Hexadecimal

From Wikipedia, the free encyclopedia

Numeral systems by culture
Hindu-Arabic numerals
Western Arabic Eastern Arabic Khmer	Indian family Brahmi Thai
Eastern Asian numerals
Chinese Japanese	Korean
Alphabetic numerals
Abjad Armenian Cyrillic Ge'ez	Hebrew Ionian/Greek Sanskrit
Other systems
Attic Etruscan Roman	Babylonian Egyptian Mayan
List of numeral system topics
Positional systems by base
Decimal (10)
2, 4, 8, 16, 32, 64, 128
3, 9, 12, 24, 30, 36, 60, more…
v • d • e</div>

In mathematics and computer science, base-16, hexadecimal, or simply hex, is a numeral system with a radix or base of 16, usually written using the symbols 0–9 and A–F or a–f. For example, the decimal numeral 79 whose binary representation is 01001111 can be written as 4F in hexadecimal (4 = 0100, F = 1111). The current hexadecimal system was first introduced to the computing world in 1963 by IBM. An earlier version, using the digits 0–9 and u–z, was used by the Bendix G-15 computer, introduced in 1956.

1 Uses
2 Representing hexadecimal
- 2.1 Verbal Representations
3 Fractions
4 Mapping to binary
5 Converting from other bases
- 5.1 Division-remainder in source base
- 5.2 Conversion via binary
6 Etymology
7 Humour
8 Cultural References
9 See also
10 References
11 External links

[edit] Uses

Hex	Bin	Dec
0	0000	0
1	0001	1
2	0010	2
3	0011	3
4	0100	4
5	0101	5
6	0110	6
7	0111	7
8	1000	8
9	1001	9
A	1010	10
B	1011	11
C	1100	12
D	1101	13
E	1110	14
F	1111	15

Hexadecimal is primarily used in computing as the most common form of expressing a guaranteeably human-readable string representation of a byte. All 256 possible values of a byte can be represented using 2 digits in hexadecimal notation. Some people assume that using 8-bit "ASCII" to represent the value of a byte should work, but this has a number of problems; firstly, there are a number of unprintable control characters, secondly, ASCII itself stops at 7 bits with the remainder being system-specific extensions, and, finally, even assuming all characters in the machine's set were displayable as something neither users nor input methods are generally prepared to handle 256 unique characters.

WTF

[edit] Representing hexadecimal

Some hexadecimal representations are indistinguishable from decimal representations (to both humans and computers). Therefore, some convention is usually used to flag them.

In typeset text, the indication is often a subscripted suffix such as 5A3₁₆, 5A3_SIXTEEN, or 5A3_HEX.

In computer programming languages (which are nearly always plain text without such typographical distinctions as subscript and superscript) a wide variety of ways of marking hexadecimal representations have appeared. These are also seen even in typeset text especially if that text relates to a programming language.

Some of the more common textual representations:

Ada and VHDL enclose hexadecimal numerals in based "numeric quotes", e.g. 16#5A3#. (Note: Ada accepts this notation for all bases from 2 through 16 and for both integer and real types.)
C and languages with a similar syntax (such as C++, C#, Java and Javascript) prefix hexadecimal numerals with 0x, e.g. 0x5A3. The leading 0 is used so that the parser can simply recognize a number, and the x stands for hexadecimal (cf. o for Octal and b for Binary). The x in 0x can be either in upper or lower case but is almost always seen written in lower case.
*nix shells use an escape character form \x0FF in expressions and 0xFF for constants.
In HTML, hexadecimal character references also use the x: ֣ should give the same as ֣ – with your browser ֣ and ֣ respectively (Hebrew accent munah). Hexadecimal color references are prefixed with #, e.g. #FFFFFF (white).
Some assemblers indicate hex by an appended h (if the numeral starts with a letter, then also with a preceding 0, to indicate that it is a number), e.g., 0A3Ch, 5A3h. The syntax may vary per assembly language. For example, the 6502 assembly language uses a $ as prefix, e.g. #$10FF (the # indicates a number).
Postscript indicates hex by a prefix 16#.
Common Lisp use the prefixes #x and #16r.
Pascal, other assemblers (AT&T, Motorola), and some versions of BASIC use a prefixed $, e.g. $5A3.
The Smalltalk programming language uses the prefix 16r. Note Smalltalk accepts the format <radix>r<digits> where radix is a number base from 2 upwards (i.e. 2r1110 is 10r14 or 16rE), with the practical limitation being within the ASCII character set range 0-9 and A-Z used to represent the digits. Some versions of Smalltalk allow fractional digits following a period character, ., enabling hexadecimal (and other base) representation of floating point numbers.
Some versions of BASIC, notably Microsoft's variants including QBasic and Visual Basic), prefix hexadecimal numerals with &H, e.g. &H5A3; others such as BBC BASIC just used & (used for octal in Microsoft's BASIC!).
TI 89 and 92 series designate 0h (ex 0hA3)
Notations such as X'5A3' are sometimes seen; PL/I uses such notation. This is the most common format for hexadecimal on IBM mainframes (zSeries) and minicomputers (iSeries) running the traditional OS's (zOS, zVSE, zVM, TPF, OS/400), and is used in Assembler, PL/1, Cobol, JCL, scripts, commands and other places. The most common exceptions are when using a language with a different native convention (C, Java, etc.). This format was common on other (and now obsolete) IBM systems as well.
Donald Knuth introduced the use of different fonts to represent radices in his book The TeXbook. In his notation, hexadecimal representations are written in a typewriter type, e.g. 5A3

Image:Hexadecimal-multiplication-table.png

There is no single agreed-upon standard, so all the above conventions are in use, sometimes even in the same paper. However, as they are quite unambiguous, little difficulty arises from this.

The most commonly used (or encountered) notations are the ones with a prefix "0x" or a subscript-base 16, for hex numbers. For example, both 0x2BAD and 2BAD₁₆ represent the decimal number 11181 (or 11181₁₀).

The choice of the letters A through F to represent the additional digits was not universal in the early history of computers. During the 1950s, some installations favored using the digits 0 through 5 with a macron to indicate the values 10-15. Users of Bendix G-15 computers used the letters U through Z.

[edit] Verbal Representations

Not only are there currently no proper digits to represent the quantities from ten to fifteen (so letters are used as a substitute), but English also lacks a proper nomenclature to name hexadecimal numbers. Names such as "thirteen" and "fourteen" are decimal-based, and even though English has a few names for non-decimal powers —pair and score for the first binary power and the first vigesimal power, respectively; dozen, gross and great gross for the first three duodecimal(note that the term duodecimal itself is derived from the decimal system) powers—, no English name currently exists for any of the hexadecimal powers (corresponding to the decimal values 16, 256, 4096, 65536, 1048576, 16777216, etc.). So people have resorted to reading hexadecimal numbers by naming their digits (or digit-letters) individually in sequence in the same way as reading phone numbers (i.e., 4DA as "four-dee-aye"). However, the letter 'A' sounds similar to '8', 'C' sounds similar to '3', and 'D' can easily be mistaken for the 'ty' suffix as in "forty"; so 4DA could be mistaken for 48. To avoid misunderstandings, some convention must be established when exchanging hexadecimal numbers verbally, at least until a proper hexadecimal nomenclature is developed (if ever). To avoid confusion, the digits A-F are commonly pronounced with the NATO phonetic alphabet ("four-delta-alpha"), the World War II era Joint Army/Navy Phonetic Alphabet ("four-dog-able"), or some approximation of one of those systems.

The following defines another convention:

To pronounce a hexadecimal number when speaking, begin the number with an "x" prefix, where then any letters A-F all have an "x" suffix. Each digit or group of digits would be pronounced sequentially, where decimal expressions MUST NOT contain any letters. By starting with 'x', this alerts the listener to the use of the 'x' prefix/suffix convention and indicates that the number is hexadecimal.

The verbal hexadecimal characters are:

x 0 1 2 3 4 5 6 7 8 9 Ax Bx Cx Dx Ex Fx/F

The letters A-F are pronounced:

 ā-eks, bē-eks, sē-eks, dē-eks, ē-eks, ef-eks/ef

The hex number '34D' could be spoken as x 3 4 Dx.

This could also be spoken as x thirty-four Dx, but not as x three-hundred-forty Dx. An exception may be made for Fx, where it is pronounced ef.

[edit] Fractions

As with other numeral systems, the hexadecimal system can be used to represent rational numbers, although recurring digits are common since 16 has only a single prime factor:


1/1	=	1	1/5	<center> =	0.3	1/9	<center> =	0.1C7	1/D	<center> =	0.13B
1/2	<center> =	0.8	1/6	<center> =	0.2A	1/A	<center> =	0.19	1/E	<center> =	0.1249
1/3	<center> =	0.5	1/7	<center> =	0.249	1/B	<center> =	0.1745D	1/F	<center> =	0.1
1/4	<center> =	0.4	1/8	<center> =	0.2	1/C	<center> =	0.15	1/10	<center> =	0.1

Because the radix 16 is a square (4²), hexadecimal fractions have an odd period much more often than decimal ones. Recurring digits occur when the denominator in lowest terms has a prime factor not found in the radix. Using hexadecimal notation, all fractions with denominators that are not a power of two will result in an infinite string of recurring digits. This makes hexadecimal (and binary) less convenient than decimal (let alone than duodecimal) to represent rational numbers, since a larger proportion of them lie outside its range of finite representation (all rational numbers finitely representable in hexadecimal are finitely representable in decimal and duodecimal, while only a fraction of those finitely representable in the latter ones are finitely representable in the former). Although hexadecimal is more efficient than either for the particular case of representing fractions with powers of two as the denominator (cf. one sixteenth is 0.1 in hexadecimal, 0.09 in duodecimal and 0.0625 in decimal).

[edit] Mapping to binary

When working with computers we often need to deal with binary data. It is much easier for humans to handle numbers in hexadecimal than in binary (since numbers require more binary digits than decimal or hexadecimal) and whilst we are more familiar with the base 10 system, it is much easier to map binary to hexadecimal than to decimal since each hexadecimal digit maps to a whole number of bits (4₁₀).

Consider converting 1111₂ to base 10. Since each position in a binary numeral can only 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

Therefore:

1111₂	= 8₁₀ + 4₁₀ + 2₁₀ + 1₁₀
	= 15₁₀

This is a very simple example which still requires the addition of 4 numbers; whereas, with some practice, 1111₂ can be mapped directly to F₁₆ in one step (see table in Representing hexadecimal). When the number is very much greater, conversion to decimal becomes much more tedious; however, when mapping to hexadecimal, it is simple to divide the binary string up in blocks of 4 positions and map each block of 4 bits to a single position hexadecimal digit. For example, consider a tedious conversion to decimal:

This article or section needs copy editing for proper spelling, grammar, usage, tone, style, and voice.
You can help by editing it now. A guide is available, as is general editing help.

</tr>

01011110101101010010₂	= 262144₁₀ + 65536₁₀ + 32768₁₀ + 16384₁₀ + 8192₁₀ + 2048₁₀ + 512₁₀ + 256₁₀ + 64₁₀ + 16₁₀ + 2₁₀
	= 387922₁₀

Compare this to the conversion to hexadecimal:

01011110101101010010₂	=	0101	1110	1011	0101	0010₂
	=	5	E	B	5	2₁₆
	=	5EB52₁₆

Conversion from hexadecimal back to binary is just as direct.

Octal is also useful as a way for humans to deal with computer data (in blocks of 3 bits instead of 4); however, hexadecimal's big advantage over octal is that exactly 2 digits represent a byte (octet). This means that with hexadecimal, you can easily see from the value of a word what the value of the individual bytes will be; conversely, if you have the values of the bytes, you can easily assemble them to get the value of a word.

[edit] Converting from other bases

[edit] 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 h_ih_i-1...h₂h₁ be the hexadecimal 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 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(d-r==0) 
                result = toChar(r);
	else 
                result = toHex( (d-r)/16 )+toChar(r);
        return result;
}

function toChar(n) {
	var alpha = "0123456789ABCDEF";
	return alpha.charAt(n);
}

Wtf>?

[edit] Conversion via binary

As computers generally work in binary the normal way for a computer to make such a conversion would be to convert to binary first and then make use of the direct mapping from binary to hexadecimal.

[edit] 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 should be "denary".)<ref>Knuth, Donald. (1969). Donald Knuth, in The Art of Computer Programming, Volume 2. ISBN 0-201-03802-1. (Chapter 17.)</ref> Schwartzman notes that the expected purely Latin form would be "sexadecimal", but then computer hackers would be tempted to shorten the word to "sex".<ref>Schwartzman, S. (1994). The Words of Mathematics: an etymological dictionary of mathematical terms used in English. ISBN 0-88385-511-9.</ref> Incidentally, the etymologically proper Greek term would be hexadecadic (although in Modern Greek deca-hexadic (δεκαεξαδικός) is more commonly used).

[edit] Humour

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". This is an example of such a joke. Since these are quickly recognisable by programmers, debugging setups sometimes initialise memory to them to help programmers see when something has not been initialised. 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. With that last H it becomes possible to write new words and sentences, such as for example 1517ADEADB17CH (is it a dead bitch).

This may be the forerunner of the modern web parlance of "1337speak"

Another example is the magic number in FAT Mach-O files and java programs, which is "CAFEBABE".

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 multiples of 12, while in the last one "0x12" in hex is 18.

0xdeadbeef is sometimes put into uninitialized memory.

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

[edit] Cultural References

In The Simpsons, on the episode Treehouse of Horror VI, where Homer enters the third dimension (Homer³), a hexadecimal string (46 72 69 6e 6b 20 52 75 6c 65 73 21) is floating in "3-D land" which, when used as character indices in the ASCII character set, translates to "Frink rules!" (excluding the quotes but including the exclamation mark).

In the TV show ReBoot there is a villainous character named Hexadecimal, who appears as a harlequin with constantly-changing masks, each with a different facial expression to represent differing emotional states.

In 1998, Subaru sold a special edition Impreza called the WRX-STi 22B. While some contend the name was derived from the use of a 2.2L motor ("22") and Bilstein brand ("B") suspension components, it has also been shown that "22B" is the hexadecimal equivalent of "555", where State Express 555 is the British American Tobacco brand that sponsored Subaru's early rally efforts.