Numeral systems by culture  

HinduArabic numerals  
Eastern Arabic Indian family Khmer 
Mongolian Thai Western Arabic 
East Asian numerals  
Chinese Counting rods Japanese 
Korean Suzhou Vietnamese 
Alphabetic numerals  
Abjad Armenian Āryabhaṭa Cyrillic 
Ge'ez Greek (Ionian) Hebrew 
Other systems  
Attic Babylonian Brahmi Egyptian Etruscan Inuit 
Mayan Quipu Roman Urnfield 
List of numeral system topics  
Positional systems by base  
Decimal (10)  
1, 2, 3, 4, 5, 8, 12, 16, 20, 60 more…  
In mathematics and computer science, hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F (or alternatively a through f) to represent values ten to fifteen. For example, the hexadecimal number 2AF3 is equal, in decimal, to (2 × 16^{3}) + (10 × 16^{2}) + (15 × 16) + 3, or 10,995.
Each hexadecimal digit represents four binary digits (bits) (also called a "nibble"), and the primary use of hexadecimal notation is as a humanfriendly representation of binary coded values in computing and digital electronics. For example, byte values can range from 0 to 255 (decimal) but may be more conveniently represented as two hexadecimal digits in the range 00 through FF. Hexadecimal is also commonly used to represent computer memory addresses.
Contents 
0_{hex}  =  0_{dec}  =  0_{oct}  0  0  0  0  
1_{hex}  =  1_{dec}  =  1_{oct}  0  0  0  1  
2_{hex}  =  2_{dec}  =  2_{oct}  0  0  1  0  
3_{hex}  =  3_{dec}  =  3_{oct}  0  0  1  1  
4_{hex}  =  4_{dec}  =  4_{oct}  0  1  0  0  
5_{hex}  =  5_{dec}  =  5_{oct}  0  1  0  1  
6_{hex}  =  6_{dec}  =  6_{oct}  0  1  1  0  
7_{hex}  =  7_{dec}  =  7_{oct}  0  1  1  1  
8_{hex}  =  8_{dec}  =  10_{oct}  1  0  0  0  
9_{hex}  =  9_{dec}  =  11_{oct}  1  0  0  1  
A_{hex}  =  10_{dec}  =  12_{oct}  1  0  1  0  
B_{hex}  =  11_{dec}  =  13_{oct}  1  0  1  1  
C_{hex}  =  12_{dec}  =  14_{oct}  1  1  0  0  
D_{hex}  =  13_{dec}  =  15_{oct}  1  1  0  1  
E_{hex}  =  14_{dec}  =  16_{oct}  1  1  1  0  
F_{hex}  =  15_{dec}  =  17_{oct}  1  1  1  1  
In situations where there is no context, a hexadecimal number might be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give the base explicitly: 159_{10} is decimal 159; 159_{16} is hexadecimal 159, which is equal to 345_{10}. Other authors prefer a text subscript, such as 159_{decimal} and 159_{hex}, or 159_{d} and 159_{h}.
In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:
%
: http://www.example.com/name%20with%20spaces
where %20
is the space (blank) character (code value 20 in hex, 32 in decimal).ode;
, where code is the 1 to 6digit hex number assigned to the character in the Unicode standard. Thus ’
represents the curled right single quote (Unicode value 2019 in hex, 8217 in decimal).#
: white, for example, is represented #FFFFFF
.^{[1]} CSS allows 3hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange).0x
for numeric constants represented in hex: 0x5A3
. Character and string constants may express character codes in hexadecimal with the prefix \x
followed by two hex digits: '\x1B'
represents the Esc control character; "\x1B[0m\x1B[25;1H"
is a string containing 11 characters (plus a trailing NUL to mark the end of the string) with two embedded Esc characters.^{[3]} To output an integer as hexadecimal with the printf function family, the format conversion code %X
or %x
is used.U+
followed by the hex value: U+20AC
is the Euro sign (€).=
, as in Espa=F1a
to send "España" (Spain). (Hexadecimal F1, equal to decimal 241, is the code number for the lower case n with tilda in the ISO/IEC 88591 character set.)FFh
or 05A3H
. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit: 0FFh
$
as a prefix: $5A3
.H'ABCD'
(for ABCD_{16}).16#5A3#
. VHDL also supports x"5A3"
.8'hFF
, where 8 is the number of bits in the value and FF is the hexadecimal constant.#05A3
16r
: 16r5A3
16#
: 16#5A3
. Binary data (such as image pixels) can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC
...#x
and #16r
.&H
: &H5A3
&
for hex.^{[4]}0h
prefix: 0h5A3
X'5A3'
are sometimes seen, such as in PL/I. This is the most common format for hexadecimal on IBM mainframes (zSeries) and midrange computers (iSeries) running traditional OS's (zOS, zVSE, zVM, TPF, OS/400), and is used in Assembler, PL/1, Cobol, JCL, scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well.:
). This, for example, is a valid IPv6 address: 2001:0db8:85a3:0000:0000:8a2e:0370:7334There is no universal convention to use lowercase or uppercase for the letter digits, and each is prevalent or preferred in particular environments by community standards or convention.
The choice of the letters A through F to represent the digits above nine was not universal in the early history of computers. During the 1950s, some installations favored using the digits 0 through 5 with a macron character ("¯") to indicate the values 10–15. Users of Bendix G15 computers used the letters U through Z. Bruce A. Martin of Brookhaven National Laboratory considered the choice of A–F "ridiculous" and in a 1968 letter to the editor of the CACM proposed an entirely new set of symbols based on the bit locations, which did not gain much acceptance.^{[6]}
There are no traditional numerals to represent the quantities from ten to fifteen—letters are used as a substitute—and most Western European languages lack nondecimal names for the numerals above ten. Even though English has names for several nondecimal powers (pair for the first binary power, score for the first vigesimal power, dozen, gross, and great gross for the first three duodecimal powers), no English name describes the hexadecimal powers (decimal 16, 256, 4096, 65536, ... ). Some people read hexadecimal numbers digit by digit like a phone number: 4DA is "fourdeeay". However, the letter A sounds like "eight", C sounds like "three", and D can easily be mistaken for the "ty" suffix: Is it 4D or forty? Other people avoid confusion by using the NATO phonetic alphabet: 4DA is "fourdeltaalfa"; or the Joint Army/Navy Phonetic Alphabet ("fourdogable"); or a similar ad hoc system.
Systems of counting on digits have been devised for both binary and hexadecimal. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 1023 on ten fingers. Another system for counting up to FF (255) is illustrated on the right; it seems to be an extension of an existing system for counting in twelves (dozens and grosses), that is common in South Asia and elsewhere.
The hexadecimal system can express negative numbers the same way as in decimal: –2A to represent –42 and so on.
However, some prefer instead to express the exact bit patterns used in the processor and consider hexadecimal values best handled as signed values. This way, the negative number –42 can be written as FFFF FFD6 in a 32bit CPU register, as C228 0000 in a 32bit FPU register or C045 0000 0000 0000 in a 64bit FPU register (assuming certain representation schemes, twoscomplement in the 32bit nonFPU instance and signmagnitude in the FPU instances.)
Most computers manipulate binary data, but it is difficult for humans to work with the 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 hexadecimal than to decimal because each hexadecimal 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 0, its value may be easily determined by its position from the right:
Therefore:
1111_{2}  = 8_{10} + 4_{10} + 2_{10} + 1_{10} 
= 15_{10} 
With surprisingly little practice, mapping 1111_{2} to F_{16} in one step becomes easy: see table in Uses. The advantage of using hexadecimal 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 hexadecimal, it is trivial to regard the binary string as 4digit groups and map each to a single hexadecimal digit.
This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
01011110101101010010_{2}  = 262144_{10} + 65536_{10} + 32768_{10} + 16384_{10} + 8192_{10} + 2048_{10} + 512_{10} + 256_{10} + 64_{10} + 16_{10} + 2_{10} 
= 387922_{10} 
Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently, and converted directly:
01011110101101010010_{2}  =  0101_{ }  1110_{ }  1011_{ }  0101_{ }  0010_{2} 
=  5  E  B  5  2_{16}  
=  5EB52_{16} 
The conversion from hexadecimal to binary is equally direct.
The octal system can also be useful as a tool for people who need to deal directly with binary computer data. Octal represents data as three bits per character, rather than four.
As with all bases there is a simple algorithm for converting a representation of a number to hexadecimal by doing integer division and remainder operations in the source base. Theoretically 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 hexadecimal, and the series h_{i}h_{i1}...h_{2}h_{1} be the hexadecimal digits representing the number.
"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 hexadecimal 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; var result; if (dr == 0) result = toChar(r); else result = toHex( (dr)/16 ) + toChar(r); return result; } function toChar(n) { const alpha = "0123456789ABCDEF"; return alpha.charAt(n); }
It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value and then performing multiplication and addition to get the final representation. I.e. to convert the number B3AD to decimal one can split the conversion into D (13_{10}), A (10_{10}), 3 (3_{10}) and B (11_{10}) then get the final result by multiplying each decimal representation by 16^{p}, where 'p' is the corresponding position from right to left, beginning with 0. In this case we have 13*(16^{0}) + 10*(16^{1}) + 3*(16^{2}) + 11*(16^{3}), which is equal 45997 in the decimal system.
Most modern computer systems with graphical user interfaces provide a builtin calculator utility, capable of performing conversions between various radixes, generally including hexadecimal.
In Microsoft Windows, the Calculator utility can be set to scientific calculator mode, which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 (octal) and 2 (binary); the bases most commonly used by programmers. In Scientific Mode, the onscreen numeric keypad includes the hexadecimal digits A through F which are active when "Hex" is selected. In hex mode, however, the Windows Calculator only supports integers.
As with other numeral systems, the hexadecimal system can be used to represent rational numbers, although recurring digits are common since sixteen (10h) has only a single prime factor (two):
^{1}⁄_{2} 

0.8  ^{1}⁄_{6} 

0.2AAAAAAAA...  ^{1}⁄_{A} 

0.1999999999...  ^{1}⁄_{E} 

0.1249249249... 
^{1}⁄_{3} 

0.5555555555...  ^{1}⁄_{7} 

0.2492492492...  ^{1}⁄_{B} 

0.1745D1745D...  ^{1}⁄_{F} 

0.1111111111... 
^{1}⁄_{4} 

0.4  ^{1}⁄_{8} 

0.2  ^{1}⁄_{C} 

0.1555555555...  ^{1}⁄_{10} 

0.1 
^{1}⁄_{5} 

0.3333333333...  ^{1}⁄_{9} 

0.1C71C71C71...  ^{1}⁄_{D} 

0.13B13B13B1...  ^{1}⁄_{11} 

0.0F0F0F0F0F... 
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: Counting in base 3 is 0, 1, 2, 10 (three). Thus, whether dividing one by two for binary or dividing one by sixteen for hexadecimal, both of these fractions are written as 0.1
. Because the radix 16 is a perfect square (4²), fractions expressed in hexadecimal 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 hexadecimal 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 hexadecimal (and binary) less convenient than decimal for representing rational numbers since a larger proportion lie outside its range of finite representation.
All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hexadecimal number with a finite number of digits 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 hexadecimal: That is, decimal 0.1 corresponds to the infinite recurring representation 0.199999999999... in hexadecimal. However, hexadecimal is more efficient than bases 12 and 60 for representing fractions with powers of two in the denominator (e.g., decimal one sixteenth is 0.1 in hexadecimal, 0.09 in duodecimal, 0;3,45 in sexagesimal and 0.0625 in decimal).
In decimal Prime factors of the base: 2, 5 
In hexadecimal Prime factors of the base: 2 

Fraction  Prime factors of the denominator 
Positional representation  Positional representation  Prime factors of the denominator 
Fraction 
1/2  2  0.5  0.8  2  1/2 
1/3  3  0.3333... = 0.3  0.5555... = 0.5  3  1/3 
1/4  2  0.25  0.4  2  1/4 
1/5  5  0.2  0.3  5  1/5 
1/6  2, 3  0.16  0.2A  2, 3  1/6 
1/7  7  0.142857  0.249  7  1/7 
1/8  2  0.125  0.2  2  1/8 
1/9  3  0.1  0.1C7  3  1/9 
1/10  2, 5  0.1  0.19  2, 5  1/A 
1/11  11  0.09  0.1745D  B  1/B 
1/12  2, 3  0.083  0.15  2, 3  1/C 
1/13  13  0.076923  0.13B  D  1/D 
1/14  2, 7  0.0714285  0.1249  2, 7  1/E 
1/15  3, 5  0.06  0.1  3, 5  1/F 
1/16  2  0.0625  0.1  2  1/10 
1/17  17  0.0588235294117647  0.0F  11  1/11 
1/18  2, 3  0.05  0.0E38  2, 3  1/12 
1/19  19  0.052631578947368421  0.0D79435E50  13  1/13 
1/20  2, 5  0.05  0.0C  2, 5  1/14 
1/21  3, 7  0.047619  0.0C3  3, 7  1/15 
1/22  2, 11  0.045  0.0BA2E8  2, B  1/16 
1/23  23  0.0434782608695652173913  0.0B21642C8590  17  1/17 
1/24  2, 3  0.0416  0.0A  2, 3  1/18 
1/25  5  0.04  0.0A3D70  5  1/19 
1/26  2, 13  0.0384615  0.09D8  2, B  1/1A 
1/27  3  0.037  0.097B425ED  3  1/1B 
1/28  2, 7  0.03571428  0.0924  2, 7  1/1C 
1/29  29  0.0344827586206896551724137931  0.08D3DCB  1D  1/1D 
1/30  2, 3, 5  0.03  0.08  2, 3, 5  1/1E 
1/31  31  0.032258064516129  0.08421  1F  1/1F 
1/32  2  0.03125  0.08  2  1/20 
1/33  3, 11  0.03  0.07C1F  3, B  1/21 
1/34  2, 17  0.02941176470588235  0.078  2, 11  1/22 
1/35  5, 7  0.0285714  0.075  5, 7  1/23 
1/36  2, 3  0.027  0.071C  2, 3  1/24 
Algebraic irrational number  In decimal  In hexadecimal 
√2 (the length of the diagonal of a unit square)  1.41421356237309...  1.6A09E667F3BCD... 
√3 (the length of the diagonal of a unit cube)  1.73205080756887...  1.BB67AE8584CAA... 
√5 (the length of the diagonal of a 1×2 rectangle)  2.2360679774997...  2.3C6EF372FE95... 
φ (phi, the golden ratio = ^{(1+√5)}⁄_{2})  1.6180339887498...  1.9E3779B97F4A... 
Transcendental irrational number  
π (pi, the ratio of circumference to diameter)  3.1415926535897932384626433 8327950288419716939937510... 
3.243F6A8885A308D313198A2E0 3707344A4093822299F31D008... 
e (the base of the natural logarithm)  2.7182818284590452...  2.B7E151628AED2A6B... 
τ (the Thue–Morse constant)  0.412454033640...  0.6996 9669 9669 6996 ... 
Number  
γ (the limiting difference between the harmonic series and the natural logarithm)  0.5772156649015328606...  0.93C467E37DB0C7A4D1B... 
Possibly the most widely used powers, powers of two, are easier to show using the 16th base. The first sixteen powers of two are shown below.
2^{x}  value 

2^{0}  1 
2^{1}  2 
2^{2}  4 
2^{3}  8 
2^{4}  10_{hex} 
2^{5}  20_{hex} 
2^{6}  40_{hex} 
2^{7}  80_{hex} 
2^{8}  100_{hex} 
2^{9}  200_{hex} 
2^{A} ()  400_{hex} 
2^{B} ()  800_{hex} 
2^{C} ()  1000_{hex} 
2^{D} ()  2000_{hex} 
2^{E} ()  4000_{hex} 
2^{F} ()  8000_{hex} 
2^{10} ()  10000_{hex} 
Since four squared is sixteen, powers of four have an even easier relation:
4^{x}  value 

4^{0}  1 
4^{1}  4 
4^{2}  10_{hex} 
4^{3}  40_{hex} 
4^{4}  100_{hex} 
4^{5}  400_{hex} 
4^{6}  1000_{hex} 
4^{7}  4000_{hex} 
4^{8}  10000_{hex} 
This also makes tetration easier when using two and four since ^{3}2 = 10_{hex}, ^{4}2 = 10000_{hex} and ^{5}2 = 1000000000...(16384 zeros)...000000000_{hex}.
The word hexadecimal is composed of hexa, derived from the Greek έξ (hex) for "six", and decimal, derived from the Latin for "tenth". Webster's Third New International online derives "hexadecimal" as an alteration of the allLatin "sexadecimal" (which appears in the earlier Bendix documentation). The earliest date attested for "hexadecimal" in MerriamWebster Collegiate online is 1954, placing it safely in the category of international scientific vocabulary (ISV). It is common in ISV to mix Greek and Latin combining forms freely. The word "sexagesimal" (for base 60) retains the Latin prefix. Donald Knuth has pointed out that the etymologically correct term is "senidenary", from the Latin term for "grouped by 16". (The terms "binary", "ternary" and "quaternary" are from the same Latin construction, and the etymologically correct term for "decimal" arithmetic is "denary".)^{[7]} Schwartzman notes that the expected form from usual Latin phrasing would be "sexadecimal", but computer hackers would be tempted to shorten that word to "sex".^{[8]} The etymologically proper Greek term would be hexadecadic (although in Modern Greek decahexadic (δεκαεξαδικός) is more commonly used).
Hexadecimal is sometimes used in programmer jokes because certain words can be formed using only hexadecimal digits. Some of these words are "dead", "beef", "babe", and with appropriate substitutions "c0ffee". Since these are quickly recognizable by programmers, debugging setups sometimes initialize memory to them to help programmers see when something has not been initialized. Some people add an H after a number if they want to show that it is written in hexadecimal. In older Intel assembly syntax, this is sometimes the case. "Hexspeak" may be the forerunner of the modern web parlance of "1337speak"
An example is the magic number in FAT MachO files and java class file structure, which is "CAFEBABE
". Singlearchitecture MachO files have the magic number "FEEDFACE
" at their beginning. "DEADBEEF
" is sometimes put into uninitialized memory. Microsoft Windows XP clears its locked index.dat files with the hex codes: "0BADF00D
". The Visual C++ remote debugger uses "BADCAB1E
" to denote a broken link to the target system.
Two common bit patterns often employed to test hardware are 01010101
and 10101010
(their corresponding hex values are 55h and AAh, respectively). The reason for their use is to alternate between off ('0') to on ('1') or vice versa when switching between these two patterns. These two values are often used together as signatures in critical PC system sectors (e.g., the hex word, 0xAA55
which on littleendian systems is 55h followed by AAh, must be at the end of a valid Master Boot Record).
The following table shows a joke in hexadecimal:
3x12=36 2x12=24 1x12=12 0x12=18
The first three are interpreted as multiplication, but in the last, "0x" signals Hexadecimal interpretation of 12, which is 18.
Another joke based on the use of a word containing only letters from the first six in the alphabet (and thus those used in hexadecimal) is...
In this case, DEAD refers to a hexadecimal number (57005 base 10), not the state of being no longer alive. Obviously, DEAD normally should not be written in allcaps (as in the preceding) as it makes it stand out, thus ruining the riddle.
A Knuth reward check is one hexadecimal dollar, or $2.56.
Similar to dozenal advocacy, there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts usually propose pronunciation and/or symbology.^{[9]} Sometimes the proposal unifies standard measures so that they are multiples of 16.^{[10]}^{[11]}^{[12]}
An example of unifying standard measures is Hexadecimal time which subdivides a day by 16 so that there are 16 "hexhours" in a day.^{[12]}
Simple key for notations used in article:
Full Text Notation  Abbreviation  Number Base 

binary  bin  2 
octal  oct  8 
decimal  dec  10 
hexadecimal  hex  16 
"\x1B[0m\x1B[25;1H"
specifies the character sequence Esc [ 0 m Esc [ 2 5 ; 1 H Nul. These are the escape sequences used on an ANSI terminal that reset the character set and color, and then move the cursor to line 25.&
to prefix octal values. (Microsoft BASIC primarily uses &O
to prefix octal, and it uses &H
to prefix hexadecimal, but the ampersand alone yields a default interpretation as an octal prefix.
