Hexadecimal

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In mathematics and computer science, hexadecimal, base-16, 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. Its primary purpose is to represent the binary code in a format easier for humans to read, and acts as a form of shorthand, in which one hexadecimal digit stands in place of four binary bits. For example, the decimal numeral 79, whose binary representation is 01001111, is 4F in hexadecimal (4 = 0100, F = 1111). IBM introduced the current hexadecimal system to the computing world; an earlier version, using the digits 0–9 and u–z, had been introduced in 1956, and had been used by the Bendix G-15 computer.

Contents 1 Uses 1.1 Hex triplets 1.2 Other common uses 2 Representing hexadecimal 3 Verbal representations 4 Fractions 5 Mapping to binary 6 Converting from other bases 6.1 Division-remainder in source base 6.2 Addition and multiplication in hexadecimal 6.3 Conversion via binary 7 Etymology 8 Humor 9 See also 10 References 11 External links

[edit] Uses

	0hex	=	0dec	=	0oct		0	0	0	0
	1hex	=	1dec	=	1oct		0	0	0	1
	2hex	=	2dec	=	2oct		0	0	1	0
	3hex	=	3dec	=	3oct		0	0	1	1
	4hex	=	4dec	=	4oct		0	1	0	0
	5hex	=	5dec	=	5oct		0	1	0	1
	6hex	=	6dec	=	6oct		0	1	1	0
	7hex	=	7dec	=	7oct		0	1	1	1
	8hex	=	8dec	=	10oct		1	0	0	0
	9hex	=	9dec	=	11oct		1	0	0	1
	Ahex	=	10dec	=	12oct		1	0	1	0
	Bhex	=	11dec	=	13oct		1	0	1	1
	Chex	=	12dec	=	14oct		1	1	0	0
	Dhex	=	13dec	=	15oct		1	1	0	1
	Ehex	=	14dec	=	16oct		1	1	1	0
	Fhex	=	15dec	=	17oct		1	1	1	1

In computing, hexadecimal is primarily used to represent bytes. Attempts to represent the 256 possible byte values by other means have led to problems: First, there are a number of unprintable control characters in the standard 7-bit ASCII character set; second, extended (8-bit) ASCII sets may be system-specific (different computers may have different extended-character sets). Finally, even if there were a standard set of printable characters for every byte value, neither users nor input methods (e.g., the standard PC keyboard) are generally prepared to handle 256 unique characters. Most hex and disk editors include columns which represent as many bytes as possible using some given set of characters; the bytes which cannot be uniquely represented, are often filled with a dot (.).

[edit] Hex triplets

HTML and CSS use hexadecimal notation (hex triplets) to specify colors on web pages, with "#" standing for hexadecimal. Twenty-four-bit color is represented in the format #RRGGBB: where RR specifies the value of the Red component of the color, GG the Green component, and BB the Blue component. For example, a shade of red that is (238,9,63) in decimal is coded as #EE093F. This syntax is borrowed from the X Window System.

Example of conversion from hexadecimal triplet to decimal triplet: Hexadecimal triplet: FFCF4B

Step 1: Separate the triplets: FF CF 4B

Step 2: Convert each hexadecimal value to a decimal representation:

• FF = 15*16 + 15*1 = 255
• CF = 12*16 + 15*1 = 207
• 4B = 4*16 + 11*1 = 75

Result: Hexadecimal triplet FFCF4B = Decimal triplet 255,207,75

[edit] Other common uses

In URLs, all characters can be coded hexadecimally. This is specified in RFC 3986. Each 2-digit (1 byte) hexadecimal sequence is preceded by a percent sign. For example, in the URL http://en.wikipedia.org/wiki/Main%20Page, the (hexadecimal) UTF-8 character code for a space (" ") is 20.

Some software programs will create unique order numbers by using a hexadecimal representation of the exact second the order was taken, based on the total number of seconds since the start of the 20th century. For example, C9BCE0F5 represents April 2, 2007 14:19:32.

Page numbers on teletext are written in hexadecimal, with available numbers being in the range of 100-8FF. However, page numbers with letters are only used for "hidden" and engineering pages.

[edit] Representing hexadecimal

Some hexadecimal representations are indistinguishable from decimal representations (to humans and computers); therefore, some convention is usually used to flag them, in typeset text often indicated by a subscripted suffix such as 5A3₁₆ or 5A3_HEX. Programming languages have traditionally used plain text (i.e. ASCII), without such typographical distinctions as subscript and superscript, and have therefore instead denoted hexadecimal numbers using a wide variety of alternative conventions.

The following are some of the most common representations:

• In many assemblers hexadecimal is denoted by an appended H or h, e.g. FFh, 0A3CH. Some allow numbers to start with a letter, some demand a leading zero for such cases.
• Other assemblers (AT&T, Motorola), Pascal, and some versions of BASIC and Forth use $ as a prefix, e.g. $5A3.
• There are also assemblers (Microchip) that use the notation H'ABCD' (for ABCD₁₆), as well as the C-family convention (see below).
• C and languages with a similar syntax (such as C++, C#, Java and JavaScript) use the prefix 0x, e.g. 0x5A3; the leading 0 denotes a number and the x stands for hexadecimal.
• Ada and VHDL enclose hexadecimal numerals in based "numeric quotes" using any base, e.g. 16#5A3#. (Ada accepts this notation for both integer and real types.)
• Modula 2 and some other languages use # as a prefix.
• The Smalltalk programming language uses the prefix 16r. (Smalltalk accepts other bases than 16; some versions also accept fractional digits in bases other than 10.)
• *nix shells use an escape character form \x0FF in expressions and 0xFF for constants.
• In HTML, hexadecimal use the notation &#ABCD;. hexadecimal color references are prefixed with #, e.g. #FFFFFF (white).
• Postscript indicates hex by a prefix 16#.
• Common Lisp use the prefixes #x and #16r.
• QBasic and Visual Basic, prefix hexadecimal numerals with &H, e.g. &H5A3
• BBC BASIC 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. This format was common on other (and now obsolete) IBM systems as well.
• Donald Knuth introduced the use of fonts to represent radices, in his book The TeXbook. There, hexadecimal representations are written in a typewriter font, e.g. 5A3

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. Perhaps the most commonly encountered notation today is the "0x" notation, due to the widespread use of the C programming language and its descendants.

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.

One solution how to write hexadecimal numbers distinctively is the use of figures that are made for the hexadecimal system but are not yet representable in Unicode.

[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 the English language, and most Western European languages, also lack 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 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 currently exists for any of the hexadecimal powers (corresponding to the decimal values 16, 256, 4096, 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.

[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:

1⁄2	=	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...

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 occur 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 will result in an infinite string of recurring digits (e.g., thirds and fifths). This makes hexadecimal (and binary) less convenient than decimal (not to mention duodecimal and sexagesimal) for the purpose of representing rational numbers, since a larger proportion of them lie outside its range of finite representation. All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal (i.e., any hexadecimal number with a finite number of digits will have a finite number of digits when expressed in those other bases), whereas only a fraction of those finitely representable in the latter ones are finitely representable in the former (e.g., decimal 0.1, that is, the fraction one tenth, corresponds to the infinite recurring representation 0.199999999999... in hexadecimal). Although hexadecimal is more efficient than those other bases 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, 0;3,45 in sexagesimal and 0.0625 in decimal).

[edit] Mapping to binary

Sometimes it is necessary to use binary data when working with computers, but it is difficult for humans to work with the large number of digits in binary. Although most humans are more 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). The following is an example of 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 example shows addition of 4 numbers; but with some practice, 1111₂ can be mapped directly to F₁₆ in one step (see table in Representing hexadecimal). The advantage of using hexadecimal rather than decimal increases with the size of the number. When the number becomes large, conversion to decimal becomes much more tedious; however, when mapping to hexadecimal, it is simple to divide the binary string into blocks of 4 positions and map each block of 4 bits to a single position 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

Conversion from hexadecimal back to binary is just as direct.

The octal system can also be useful as a tool for people who need to deal directly with binary computer data, as in reading and understanding it. Compared to hexadecimal, octal represents data in blocks of 3 bits each, rather than 4.

One advantage of hexadecimal is that every unique 2-digit pair (or octet) always represents the same byte value. To "translate" a hexadecimal value into bytes, one needs only to separate the value into individual 2-digit groups, translate each group into its respective byte value, and then combine the results together to form an accurate translation of the entire original hexadecimal word. Conversely, bytes can also be easily translated into hexadecimal values by translating each byte individually into its hexadecimal 2-digit value, and then recombining the hexadecimal values into a "word". The resulting "word" will be an accurate hexadecimal representation of the original string of bytes.

[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.

1. i := 1
2. h_i := d mod 16
3. d := (d-h_i) / 16
4. 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);
}

[edit] Addition and multiplication in hexadecimal

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.

[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 (by doing multiplication and addition in binary) 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".)^[1] Schwartzman notes that the expected purely Latin form would be "sexadecimal", but then computer hackers would be tempted to shorten the word to "sex".^[2] Incidentally, the etymologically proper Greek term would be hexadecadic (although in Modern Greek deca-hexadic (δεκαεξαδικός) is more commonly used).

[edit] 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, 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] See also

[edit] References

[edit] External links

• A Guide to Network Math
• Online Converter for Decimal/Hexadecimal Numerals (JavaScript, GPL)
• Intuitor Hex Headquarters - A site dedicated to changing the traditional base 10 (decimal) standard to hexadecimal.
• The hexadecimal metric system  at hexadecimal.florencetime.net
• base42 - A site proposing an alternate symbol set for hexadecimal numbers.
• Leet Key, a Firefox extension that supports ASCII/Hex conversions and typing
• (Hexadecimal) web colors explained

• Hex/ASCII 'translation' service