HEXADECIMAL SYSTEM

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 ASCII to 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:

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.

See also

• Base32
• Base64
• Bubble Babble
• 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

References

1. ^ Knuth, Donald. (1969). Donald Knuth, in The Art of Computer Programming, Volume 2. ISBN 0-201-03802-1. (Chapter 17.)
2. ^ Schwartzman, S. (1994). The Words of Mathematics: an etymological dictionary of mathematical terms used in English. ISBN 0-88385-511-9.

External links

• Intuitor Hex Headquarters - A site dedicated to changing the traditional base 10 (decimal) standard to hexadecimal.
• The hexadecimal metric system at hexadecimal.florencetime.net
• 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