Hexadecimal

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Hexadecimal

In mathematics and computer science, base-, 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. 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.

For example, the decimal numeral 79 whose binary representation is 01001111 can be written as 4F in hexadecimal (4 = 0100, F = 1111). 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 έξι (hexi) 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". Schwartzman notes: “Since "hexadecimal" is a rather long word, it is sometimes abbreviated "hex". The word "hexadecimal" is unusual because Greek and Latin elements are combined; the expected purely Latin form would be "sexadecimal", but then computer hackers would be tempted to shorten the word to "sex".” 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".)

Several years ago an alternate, unambiguous set of hexadecimal digits was proposed. (Cf. Hexadecimal time)

Representing hexadecimal

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

Some hexadecimal numbers are indistinguishable from a decimal number (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 numbers 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. 0 for Octal). 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".
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 "r" 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 bases of) floating point numbers to be represented.
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.
Donald Knuth introduced the use of different fonts to represent radices in his book The TeXbook. In his notation, hexadecimal numbers are represented in a typewriter type, 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.

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 computers used the letters U through Z.

Uses

A common use of hexadecimal numerals is found in HTML and CSS. They use hexadecimal notation (hex triplets) to specify colours on web pages; there is just the # symbol, not a separate symbol 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

Separate the triplets: FF CF 4B

Convert each hexadecimal value to a decimal number:

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

Hexadecimal triplet FFCF4B = Decimal triplet 255,207,75

Hexadecimal is used also in more generic computing, as the most commonly found form of expressing a guaranteeably human-readable string representation of a byte. All the possible values of a byte (256 values) can be represented using the hexadecimal system. Some people assume that using 8-bit ASCIIto represent the value of a byte should work, but it does not, because ASCII has some unprintable characters (also called control characters) and therefore is not good for this purpose.

In URLs, special characters can be coded hexadecimally, with a percent sign used to introduce each byte; e.g., http://en.wikipedia.org/wiki/Main%20Page

The canonical written form of numeric IPv6 addresses represents each group of 16 bits as a separate hexadecimal number, to ease reading and transcription of the 128-bit addresses.

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

In October 1996, Simon Plouffe, Peter Borwein and Jonathan Borwein created an equation that allows the nth digit of pi in hexadecimal to be calculated, without knowing all (or indeed any) of the previous digits. The equation is given by: $\pi\ = \sum_{n=0}^\infty \left ( \frac{4}{8n+1} - \frac{2}{8n+4} - \frac{1}{8n+5} - \frac{1}{8n+6} \right ) \times \left ( \frac{1}{16} \right )^n$

Fractions

As with other numeral systems, the hexadecimal system can be used in forming vulgar fractions, although recurring digits are common since 16 has only a single prime factor:


1/ 0x1	=	0x1	1/ 0x5	=	0x0.3	1/ 0x9	=	0x0.1C7	1/ 0xD	=	0x0.13B
1/ 0x2	=	0x0.8	1/ 0x6	=	0x0.2A	1/ 0xA	=	0x0.19	1/ 0xE	=	0x0.1249
1/ 0x3	=	0x0.5	1/ 0x7	=	0x0.249	1/ 0xB	=	0x0.1745D	1/ 0xF	=	0x0.1
1/ 0x4	=	0x0.4	1/ 0x8	=	0x0.2	1/ 0xC	=	0x0.15	1/ 0x10	=	0x0.1

Because the radix 16 is a square (4²), hexadecimal fractions have an odd period much more often than decimal ones. Recurring decimals occur when the denominator in lowest terms has a prime factor not found in the radix. In the case of hexadecimal numbers, all fractions with denominators that are not a power of two will result in a recurring decimal.

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". 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 a hexadecimal number. 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).

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.

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 (just think of lots of '0's and '1's) 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 (base 2) number can only be 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 binary number is very much greater, conversion to decimal becomes much more tedious; however, when mapping to hexadecimal, it is simple to divide the binary number up in blocks of 4 positions and map each block of 4 bits to a single position hexadecimal number. For example a tedious conversion to decimal:

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

Compared 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.

Converting from other bases

Division-remainder in source base

As with all bases there is a simple algorithm for converting 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 decimal number to convert, and the series h_ih_i-1...h₂h₁ be the hexadecimal digits representing the number.

1. H₁ := d mod 16
2. D := (d-h₁) / 16
3. If d==0 (return series h_i)
else go to 1

"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 (maybe other uses that may be thought of). 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);}
	else {return toHex( (d-r)/16 )+toChar(r);}
}
function toChar(n) {
	var alpha = "0123456789ABCDEF";
	return alpha.charAt(n);
}

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 full hexadecimal number.

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.

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.

External links

Intuitor Hex Headquarters - A site dedicated to changing the traditional base 10 (decimal) standard to hexadecimal.
Simple Conversion Methods
Leet Key, a Firefox extension that supports ASCII/Hex conversions and typing
Bits of Meaning (pdf) - Introduction to Computer Arithmetic for Bendix G-15 computer
Hexadecimal basics
Hexadecimal Numbers Guide
Hexadecimal Colors from Math Is Fun

Computer arithmetic | Positional numeral systems

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