
|-
!214
|11010110||326||d6
|-
!215
|11010111||327||d7
|-
!216
|11011000||330||d8
|-
!217
|11011001||331||d9
|-
!218
|11011010||332||da
|-
!219
|11011011||333||db
|-
!220
|11011100||334||dc
|-
!221
|11011101||335||dd
|-
!222
|11011110||336||de
|-
!223
|11011111||337||df
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 224–239
! 10 !! 2 !! 8 !! 16
|-
!224
|11100000||340||e0
|-
!225
|11100001||341||e1
|-
!226
|11100010||342||e2
|-
!227
|11100011||343||e3
|-
!228
|11100100||344||e4
|-
!229
|11100101||345||e5
|-
!230
|11100110||346||e6
|-
!231
|11100111||347||e7
|-
!232
|11101000||350||e8
|-
!233
|11101001||351||e9
|-
!234
|11101010||352||ea
|-
!235
|11101011||353||eb
|-
!236
|11101100||354||ec
|-
!237
|11101101||355||ed
|-
!238
|11101110||356||ee
|-
!239
|11101111||357||ef
|}
</div>
<div style=display:inline-grid>
{| class="wikitable"
|+ 240–255
! 10 !! 2 !! 8 !! 16
|-
!240
|11110000||360||f0
|-
!241
|11110001||361||f1
|-
!242
|11110010||362||f2
|-
!243
|11110011||363||f3
|-
!244
|11110100||364||f4
|-
!245
|11110101||365||f5
|-
!246
|11110110||366||f6
|-
!247
|11110111||367||f7
|-
!248
|11111000||370||f8
|-
!249
|11111001||371||f9
|-
!250
|11111010||372||fa
|-
!251
|11111011||373||fb
|-
!252
|11111100||374||fc
|-
!253
|11111101||375||fd
|-
!254
|11111110||376||fe
|-
!255
|11111111||377||ff
|}
</div>

==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]]