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Number of digits of a numeral system
In a positional numeral system, the radix (pl.: 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 (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.[1]
Radix is a Latin word for "root". Root can be considered a synonym for base, in the arithmetical sense.
In numeral systems[edit]Generally, in a system with radix b (b > 1), a string of digits d1 ... dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, where 0 ≤ di < b.[1] 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.[2]
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 5 × 122 + 9 × 121 + 10 × 120 = 838 in base 10.
Commonly used numeral systems include:
Base/radix Name Description 2 Binary numeral system 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 Octal system Used occasionally in computing. The eight digits are "0"–"7" and represent 3 bits (23). 10 Decimal system Used by humans in the wide majority of cultures. Its ten digits are "0"–"9". Used in most mechanical counters. 12 Duodecimal (dozenal) system 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 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 system 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 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 Sexagesimal system Originally used in modified form in ancient Sumer and passed to the Babylonians.[3] 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.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 11110002. 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
Radices are usually natural numbers. However, other positional systems are possible, for example, golden ratio base (whose radix is a non-integer algebraic number),[5] and negative base (whose radix is negative).[6] 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 1 × (−10)1 + 9 × (−10)0 = −1.
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.[7]
The first row in the tables is the base written in decimal.
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 ffLook up
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