Hexadecimal

In mathematics and computer science, 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 = 01000000, F = 1111).

It is a useful system in computers because there is an easy mapping from four bits to a single hex digit. A byte can be represented as two consecutive hexadecimal digits.

It was IBM that decided on the prefix of "hexa" rather than the proper Latin but more politically incorrect 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 "ten". 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 pure Latin "sexidecimal", 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".

Since several years an alternate, unambiguous set of hexadecimal digits is 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# and Java) 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.
• In HTML, hexadecimal character references also use the x: ֣ should give the same as ֣ – with your browser ֣ and ֣ respectively (Hebrew accent munah).
• 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 "<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 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!).
• 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).

A hexadecimal multiplication table

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.

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.

Fractions

As with other numeral systems, the hexadecimal system can be used in forming vulgar fractions, although recurring digits are common:

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 |align=right| 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.

Many versions of the 32-bit PowerPC chipsets are initialized with a value of "DEADBEEF" in the GPRs during power-on; this has aided in the debugging of bootloader and kernel crashes during early boot, since memory which has never been written to will retain the value. Although the value chosen is clearly humourous, this is actually an example of a "magic number" — in this case, one which is deliberately added as a marker so that the effects of particular parts of the code can easily be traced.

A Knuth_reward_check is one hexadecimal dollar, or $2.56.

Mapping to binary

When working with computers we often need to deal with binary data. It is much easier to handle in hexadecimal or octal 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 or octal than to decimal since each hexadecimal or octal digit maps to a whole number of bits (4 for hexadecimal 3 for octal). Hexadecimal's big advantage over octal is that exactly 2 digits represent a byte. 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.

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 very 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 similarly easy to map to binary, only in blocks of 3 bits instead of 4.

See also

• Base32
• Base64
• Numeral system — a list of other base systems
• Hexadecimal time
• Hexspeak
• Nibble — one hexadecimal digit can exactly represent one "nibble"

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

Calculators

• Hex/Unsigned Decimal/Binary Converter (integer only)
• Hex/Unsigned Decimal Converter
• Virtual Calc 2000 - Arbitrary Precision Calculator will do floating-point hexadecimal arithmetic