Radix#Conversion among bases
{{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 digits, including the digit zero, used 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 standard positional numeral system, a number is conventionally written as {{nowrap|(x)y}} with x as the 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. For example, (100)10 is equivalent to 100 (the decimal system is implied in the latter) and represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four.
Etymology
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|d1 ... dn}} denotes the number {{nowrap|d1bn−1 + d2bn−2 + ... + dnb0}}, where {{nowrap|0 ≤ di < b}}.
{{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
}} 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 b1s' place, a b2s' place, etc.{{Cite web|url=https://experimonkey.com/facts/computer-science/binary|title=Binary|website=experimonkey.com|access-date=2023-05-14}}
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 × 122 + 9 × 121 + 10 × 120}} = 838 in base 10.
Commonly used numeral systems include:
| class="wikitable sortable"
! Base/radix ! Name ! Description |
| 2
| Used internally by nearly all computers. The two digits are "0" and "1", expressed from switches displaying OFF and ON, respectively. Used in most electric counters. |
| 8
| Used occasionally in computing. The eight digits are "0"–"7" and represent 3 bits (23). |
| 10
| Used by humans in the wide majority of cultures. Its ten digits are "0"–"9". Used in most mechanical counters. |
| 12
| Sometimes advocated due to divisibility by 2, 3, 4, and 6. It was traditionally used as part of quantities expressed in dozens and grosses. |
| 16
| 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
| Traditional numeral system in several cultures, still used by some for counting. Historically also known as the score system in English, now most famous in the phrase "four score and seven years ago" in the Gettysburg Address. |
| 36
|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 letters A–Z (the ISO basic Latin alphabet). Each base36 digit needs less than 6 bits of information to be represented. |
| 60
| Originally used in modified form in ancient Sumer and passed to the Babylonians. {{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 }} Used today as the basis of modern 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 7816 is binary {{gaps|111|1000}}2. 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
:
where m is a nonnegative integer and the r's are integers such that
:0 < rm < b and 0 ≤ ri < b for i = 0, 1, ... , m − 1.{{harvtxt|McCoy|1968|p=75}}
Radices are usually natural numbers. However, other positional systems are possible, for example, golden ratio base (whose radix is a non-integer algebraic number),
{{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
}} and negative base (whose radix is negative).
{{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
}}
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)1 + 9 × (−10)0}} = −1.
Table of bases
Different bases are especially used in connection with computers.
The commonly used bases are 10 (decimal), 2 (binary), 8 (octal), and 16 (hexadecimal).
A byte with 8 bits can represent values from 0 to 255, often expressed with leading zeros in base 2, 8 or 16 to give the same length.{{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}}
The first row in the tables is the base written in decimal.
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
See also
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]