# Dictionary Definition

hexadecimal adj : of or pertaining to a number
system having 16 as its base [syn: hex]

# User Contributed Dictionary

## English

### Pronunciation

- /ˌhɛksəˈdɛsəməl/
- /%hEks@"dEs@m@l/

### Noun

hexadecimal (uncountable)- In the context of "arithmetic|computing": A number system with base 16, using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F, useful in computing as numbers in hexadecimal can be stored in four bits. Informal short form used in computing: hex

#### Synonyms

- (arithmetic, computing): base 16
- sexidecimal (pure Latin)
- sexadecimal

#### Translations

number system with base 16

- Bosnian: heksadecimalni sistem
- Danish: hexadecimale talsystem , sekstentalssystem
- Dutch: heksadecimale talstelsel
- Finnish: heksadesimaalijärjestelmä (the system); heksadesimaaliluku (a number in the system)
- French: système hexadécimal
- German: Hexadezimalsystem , Sedezimalsystem
- Hebrew:
- Italian: sistema numerico esadecimale
- Japanese: 16進数 (じゅうろくしんすう, jūrokushinsū)
- Portuguese: sistema hexadecimal
- Russian: шестнадцатеричная система (šestnadcat'eríčnaja sist'éma)
- Serbian:
- Cyrillic: хексадецимални систем
- Romann: heksadecimalni sistem

### Adjective

hexadecimal (no or )- Of a number, expressed in hexadecimal.

#### Synonyms

- (arithmetic, computing): base-16

#### Translations

expressed in hexadecimal

- Bosnian: heksadecimalan
- Danish: hexadecimal
- Dutch: hexadecimaal, hexadecimale
- Esperanto: deksesuma
- Finnish: heksadesimaalinen
- French: hexadécimal
- German: hexadezimal
- Hebrew: ,
- Russian: шестнадцатеричный (šestnadcat'eríčnyj)
- Serbian:
- Cyrillic: хексадецималан
- Roman: heksadecimalan

##### Translations to be checked

The translations for the adjective and noun seem
to have been mixed up. Put the translations in the appropriate
table above.

- ttbc Chinese: 十六进制
- ttbc Norwegian: sekstentallsystemet

# Extensive Definition

In mathematics and computer
science, hexadecimal (also base-,
hexa, or hex) is a 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 a through f) to represent values ten to
fifteen.

Its primary use is as a human friendly
representation of binary coded
values, so it is often used in digital electronics and computer
engineering. Since each hexadecimal digit represents four binary
digits (bits)—also called a
nibble—it is a compact
and easily translated shorthand to express values in
base
two.

IBM introduced the
current hexadecimal system to the computing world; an earlier
version, using the digits 0–9 and U–Z, was
introduced in 1956 by the Bendix G-15
computer.

## Uses

In digital computing, hexadecimal is primarily used to represent bytes. Attempts to represent the 256 possible byte values by other means have led to problems. Directly representing each possible byte value with a single character representation runs into unprintable control characters in the ASCII character set. Even if a standard set of printable characters were devised for every byte value, neither users nor input hardware are equipped to handle 256 unique characters. Most hex editing software displays each byte as a single character, but unprintable characters are usually substituted with period or blank.In URLs, all characters
can be coded using hexadecimal. Each 2-digit (1 byte) hexadecimal
sequence is preceded by a percent sign. For example, the
URL http://en.wikipedia.org/wiki/Main%20Page
substitutes a space (which is not allowed in URLs) with the hex
code for a space (%20).

## Representing hexadecimal

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 unambiguously expressing values. In mathematics, a subscript is often used on each number explicitly giving the base: 15910 is decimal 159; 15916 is hexadecimal 159 which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex.In linear text systems, such as those used in
most computer programming environments, a variety of methods have
arisen:

- In URLs, character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the space (blank) character, code 20 hex, or 32 decimal.
- In HTML, characters can be expressed as hexadecimal using the notation ꯍ. Color references are expressed in hex prefixed with #: #FFFFFF which gives white.
- The C programming language (and its syntactical descendants) use the prefix 0x: 0x5A3 Character and string constants may express character codes in hexadecimal with the prefix \x followed by two hex digits: '\x1B' (specifies the Esc control character), "\x1B[0m\x1B[25;1H" is a string containing 11 characters (not including an implied trailing NUL). To output a value as hexadecimal with the printf function family, the format conversion code %X or %x is used.
- In the Unicode standard, a character value is represented with U+ followed by the hex value: U+20AC is the Euro sign (€).
- MIME (e-mail extensions) quoted-printable characters by code inside a text/plain MIME-part body prefix non-printable ASCII characters with an equal to sign =, as in Espa=D1a to send "España" (Spain).
- In Intel-derived assembly languages, hexadecimal is indicated with a suffixed H or h: FFh or 0A3CH. Some implementations require a leading zero when the first character is not a digit: 0FFh
- Other assembly languages (6502, AT&T, Motorola), Pascal, and some versions of BASIC (Commodore) and Forth use $ as a prefix: $5A3.
- Some assembly languages (Microchip) use the notation H'ABCD' (for ABCD16).
- *nix (UNIX and related) shells use an escape character form \x0FF in expressions and 0xFF for constants.
- Ada and VHDL enclose hexadecimal numerals in based "numeric quotes": 16#5A3#
- Verilog represents hexadecimal constants in the form 8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant.
- Modula 2 and some other languages use # as a prefix: #01AF
- The Smalltalk programming language uses the prefix 16r: 16r6EF7
- Postscript indicates hex with prefix 16#: 16#ABCD. Binary data (such as image pixels) can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC...
- Common Lisp use the prefixes #x and #16r.
- QBasic and Visual Basic, prefix hexadecimal numerals with &H: &H5A3
- BBC BASIC uses & for hex.
- TI-89 and 92 series uses 0h: 0hA3
- Notations such as X'5A3' are sometimes seen, such as in PL/I. This is the most common format for hexadecimal on IBM mainframes (zSeries) and minicomputers (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.
- Donald Knuth introduced the use of particular typeface to represent a particular radix in his book The TeXbook. There, hexadecimal representations are written in a typewriter typeface: 5A3

There is no universal convention to use lowercase
or uppercase for the letter digits, and each is prevalent or
preferred by 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 G-15
computers used the letters U through Z. Bruce A.
Martin of
Brookhaven National Laboratory considered the choice of A-F
"ridiculous" and in 1968 proposed in a letter to the editor of the
ACM an entirely new set of symbols based on the bit locations,
which did not gain much acceptance.

## Verbal representations

Not only are there no digits to represent the quantities from ten to fifteen—so letters are used as a substitute—but most Western European languages also lack a nomenclature to name hexadecimal numbers. "Thirteen" and "fourteen" are decimal-based, and even though English has names for several non-decimal powers: pair for the first binary power; score for the first vigesimal power; dozen, gross, and great gross for the first three duodecimal powers. However, no English name describes the hexadecimal powers (corresponding to the decimal values 16, 256, 4096, 65536, ...). Some people read hexadecimal numbers digit by digit like a phone number: 4DA is "four-dee-aye". However, the letter 'A' sounds similar to eight, 'C' sounds similar to 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 "four-delta-alpha". Similarly, some use the Joint Army/Navy Phonetic Alphabet ("four-dog-able"), or a similar ad hoc system.## Signs

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 unsigned values. This way, the
negative number –42 can be written as FFFF FFD6 in a
32-bit CPU
register, as C228 0000 in a 32-bit FPU
register or C045 0000 0000 0000 in a
64-bit FPU register.

## Fractions

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):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).

## Binary translation

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 (410). This example converts 11112 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:- 00012 = 110
- 00102 = 210
- 01002 = 410
- 10002 = 810

11112 = 810 + 410 + 210 + 110

= 1510

With surprisingly little practice, mapping 11112
to F16 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 4 digit
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.

010111101011010100102 = 26214410 + 6553610 +
3276810 + 1638410 + 819210 + 204810 + 51210 + 25610 + 6410 + 1610 +
210

= 38792210

Compare this to the conversion to hexadecimal,
where each group of four digits can be considered independently,
and converted directly:

010111101011010100102 = 0101
1110 1011 0101 00102

= 5 E B 5 216

= 5EB5216

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.

## Converting from other bases

### Division-remainder in source base

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 hihi-1...h2h1 be the hexadecimal digits representing
the number.

- i := 1
- hi := d mod 16
- d := (d-hi) / 16
- If d = 0 (return series hi) 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 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)

function toChar(n)

### Addition and multiplication

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 (1310), A (1010), 3 (310) and B (1110) then get the final result by multiplying each decimal representation by 16p, where 'p' is the corresponding position from right to left, beginning with 0. In this case we have 13*(160) + 10*(161) + 3*(162) + 11*(163), which is equal 45997 in decimal system.### Conversion via binary

As most computers work in binary, the normal way for a computer to make such a conversion would be to convert to binary first (by doing multiplication and addition in binary) and then make use of the direct mapping from binary to hexadecimal.### Tools for conversion

Most modern computer systems with graphical user interfaces provide a built-in 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 on screen numeric
keypad includes the hexadecimal digits A through F which are
active when "Hex" is selected.

## Cultural

### Etymology

It was IBM that decided on the prefix of "hexa" rather than the proper Latin prefix of "sexa". The word "hexadecimal" is strange in that hexa is derived from the Greek έξ (hex) for "six" and decimal is derived from the Latin for "tenth". It may have been derived from the Latin root, but Greek deka is so similar to the Latin decem that some would not consider this nomenclature inconsistent. An older term was the incorrect Latin-like "sexidecimal" (correct Latin is "sedecim" for 16), but that was changed because some people thought it too risqué, and it also had an alternative meaning of "base 60". However, the word "sexagesimal" (base 60) retains the prefix. The earlier Bendix documentation used the term "sexadecimal". 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".) Schwartzman notes that the expected purely Latin form would be "sexadecimal", but then computer hackers would be tempted to shorten the word to "sex". Incidentally, the etymologically proper Greek term would be hexadecadic (although in Modern Greek deca-hexadic (δεκαεξαδικός) is more commonly used).### Common patterns and humor

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 Mach-O files and java
programs, which is "CAFEBABE". Single-architecture Mach-O files
have the magic number "FEEDFACE" at their beginning.

A Knuth
reward check is one hexadecimal dollar, or $2.56.

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.

0xdeadbeef is
sometimes put into uninitialized memory.

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...

- If only DEAD people understand hexadecimal, how many people understand hexadecimal?

Microsoft Windows XP clears its locked index.dat
files with the hex codes: "0BADF00D".

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 little-endian systems is 55h
followed by AAh, must at the end of a valid Master
Boot Record).

### Primary numeral system

There have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts usually propose pronunciation and/or symbology. Sometimes the proposal unifies standard measures so that they are multiples of 16.An example of unifying standard measures is
Hexadecimal
time which subdivides a day by 16 so that there are 16
"hexhours" in a day.

## See also

- Base 32
- Base 64
- Web colours
- Hex editor
- Hexadecimal time
- Hexspeak
- Nibble — one hexadecimal digit can exactly represent one "nibble"
- Numeral system — a list of other base systems
- Binary numeral system
- HTML
- Bubble Babble

## References

## External links

### Hex conversion utilities or pages

- Online Converter for Decimal/Hexadecimal Numerals (JavaScript, GPL)
- Online ASCII/Hexadecimal converter (PHP)
- Hex/ASCII 'translation' service
- Leet Key, a Firefox extension that supports ASCII/Hex conversions and typing

hexadecimal in Arabic: نظام عد سداسي عشر

hexadecimal in Bosnian: Heksadecimalni
sistem

hexadecimal in Breton: Diazez c'hwezekred

hexadecimal in Bulgarian: Шестнадесетична бройна
система

hexadecimal in Catalan: Sistema
hexadecimal

hexadecimal in Czech: Hexadecimální číslo

hexadecimal in Danish: Hexadecimale
talsystem

hexadecimal in German: Hexadezimalsystem

hexadecimal in Modern Greek (1453-): Δεκαεξαδικό
σύστημα αρίθμησης

hexadecimal in Spanish: Sistema
hexadecimal

hexadecimal in Esperanto: Deksesuma
sistemo

hexadecimal in Basque: Zenbaki-sistema
hamaseitar

hexadecimal in French: Système hexadécimal

hexadecimal in Galician: Código
hexadecimal

hexadecimal in Korean: 십육진법

hexadecimal in Croatian: Heksadekadski brojevni
sustav

hexadecimal in Indonesian: Heksadesimal

hexadecimal in Icelandic: Sextánundakerfi

hexadecimal in Italian: Sistema numerico
esadecimale

hexadecimal in Hebrew: בסיס הקסדצימלי

hexadecimal in Haitian: Sistèm ekzadesimal

hexadecimal in Hungarian: Tizenhatos
számrendszer

hexadecimal in Malay (macrolanguage): Nombor
perenambelasan

hexadecimal in Dutch: Hexadecimaal

hexadecimal in Japanese: 十六進法

hexadecimal in Norwegian:
Sekstentallsystemet

hexadecimal in Norwegian Nynorsk:
Sekstentalssystemet

hexadecimal in Polish: Szesnastkowy system
liczbowy

hexadecimal in Portuguese: Sistema
hexadecimal

hexadecimal in Romanian: Sistem
hexazecimal

hexadecimal in Russian: Шестнадцатеричная
система счисления

hexadecimal in Simple English: Hexadecimal
numeral system

hexadecimal in Slovak: Šestnástková
sústava

hexadecimal in Slovenian: Šestnajstiški
številski sistem

hexadecimal in Serbian: Хексадецимални
систем

hexadecimal in Serbo-Croatian: Heksadecimalni
sistem

hexadecimal in Finnish:
Heksadesimaalijärjestelmä

hexadecimal in Swedish: Hexadecimala
talsystemet

hexadecimal in Thai: เลขฐานสิบหก

hexadecimal in Vietnamese: Hệ thập lục
phân

hexadecimal in Turkish: Heksadesimal

hexadecimal in Ukrainian: Шістнадцяткова система
числення

hexadecimal in Yiddish: העקס

hexadecimal in Chinese: 十六进制