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the (303), and (123), hex (112), for (82), with (47), from (44), #decimal (43), base (40), number (38), #binary (36), digits (34), system (32), numbers (31), archived (30), hexadecimal (29), used (29), can (29), edit (28), that (28), retrieved (27), original (27), notation (27), representation (25), are (22), digit (22), pdf (21), letters (21), conversion (20), each (18), other (18), value (18), this (17), its (17), bit (17), character (17), use (16), example (16), one (15), have (15), languages (14), computer (14), standard (14), written (14), prefix (14), using (13), common (13), some (13), numeral (12), systems (12), also (12), was (11), symbols (11), than (11), such (11), code (10), ibm (10), programming (10), values (10), octal (10), data (10), when (10), pages (9), two (9), sixteen (9), 159 (9), any (9), not (9), most (9), non (8), new (8), sexadecimal (8), 1968 (8), basic (8), uses (8), point (8), bases (8), wikipedia (7), terms (7), may (7), arithmetic (7), see (7), 2016 (7), 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Text of the page (random words):
a conversion specifiers this notation can be produced by implementations of the printf family of functions following the c99 specification 23 and single unix specification ieee std 1003 1 posix standard 24 verbal representation edit hex finger counting scheme most european languages lack non decimal based words for some of the numerals eleven to fifteen some people read hex numbers digit by digit like a phone number or using the nato phonetic alphabet the joint army navy phonetic alphabet or a similar ad hoc system in the wake of the adoption of hex among ibm system 360 programmers magnuson 1968 25 suggested a pronunciation guide that gave short names to the letters of hex for instance a was pronounced ann b bet c chris etc 25 another naming system was published online by rogers 2007 26 that tries to make the verbal representation distinguishable in any case even when the actual number does not contain numbers a f examples are listed in the tables below yet another naming system was elaborated by babb 2015 based on a joke in silicon valley 27 others have proposed using the verbal morse code conventions to express four bit hex digits with dit and dah representing zero and one respectively so that 0000 is voiced as dit dit dit dit dah dit dit dah voices the digit with a value of nine and dah dah dah dah voices the hex digit for decimal 15 systems of counting on digits have been devised for both binary and hex arthur c clarke suggested using each finger as an on off bit allowing finger counting from zero to 1023 10 on ten fingers 28 another system for counting up to ff 16 255 10 is illustrated on the right magnuson 1968 25 naming method hex name decimal a ann 10 b bet 11 c chris 12 d dot 13 e ernest 14 f frost 15 1a annteen 26 a0 annty 160 5b fifty bet 91 a01c annty christeen 40 988 1ad0 annteen dotty 6 864 3a7d thirty ann seventy dot 14 973 rogers 2007 26 naming method hex name decimal a ten 10 b eleven 11 c twelve 12 d draze 13 e eptwin 14 f fim 15 10 tex 16 11 oneteek 17 1f fimteek 31 50 fiftek 80 c0 twelftek 192 100 hundrek 256 1000 thousek 4 096 3e thirtek eptwin 62 e1 eptek one 225 c4a twelve hundrek fourtek ten 3 146 1743 one thousek seven hundrek fourtek three 5 955 conversion edit binary conversion edit the programmable rpn calculator hp 16c computer scientist from 1982 was designed for programmers one of its key features was the conversion between different numeral systems note hex number in display most computers manipulate binary data but it is difficult for humans to work with a large number of digits for even a relatively small binary number although most humans are familiar with the base 10 system it is much easier to map binary to hex than to decimal because each hex digit maps to a whole number of bits 4 10 this example converts 1111 2 to base ten since each position in a binary numeral can contain either a 1 or a 0 its value may be easily determined by its position from the right 0001 2 1 10 0010 2 2 10 0100 2 4 10 1000 2 8 10 therefore 1111 2 8 10 4 10 2 10 1 10 15 10 with little practice mapping 1111 2 to f 16 in one step becomes easy the advantage of using hex rather than decimal increases rapidly with the size of the number when the number becomes large conversion to decimal is very tedious however when mapping to hex it is trivial to regard the binary string as 4 digit groups and map each to a single hex digit 29 this example shows the conversion of a binary number to decimal mapping each digit to the decimal value and adding the results 1001011100 2 512 10 64 10 16 10 8 10 4 10 604 10 compare this to the conversion to hex where each group of four digits can be considered independently and converted directly 1001011100 2 0010 0101 1100 2 2 5 c 16 25c 16 the conversion from hex to binary is equally direct 29 other simple conversions edit although quaternary base 4 is little used it can easily be converted to and from hex or binary each hex digit corresponds to a pair of quaternary digits and each quaternary digit corresponds to a pair of binary digits in the above example 2 5 c 16 02 11 30 4 the octal base 8 system can also be converted with relative ease although not quite as trivially as with bases 2 and 4 each octal digit corresponds to three binary digits rather than four therefore we can convert between octal and hex via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four division remainder in source base edit as with all bases there is a simple algorithm for converting a representation of a number to hex by doing integer division and remainder operations in the source base in theory this is possible from any base but for most humans only decimal and for most computers only binary which can be converted by far more efficient methods can be easily handled with this method let d be the number to represent in hex and the series h i h i 1 h 2 h 1 be the hex digits representing the number i 1 h i d mod 16 d d h i 16 if d 0 return series h i else increment i and go to step 2 16 may be replaced with any other base that may be desired the following is a javascript implementation of the above algorithm for converting any number to a hex in string representation its purpose is to illustrate the above algorithm to work with data seriously however it is much more advisable to work with bitwise operators function tohex d var r d 16 if d r 0 return tochar r return tohex d r 16 tochar r function tochar n const alpha 0123456789abcdef return alpha charat n conversion through addition and multiplication edit a hex multiplication table it is also possible to make the conversion by assigning each place in the source base the hex representation of its place value before carrying out multiplication and addition to get the final representation for example to convert the number b3ad to decimal one can split the hex number into its digits b 11 10 3 3 10 a 10 10 and d 13 10 and then get the final result by multiplying each decimal representation by 16 p p being the corresponding hex digit position counting from right to left beginning with 0 in this case we have that b3ad 11 16 3 3 16 2 10 16 1 13 16 0 which is 45997 in base 10 tools for conversion edit hexidecimal chart on the control panel of an ibm system 360 many computer systems provide a calculator utility capable of performing conversions between the various radices frequently including hex in microsoft windows the calculator on its programmer mode allows conversions between hex and other common programming bases elementary arithmetic edit elementary operations such as division can be carried out indirectly through conversion to an alternate numeral system such as the commonly used decimal system or the binary system where each hex digit corresponds to four binary digits alternatively one can also perform elementary operations directly within the hex system itself by relying on its addition multiplication tables and its corresponding standard algorithms such as long division and the traditional subtraction algorithm real numbers edit rational numbers edit as with other numeral systems the hex system can be used to represent rational numbers although repeating expansions are common since sixteen 10 16 has only a single prime factor two for any base 0 1 or 1 10 is always equivalent to one divided by the representation of that base value in its own number system thus whether dividing one by two for binary or dividing one by sixteen for hex both of these fractions are written as 0 1 because the radix 16 is a perfect square 4 2 fractions expressed in hex have an odd period much more often than decimal ones and there are no cyclic numbers other than trivial single digits recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix thus when using hex notation all fractions with denominators that are not a power of two result in an infinite string of recurring digits such as thirds and fifths this makes hex and binary less convenient than decimal for representing rational numbers since a larger proportion lies outside its range of finite representation all rational numbers finitely representable in hex are also finitely representable in decimal duodecimal and sexagesimal that is any hex number with a finite number of digits also has a finite number of digits when expressed in those other bases conversely only a fraction of those finitely representable in the latter bases are finitely representable in hex for example decimal 0 1 corresponds to the infinite recurring representation 0 1 9 in hex however hex is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator for example 0 0625 10 one sixteenth is equivalent to 0 1 16 0 09 12 and 0 3 45 60 n decimal prime factors of base b 10 2 5 b 1 9 3 b 1 11 11 hexadecimal prime factors of base b 16 10 10 2 b 1 15 10 f 3 5 b 1 17 10 11 11 reciprocal prime factors positional representation decimal positional representation hex prime factors reciprocal 2 1 2 2 0 5 0 8 2 1 2 3 1 3 3 0 3333 0 3 0 5555 0 5 3 1 3 4 1 4 2 0 25 0 4 2 1 4 5 1 5 5 0 2 0 3 5 1 5 6 1 6 2 3 0 1 6 0 2 a 2 3 1 6 7 1 7 7 0 142857 0 249 7 1 7 8 1 8 2 0 125 0 2 2 1 8 9 1 9 3 0 1 0 1c7 3 1 9 10 1 10 2 5 0 1 0 1 9 2 5 1 a 11 1 11 11 0 09 0 1745d b 1 b 12 1 12 2 3 0 08 3 0 1 5 2 3 1 c 13 1 13 13 0 076923 0 13b d 1 d 14 1 14 2 7 0 0 714285 0 1 249 2 7 1 e 15 1 15 3 5 0 0 6 0 1 3 5 1 f 16 1 16 2 0 0625 0 1 2 1 10 17 1 17 17 0 0588235294117647 0 0f 11 1 11 18 1 18 2 3 0 0 5 0 0 e38 2 3 1 12 19 1 19 19 0 052631578947368421 0 0d79435e5 13 1 13 20 1 20 2 5 0 05 0 0 c 2 5 1 14 21 1 21 3 7 0 047619 0 0c3 3 7 1 15 22 1 22 2 11 0 0 45 0 0 ba2e8 2 b 1 16 23 1 23 23 0 0434782608695652173913 0 0b21642c859 17 1 17 24 1 24 2 3 0 041 6 0 0 a 2 3 1 18 25 1 25 5 0 04 0 0a3d7 5 1 19 26 1 26 2 13 0 0 384615 0 0 9d8 2 d 1 1a 27 1 27 3 0 037 0 097b425ed 3 1 1b 28 1 28 2 7 0 03 571428 0 0 924 2 7 1 1c 29 1 29 29 0 0344827586206896551724137931 0 08d3dcb 1d 1 1d 30 1 30 2 3 5 0 0 3 0 0 8 2 3 5 1 1e 31 1 31 31 0 032258064516129 0 08421 1f 1 1f 32 1 32 2 0 03125 0 08 2 1 20 33 1 33 3 11 0 03 0 07c1f 3 b 1 21 34 1 34 2 17 0 0 2941176470588235 0 0 78 2 11 1 22 35 1 35 5 7 0 0 285714 0 075 5 7 1 23 36 1 36 2 3 0 02 7 0 0 71c 2 3 1 24 37 1 37 37 0 027 0 06eb3e453 25 1 25 38 1 38 2 19 0 0 2631578947368421 0 0 435e50d79435e4ac62b4 2 13 1 26 39 1 39 3 13 0 0 256410 0 0 69 3 d 1 27 40 1 40 2 5 0 025 0 0 6 2 5 1 28 41 1 41 41 0 0 2439 0 0 63e9538d283b5b62fb8 29 1 29 42 1 42 2 3 7 0 0 238095 0 0 618 2 3 7 1 2a 43 1 43 43 0 0 23255813953488372093 0 0 5f417d 2b 1 2b 44 1 44 2 11 0 02 27 0 05 d1745 2 b 1 2c 45 1 45 3 5 0 0 2 0 05b 3 5 1 2d 46 1 46 2 23 0 0 2173913043478765869567 0 0 590b21642c9c4ef44a9 2 17 1 2e 47 1 47 47 0 0 212765957446808510638297872340425531914893617 0 0 572620ae4c415c9882b931 2f 1 2f 48 1 48 2 3 0 0208 3 0 0 5 2 3 1 30 irrational numbers edit the table below gives the expansions of some common irrational numbers in decimal and hex number positional representation decimal hex 2 the length of the diagonal of a unit square 1 414 213 562 373 095 048 1 6a09e667f3bcd 3 the length of the diagonal of a unit cube 1 732 050 807 568 877 293 1 bb67ae8584caa 5 the length of the diagonal of a 1 2 rectangle 2 236 067 977 499 789 696 2 3c6ef372fe95 φ phi the golden ratio 1 5 2 1 618 033 988 749 894 848 1 9e3779b97f4a π pi the ratio of circumference to diameter of a circle 3 141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 3 243f6a8885a308d313198a2e0 3707344a4093822299f31d008 e the base of the natural logarithm 2 718 281 828 459 045 235 2 b7e151628aed2a6b τ the thue morse constant 0 412 454 033 640 107 597 0 6996 9669 9669 6996 γ the limiting difference between the harmonic series and the natural logarithm 0 577 215 664 901 532 860 0 93c467e37db0c7a4d1b powers edit the first 16 powers of 2 are below as hex to show relative simplicity compared to decimal representation 2 x hex decimal 2 0 1 1 2 1 2 2 2 2 4 4 2 3 8 8 2 4 10 16 2 5 20 32 2 6 40 64 2 7 80 128 2 8 100 256 2 9 200 512 2 10 400 1 024 2 11 800 2 048 2 12 1 000 4 096 2 13 2 000 8 192 2 14 4 000 16 384 2 15 8 000 32 768 2 16 10 000 65 536 cultural history edit the traditional chinese units of measurement were base 16 for example one jīn 斤 in the old system equals sixteen taels the suanpan chinese abacus can be used to perform hex calculations such as additions and subtractions 30 as with the duodecimal system there have been occasional attempts to promote hex as the preferred numeral system these attempts often propose specific pronunciation and symbols for the individual numerals 31 some proposals unify standard measures so that they are multiples of 16 32 33 an early such proposal was put forward by john w nystrom in project of a new system of arithmetic weight measure and coins proposed to be called the tonal system with sixteen to the base published in 1862 34 nystrom among other things suggested hexadecimal time which subdivides a day by 16 so that there are 16 hours or 10 tims pronounced tontim in a day 35 look up hexadecimal in wiktionary the free dictionary the word hexadecimal is first recorded in 1952 36 it is macaronic in the sense that it combines greek ἕξ hex six with latinate decimal the all latin alternative sexadecimal compare the word sexagesimal for base 60 is older and sees at least occasional use from the late 19th century 37 it is still in use in the 1950s in bendix documentation schwartzman 1994 argues that use of sexadecimal may have been avoided because of its suggestive abbreviation to sex 38 many western languages since the 1960s have adopted terms equivalent in formation to hexadecimal e g french hexadécimal italian esadecimale romanian hexazecimal serbian хексадецимални etc but others have introduced terms which substitute native words for sixteen e g greek δεκαεξαδικός icelandic sextándakerfi russian шестнадцатеричной etc terminology and notation did not become settled until the end of the 1960s in 1969 donald knuth argued that the etymologically correct term would be senidenary or possibly sedenary a latinate term intended to convey grouped by 16 modelled on binary ternary quaternary etc according to knuth s argument the correct terms for decimal and octal arithmetic would be denary and octonary respectively 39 alfred b taylor used senidenary in his mid 1800s work on alternative number bases although he rejected base 16 because of its incommodious number of digits 40 41 the now current notation using the letters a to f establishes itself as the de facto standard beginning in 1966 in the wake of the publication of the fortran iv manual for ibm system 360 which unlike earlier variants of fortran recognizes a standard for entering hexadecimal constants 42 as noted above alternative notations were used by nec 1960 and the pacific data systems 1020 1964 the standard adopted by ibm seems to have become widely adopted by 1968 when bru...