In mathematics and computing, hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values zero to nine, and "A"–"F" (or alternatively "a"–"f") to represent values ten to fifteen.
Hexadecimal numerals are widely used by computer system designers and programmers, as they provide a more human-friendly representation of binary-coded values. Each hexadecimal digit represents four binary digits, also known as a nibble, which is half a byte. For example, a single byte can have values ranging from 0000 0000 to 1111 1111 in binary form, which can be more conveniently represented as 00 to FF in hexadecimal.
In mathematics, a subscript is typically used to specify the radix. For example the decimal value 10,995 would be expressed in hexadecimal as 2AF3_{16}. In programming, a number of notations are used to support hexadecimal representation, usually involving a prefix or suffix. The prefix 0x
is used in C and related languages, which would denote this value by 0x2AF3
.
Hexadecimal is used in the transfer encoding Base16, in which each byte of the plaintext is broken into two 4-bit values and represented by two hexadecimal digits.
0_{hex} | = | 0_{dec} | = | 0_{oct} | 0 | 0 | 0 | 0 | |
1_{hex} | = | 1_{dec} | = | 1_{oct} | 0 | 0 | 0 | 1 | |
2_{hex} | = | 2_{dec} | = | 2_{oct} | 0 | 0 | 1 | 0 | |
3_{hex} | = | 3_{dec} | = | 3_{oct} | 0 | 0 | 1 | 1 | |
4_{hex} | = | 4_{dec} | = | 4_{oct} | 0 | 1 | 0 | 0 | |
5_{hex} | = | 5_{dec} | = | 5_{oct} | 0 | 1 | 0 | 1 | |
6_{hex} | = | 6_{dec} | = | 6_{oct} | 0 | 1 | 1 | 0 | |
7_{hex} | = | 7_{dec} | = | 7_{oct} | 0 | 1 | 1 | 1 | |
8_{hex} | = | 8_{dec} | = | 10_{oct} | 1 | 0 | 0 | 0 | |
9_{hex} | = | 9_{dec} | = | 11_{oct} | 1 | 0 | 0 | 1 | |
A_{hex} | = | 10_{dec} | = | 12_{oct} | 1 | 0 | 1 | 0 | |
B_{hex} | = | 11_{dec} | = | 13_{oct} | 1 | 0 | 1 | 1 | |
C_{hex} | = | 12_{dec} | = | 14_{oct} | 1 | 1 | 0 | 0 | |
D_{hex} | = | 13_{dec} | = | 15_{oct} | 1 | 1 | 0 | 1 | |
E_{hex} | = | 14_{dec} | = | 16_{oct} | 1 | 1 | 1 | 0 | |
F_{hex} | = | 15_{dec} | = | 17_{oct} | 1 | 1 | 1 | 1 |
In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give the base explicitly: 159_{10} is decimal 159; 159_{16} is hexadecimal 159, which is equal to 345_{10}. Some authors prefer a text subscript, such as 159_{decimal} and 159_{hex}, or 159_{d} and 159_{h}.
In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:
%
: http://www.example.com/name%20with%20spaces
where %20
is the space (blank) character, ASCII code point 20 in hex, 32 in decimal.ode;
, where the x denotes that code is a hex code point (of 1- to 6-digits) assigned to the character in the Unicode standard. Thus ’
represents the right single quotation mark (’), Unicode code point number 2019 in hex, 8217 (thus ’
in decimal).^{[1]}U+
followed by the hex value, e.g. U+20AC
is the Euro sign (€).#
: white, for example, is represented #FFFFFF
.^{[2]} CSS allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange: ).0x
for numeric constants represented in hex: 0x5A3
. Character and string constants may express character codes in hexadecimal with the prefix \x
followed by two hex digits: '\x1B'
represents the Esc control character; "\x1B[0m\x1B[25;1H"
is a string containing 11 characters (plus a trailing NUL to mark the end of the string) with two embedded Esc characters.^{[3]} To output an integer as hexadecimal with the printf function family, the format conversion code %X
or %x
is used.=
, as in Espa=F1a
to send "España" (Spain). (Hexadecimal F1, equal to decimal 241, is the code number for the lower case n with tilde in the ISO/IEC 8859-1 character set.)FFh
or 05A3H
. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write 0FFh
instead of FFh
$
as a prefix: $5A3
.H'ABCD'
(for ABCD_{16}). Similarly, Fortran 95 uses Z'ABCD'.16#5A3#
. For bit vector constants VHDL uses the notation x"5A3"
.^{[5]}8'hFF
, where 8 is the number of bits in the value and FF is the hexadecimal constant.16r
: 16r5A3
16#
: 16#5A3
. For PostScript, binary data (such as image pixels) can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC
...#x
and #16r
. Setting the variables *read-base*^{[6]} and *print-base*^{[7]} to 16 can also be used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16.&H
: &H5A3
&
for hex.^{[9]}0h
prefix: 0h5A3
16r
to denote hexadecimal numbers: 16r5a3
. Binary, quaternary (base-4) and octal numbers can be specified similarly.X'5A3'
, and is used in Assembler, PL/I, COBOL, JCL, scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well. Occasionally quotation marks were used instead of apostrophes.:
). This, for example, is a valid IPv6 address: 2001:0db8:85a3:0000:0000:8a2e:0370:7334; this can be abbreviated as 2001:db8:85a3::8a2e:370:7334. By contrast, IPv4 addresses are usually written in decimal.{3F2504E0-4F89-41D3-9A0C-0305E82C3301}
.There is no universal convention to use lowercase or uppercase for the letter digits, and each is prevalent or preferred in particular environments by community standards or convention.
The use of the letters A through F to represent the digits above 9 was not universal in the early history of computers.
There are no traditional numerals to represent the quantities from ten to fifteen – letters are used as a substitute – and most European languages lack non-decimal names for the numerals above ten. Even though English has names for several 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 describes the hexadecimal powers (decimal 16, 256, 4096, 65536, ... ). Some people read hexadecimal numbers digit by digit like a phone number, or using the NATO phonetic alphabet, the Joint Army/Navy Phonetic Alphabet, or a similar ad hoc system.
Systems of counting on digits have been devised for both binary and hexadecimal. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 1023_{10} on ten fingers.^{[18]} Another system for counting up to FF_{16} (255_{10}) is illustrated on the right.
The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −42_{10} and so on.
Hexadecimal can also be used to express the exact bit patterns used in the processor, so a sequence of hexadecimal digits may represent a signed or even a floating point value. This way, the negative number −42_{10} can be written as FFFF FFD6 in a 32-bit CPU register (in two's-complement), as C228 0000 in a 32-bit FPU register or C045 0000 0000 0000 in a 64-bit FPU register (in the IEEE floating-point standard).
Just as decimal numbers can be represented in exponential notation, so too can hexadecimal numbers. By convention, the letter P (or p, for "power") represents times two raised to the power of, whereas E (or e) serves a similar purpose in decimal as part of the E notation. The number after the P is decimal and represents the binary exponent.
Usually the number is normalised so that the leading hexadecimal digit is 1 (unless the value is exactly 0).
Example: 1.3DEp42 represents 1.3DE_{16} × 2^{42}.
Hexadecimal exponential notation is required by the IEEE 754-2008 binary floating-point standard. This notation can be used for floating-point literals in the C99 edition of the C programming language.^{[19]} Using the %a or %A conversion specifiers, this notation can be produced by implementations of the printf family of functions following the C99 specification^{[20]} and Single Unix Specification (IEEE Std 1003.1) POSIX standard.^{[21]}
Most computers manipulate binary data, but it is difficult for humans to work with the large number of digits for even a relatively small binary number. Although most humans are 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 (4_{10}). This example converts 1111_{2} to base ten. Since each position in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from the right:
Therefore:
1111_{2} | = 8_{10} + 4_{10} + 2_{10} + 1_{10} |
= 15_{10} |
With little practice, mapping 1111_{2} to F_{16} in one step becomes easy: see table in Written representation. The advantage of using hexadecimal rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hexadecimal, it is trivial to regard the binary string as 4-digit groups and map each to a single hexadecimal digit.
This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
(01011110101101010010)_{2} | = 262144_{10} + 65536_{10} + 32768_{10} + 16384_{10} + 8192_{10} + 2048_{10} + 512_{10} + 256_{10} + 64_{10} + 16_{10} + 2_{10} |
= 387922_{10} |
Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently, and converted directly:
(01011110101101010010)_{2} | = | 0101_{ } | 1110_{ } | 1011_{ } | 0101_{ } | 0010_{2} |
= | 5 | E | B | 5 | 2_{16} | |
= | 5EB52_{16} |
The conversion from hexadecimal to binary is equally direct.
Although quaternary (base 4) is little used, it can easily be converted to and from hexadecimal or binary. Each hexadecimal digit corresponds to a pair of quaternary digits and each quaternary digit corresponds to a pair of binary digits. In the above example 5 E B 5 2_{16} = 11 32 23 11 02_{4}.
The octal (base 8) system can also be converted with relative ease, although not quite as trivially as with bases 2 and 4. Each octal digit corresponds to three binary digits, rather than four. Therefore we can convert between octal and hexadecimal via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four.
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. In theory, 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_{i}h_{i−1}...h_{2}h_{1} be the hexadecimal digits representing the number.
"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; if (d - r == 0) { return toChar(r); } return toHex( (d - r)/16 ) + toChar(r); } function toChar(n) { const alpha = "0123456789ABCDEF"; return alpha.charAt(n); }
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. That is, to convert the number B3AD to decimal one can split the hexadecimal number into its digits: B (11_{10}), 3 (3_{10}), A (10_{10}) and D (13_{10}), and then get the final result by multiplying each decimal representation by 16^{p}, where p is the corresponding hex digit position, counting from right to left, beginning with 0. In this case we have B3AD = (11 × 16^{3}) + (3 × 16^{2}) + (10 × 16^{1}) + (13 × 16^{0}), which is 45997 base 10.
Most modern computer systems with graphical user interfaces provide a built-in calculator utility, capable of performing conversions between various radices, in general including hexadecimal.
In Microsoft Windows, the Calculator utility can be set to Scientific mode (called Programmer mode in some versions), which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 (octal) and 2 (binary), the bases most commonly used by programmers. In Scientific Mode, the on-screen numeric keypad includes the hexadecimal digits A through F, which are active when "Hex" is selected. In hex mode, however, the Windows Calculator supports only integers.
As with other numeral systems, the hexadecimal system can be used to represent rational numbers, although repeating expansions are common since sixteen (10_{16}) has only a single prime factor; two.
For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two for binary or dividing one by sixteen for hexadecimal, both of these fractions are written as 0.1
. Because the radix 16 is a perfect square (4^{2}), 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 are exhibited 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 result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than decimal for representing rational numbers since a larger proportion lie outside its range of finite representation.
All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hexadecimal number with a finite number of digits also has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hexadecimal. For example, decimal 0.1 corresponds to the infinite recurring representation 0.19 in hexadecimal. However, hexadecimal is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.0625_{10} (one sixteenth) is equivalent to 0.1_{16}, 0.09_{12}, and 0;3,45_{60}.
n | Decimal Prime factors of base, b = 10: 2, 5; b − 1 = 9: 3; b + 1 = 11: 11 |
Hexadecimal Prime factors of base, b = 16_{10} = 10: 2; b − 1 = 15_{10} = F: 3, 5; b + 1 = 17_{10} = 11: 11 | ||||
---|---|---|---|---|---|---|
Fraction | Prime factors | Positional representation | Positional representation | Prime factors | Fraction(1/n) | |
2 | 1/2 | 2 | 0.5 | 0.8 | 2 | 1/2 |
3 | 1/3 | 3 | 0.3333... = 0.3 | 0.5555... = 0.5 | 3 | 1/3 |
4 | 1/4 | 2 | 0.25 | 0.4 | 2 | 1/4 |
5 | 1/5 | 5 | 0.2 | 0.3 | 5 | 1/5 |
6 | 1/6 | 2, 3 | 0.16 | 0.2A | 2, 3 | 1/6 |
7 | 1/7 | 7 | 0.142857 | 0.249 | 7 | 1/7 |
8 | 1/8 | 2 | 0.125 | 0.2 | 2 | 1/8 |
9 | 1/9 | 3 | 0.1 | 0.1C7 | 3 | 1/9 |
10 | 1/10 | 2, 5 | 0.1 | 0.19 | 2, 5 | 1/A |
11 | 1/11 | 11 | 0.09 | 0.1745D | B | 1/B |
12 | 1/12 | 2, 3 | 0.083 | 0.15 | 2, 3 | 1/C |
13 | 1/13 | 13 | 0.076923 | 0.13B | D | 1/D |
14 | 1/14 | 2, 7 | 0.0714285 | 0.1249 | 2, 7 | 1/E |
15 | 1/15 | 3, 5 | 0.06 | 0.1 | 3, 5 | 1/F |
16 | 1/16 | 2 | 0.0625 | 0.1 | 2 | 1/10 |
17 | 1/17 | 17 | 0.0588235294117647 | 0.0F | 11 | 1/11 |
18 | 1/18 | 2, 3 | 0.05 | 0.0E38 | 2, 3 | 1/12 |
19 | 1/19 | 19 | 0.052631578947368421 | 0.0D79435E5 | 13 | 1/13 |
20 | 1/20 | 2, 5 | 0.05 | 0.0C | 2, 5 | 1/14 |
21 | 1/21 | 3, 7 | 0.047619 | 0.0C3 | 3, 7 | 1/15 |
22 | 1/22 | 2, 11 | 0.045 | 0.0BA2E8 | 2, B | 1/16 |
23 | 1/23 | 23 | 0.0434782608695652173913 | 0.0B21642C859 | 17 | 1/17 |
24 | 1/24 | 2, 3 | 0.0416 | 0.0A | 2, 3 | 1/18 |
25 | 1/25 | 5 | 0.04 | 0.0A3D7 | 5 | 1/19 |
26 | 1/26 | 2, 13 | 0.0384615 | 0.09D8 | 2, D | 1/1A |
27 | 1/27 | 3 | 0.037 | 0.097B425ED | 3 | 1/1B |
28 | 1/28 | 2, 7 | 0.03571428 | 0.0924 | 2, 7 | 1/1C |
29 | 1/29 | 29 | 0.0344827586206896551724137931 | 0.08D3DCB | 1D | 1/1D |
30 | 1/30 | 2, 3, 5 | 0.03 | 0.08 | 2, 3, 5 | 1/1E |
31 | 1/31 | 31 | 0.032258064516129 | 0.08421 | 1F | 1/1F |
32 | 1/32 | 2 | 0.03125 | 0.08 | 2 | 1/20 |
33 | 1/33 | 3, 11 | 0.03 | 0.07C1F | 3, B | 1/21 |
34 | 1/34 | 2, 17 | 0.02941176470588235 | 0.078 | 2, 11 | 1/22 |
35 | 1/35 | 5, 7 | 0.0285714 | 0.075 | 5, 7 | 1/23 |
36 | 1/36 | 2, 3 | 0.027 | 0.071C | 2, 3 | 1/24 |
The table below gives the expansions of some common irrational numbers in decimal and hexadecimal.
Number | Positional representation | |
---|---|---|
Decimal | Hexadecimal | |
√2 (the length of the diagonal of a unit square) | 1.414213562373095048... | 1.6A09E667F3BCD... |
√3 (the length of the diagonal of a unit cube) | 1.732050807568877293... | 1.BB67AE8584CAA... |
√5 (the length of the diagonal of a 1×2 rectangle) | 2.236067977499789696... | 2.3C6EF372FE95... |
φ (phi, the golden ratio = (1+√5)/2) | 1.618033988749894848... | 1.9E3779B97F4A... |
π (pi, the ratio of circumference to diameter of a circle) | 3.141592653589793238462643 383279502884197169399375105... |
3.243F6A8885A308D313198A2E0 3707344A4093822299F31D008... |
e (the base of the natural logarithm) | 2.718281828459045235... | 2.B7E151628AED2A6B... |
τ (the Thue–Morse constant) | 0.412454033640107597... | 0.6996 9669 9669 6996... |
γ (the limiting difference between the harmonic series and the natural logarithm) |
0.577215664901532860... | 0.93C467E37DB0C7A4D1B... |
Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.
2^{x} | Value | Value (Decimal) |
---|---|---|
2^{0} | 1 | 1 |
2^{1} | 2 | 2 |
2^{2} | 4 | 4 |
2^{3} | 8 | 8 |
2^{4} | 10_{hex} | 16_{dec} |
2^{5} | 20_{hex} | 32_{dec} |
2^{6} | 40_{hex} | 64_{dec} |
2^{7} | 80_{hex} | 128_{dec} |
2^{8} | 100_{hex} | 256_{dec} |
2^{9} | 200_{hex} | 512_{dec} |
2^{A} (2^{10dec}) | 400_{hex} | 1024_{dec} |
2^{B} (2^{11dec}) | 800_{hex} | 2048_{dec} |
2^{C} (2^{12dec}) | 1000_{hex} | 4096_{dec} |
2^{D} (2^{13dec}) | 2000_{hex} | 8192_{dec} |
2^{E} (2^{14dec}) | 4000_{hex} | 16,384_{dec} |
2^{F} (2^{15dec}) | 8000_{hex} | 32,768_{dec} |
2^{10} (2^{16dec}) | 10000_{hex} | 65,536_{dec} |
The word hexadecimal is composed of hexa-, derived from the Greek ἕξ (hex) for six, and -decimal, derived from the Latin for tenth. Webster's Third New International online derives hexadecimal as an alteration of the all-Latin sexadecimal (which appears in the earlier Bendix documentation). The earliest date attested for hexadecimal in Merriam-Webster Collegiate online is 1954, placing it safely in the category of international scientific vocabulary (ISV). It is common in ISV to mix Greek and Latin combining forms freely. The word sexagesimal (for base 60) retains the Latin prefix. Donald Knuth has pointed out that the etymologically correct term is senidenary (or possibly, sedenary), from the Latin term for grouped by 16. (The terms binary, ternary and quaternary are from the same Latin construction, and the etymologically correct terms for decimal and octal arithmetic are denary and octonary, respectively.)^{[22]} Alfred B. Taylor used senidenary in his mid-1800s work on alternative number bases, although he rejected base 16 because of its "incommodious number of digits".^{[23]}^{[24]} Schwartzman notes that the expected form from usual Latin phrasing would be sexadecimal, but computer hackers would be tempted to shorten that word to sex.^{[25]} The etymologically proper Greek term would be hexadecadic / ἑξαδεκαδικός / hexadekadikós (although in Modern Greek, decahexadic / δεκαεξαδικός / dekaexadikos is more commonly used).
In hexadecimal, numbers with nondecreasing digits are called plaindrones, those with nonincreasing digits are called nialpdromes, those with descending digits are called katadromes, and those with ascending digits are called metadromes.^{[26]}^{[27]}
The traditional Chinese units of measurement were base-16. For example, one jīn (斤) in the old system equals sixteen taels. The suanpan (Chinese abacus) can be used to perform hexadecimal calculations.
As with the duodecimal system, there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals.^{[28]} Some proposals unify standard measures so that they are multiples of 16.^{[29]}^{[30]}^{[31]}
An example of unified standard measures is hexadecimal time, which subdivides a day by 16 so that there are 16 "hexhours" in a day.^{[31]}
Base16 (as a proper name without a space) can also refer to a binary to text encoding belonging to the same family as Base32, Base58, and Base64.
In this case, data is broken into 4-bit sequences, and each value (between 0 and 15 inclusively) is encoded using 16 symbols from the ASCII character set. Although any 16 symbols from the ASCII character set can be used, in practice the ASCII digits '0'-'9' and the letters 'A'-'F' (or the lowercase 'a'-'f') are always chosen in order to align with standard written notation for hexadecimal numbers.
There are several advantages of Base16 encoding:
The main disadvantages of Base16 encoding are:
Support for Base16 encoding is ubiquitous in modern computing. It is the basis for the W3C standard for URL Percent Encoding, where a character is replaced with a percent sign "%" and its Base16-encoded form. Most modern programming languages directly include support for formatting and parsing Base16-encoded numbers.
"\x1B[0m\x1B[25;1H"
specifies the character sequence Esc [ 0 m Esc [ 2 5 ; 1 H Nul. These are the escape sequences used on an ANSI terminal that reset the character set and color, and then move the cursor to line 25.
&
to prefix octal values. (Microsoft BASIC primarily uses &O
to prefix octal, and it uses &H
to prefix hexadecimal, but the ampersand alone yields a default interpretation as an octal prefix.
This base is used because a group of four bits can represent any one of sixteen different numbers (zero to fifteen). By assigning a symbol to each of these combinations we arrive at a notation called sexadecimal (usually hex in conversation because nobody wants to abbreviate sex). The symbols in the sexadecimal language are the ten decimal digits and, on the G-15 typewriter, the letters u, v, w, x, y and z. These are arbitrary markings; other computers may use different alphabet characters for these last six digits.
[…] They can be used interchangeable in present or future designs to offer designers a choice between two indicator fonts. The '46A, '47A, 'LS47, and 'LS48 compose the 6 and the 9 without tails and the '246, '247, 'LS247, and 'LS248 compose the 6 and the 0 with tails. Composition of all other characters, including display patterns for BCD inputs above nine, is identical. […] Display patterns for BCD input counts above 9 are unique symbols to authenticate input conditions. […]
In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically "0" (zero) and "1" (one).
The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices.
Digital Color MeterDigital Color Meter is a utility for measuring and displaying the color values of pixels displayed on the screen of a Macintosh computer.
The utility presents a "window" onto the screen which includes a cursor which by default is 1 × 1 pixel in size. The color displayed in that pixel is shown as a color value which may be represented as decimal or hexadecimal RGB triplets, CIE 1931, CIE 1976 or CIELAB triplets or a Tristimulus triplet.
The displayed color could be copied either as a solid color or as the color value which represents it, to be used in other applications (for instance an RGB triplet may be used in a color specification to be used on a World Wide Web page).
Eight OnesEO, or Eight Ones, is an 8-bit EBCDIC character code represented as all ones (binary 1111 1111, hexadecimal FF).
When translated from the EBCDIC character set to code pages with the C1 control code set, it is typically mapped to hexadecimal code 9F, in order to provide a unique character mapping in both directions.
Hex dumpIn computing, a hex dump is a hexadecimal view (on screen or paper) of computer data, from RAM or from a computer file or storage device. Looking at a hex dump of data is usually done in the context of either debugging or reverse engineering, although it is rare in modern times to need to look at a hex dump while debugging.
In a hex dump, each byte (8-bits) is represented as a two-digit hexadecimal number. Hex dumps are commonly organized into rows of 8 or 16 bytes, sometimes separated by whitespaces. Some hex dumps have the hexadecimal memory address at the beginning and/or a checksum byte at the end of each line.
Although the name implies the use of base-16 output, some hex dumping software may have options for base-8 (octal) or base-10 (decimal) output. Some common names for this program function are hexdump, od, xxd and simply dump or even D.
Hex editorA hex editor (or binary file editor or byte editor) is a type of computer program that allows for manipulation of the fundamental binary data that constitutes a computer file. The name 'hex' comes from 'hexadecimal': a standard numerical format for representing binary data. A typical computer file occupies multiple areas on the platter(s) of a disk drive, whose contents are combined to form the file. Hex editors that are designed to parse and edit sector data from the physical segments of floppy or hard disks are sometimes called sector editors or disk editors.
Hexadecimal timeHexadecimal time is the representation of the time of day as a hexadecimal number in the interval [0,1).
The day is divided into 1016 (1610) hexadecimal hours, each hour into 10016 (25610) hexadecimal minutes, and each minute into 1016 (1610) hexadecimal seconds.
IBM hexadecimal floating pointIBM System/360 computers, and subsequent machines based on that architecture (mainframes), support a hexadecimal floating-point format (HFP).In comparison to IEEE 754 floating-point, the IBM floating-point format has a longer significand, and a shorter exponent. All IBM floating-point formats have 7 bits of exponent with a bias of 64. The normalized range of representable numbers is from 16−65 to 1663 (approx. 5.39761 × 10−79 to 7.237005 × 1075).
The number is represented as the following formula: (−1)sign × 0.significand × 16exponent−64.
KEISKEIS is a stateful EBCDIC charset used in Hitachi mainframe systems. KEIS is an acronym for "Kanji processing Extended Information System".
MAC addressA media access control address (MAC address) of a device is a unique identifier assigned to a network interface controller (NIC). For communications within a network segment, it is used as a network address for most IEEE 802 network technologies, including Ethernet, Wi-Fi, and Bluetooth. Within the Open Systems Interconnection (OSI) model, MAC addresses are used in the medium access control protocol sublayer of the data link layer. As typically represented, MAC addresses are recognizable as six groups of two hexadecimal digits, separated by hyphens, colons, or no separator.
A MAC address may be referred to as the burned-in address, and is also known as an Ethernet hardware address, hardware address, and physical address.
A network node with multiple NICs must have a unique MAC address for each. Sophisticated network equipment such as a multilayer switch or router may require one or more permanently assigned MAC addresses.
MAC addresses are most often assigned by the manufacturer of network interface cards. Each is stored in hardware, such as the card's read-only memory or by a firmware mechanism. A MAC address typically includes the manufacturer's organizationally unique identifier (OUI). MAC addresses are formed according to the principles of two numbering spaces based on Extended Unique Identifiers (EUI) managed by the Institute of Electrical and Electronics Engineers (IEEE): EUI-48, which replaces the obsolete term MAC-48, and EUI-64.
NibbleIn computing, a nibble (occasionally nybble or nyble to match the spelling of byte) is a four-bit aggregation, or half an octet. It is also known as half-byte or tetrade. In a networking or telecommunication context, the nibble is often called a semi-octet, quadbit, or quartet. A nibble has sixteen (24) possible values. A nibble can be represented by a single hexadecimal digit and called a hex digit.A full byte (octet) is represented by two hexadecimal digits; therefore, it is common to display a byte of information as two nibbles. Sometimes the set of all 256 byte values is represented as a 16×16 table, which gives easily readable hexadecimal codes for each value.
Four-bit computer architectures use groups of four bits as their fundamental unit. Such architectures were used in early microprocessors, pocket calculators and pocket computers. They continue to be used in some microcontrollers.
Numeric character referenceA numeric character reference (NCR) is a common markup construct used in SGML and SGML-derived markup languages such as HTML and XML. It consists of a short sequence of characters that, in turn, represents a single character. Since WebSgml, XML and HTML 4, the code points of the Universal Character Set (UCS) of Unicode are used. NCRs are typically used in order to represent characters that are not directly encodable in a particular document (for example, because they are international characters that don't fit in the 8-bit character set being used, or because they have special syntactic meaning in the language). When the document is interpreted by a markup-aware reader, each NCR is treated as if it were the character it represents.
OctalThe octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping consecutive binary digits into groups of three (starting from the right). For example, the binary representation for decimal 74 is 1001010. Two zeroes can be added at the left: (00)1 001 010, corresponding the octal digits 1 1 2, yielding the octal representation 112.
In the decimal system each decimal place is a power of ten. For example:
In the octal system each place is a power of eight. For example:
By performing the calculation above in the familiar decimal system we see why 112 in octal is equal to 64+8+2 = 74 in decimal.
Quaternary numeral systemQuaternary is the base-4 numeral system. It uses the digits 0, 1, 2 and 3 to represent any real number.
Four is the largest number within the subitizing range and one of two numbers that is both a square and a highly composite number (the other being 36), making quaternary a convenient choice for a base at this scale. Despite being twice as large, its radix economy is equal to that of binary. However, it fares no better in the localization of prime numbers (the next best being the primorial base six, senary).
Quaternary shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation (almost unique) and the characteristics of the representations of rational numbers and irrational numbers. See decimal and binary for a discussion of these properties.
RadixIn digital numeral systems, the radix or base is the number of unique digits, including the digit zero, used to represent numbers in a positional numeral system. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ten, because it uses the ten digits from 0 through 9.
In any standard positional numeral system, a number is conventionally written as (x)y with x as the string of digits and y as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (100)dec = 100 (in the decimal system) represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four.
SuanpanThe suanpan (simplified Chinese: 算盘; traditional Chinese: 算盤; pinyin: suànpán), also spelled suan pan or souanpan) is an abacus of Chinese origin first described in a 190 CE book of the Eastern Han Dynasty, namely Supplementary Notes on the Art of Figures written by Xu Yue. However, the exact design of this suanpan is not known.
Usually, a suanpan is about 20 cm (8 in) tall and it comes in various widths depending on the application. It usually has more than seven rods. There are two beads on each rod in the upper deck and five beads on each rod in the bottom deck. This configuration is used for both decimal and hexadecimal computation. The beads are usually rounded and made of a hardwood. The beads are counted by moving them up or down towards the beam. The suanpan can be reset to the starting position instantly by a quick jerk around the horizontal axis to spin all the beads away from the horizontal beam at the center.
Suanpans can be used for functions other than counting. Unlike the simple counting board used in elementary schools, very efficient suanpan techniques have been developed to do multiplication, division, addition, subtraction, square root and cube root operations at high speed.
The modern suanpan has 4+1 beads, colored beads to indicate position and a clear-all button. When the clear-all button is pressed, two mechanical levers push the top row beads to the top position and the bottom row beads to the bottom position, thus clearing all numbers to zero. This replaces clearing the beads by hand, or quickly rotating the suanpan around its horizontal center line to clear the beads by centrifugal force.
Tiger (hash function)In cryptography, Tiger is a cryptographic hash function designed by Ross Anderson and Eli Biham in 1995 for efficiency on 64-bit platforms. The size of a Tiger hash value is 192 bits. Truncated versions (known as Tiger/128 and Tiger/160) can be used for compatibility with protocols assuming a particular hash size. Unlike the SHA-2 family, no distinguishing initialization values are defined; they are simply prefixes of the full Tiger/192 hash value.
Tiger2 is a variant where the message is padded by first appending a byte with the hexadecimal value of 0x80 as in MD4, MD5 and SHA, rather than with the hexadecimal value of 0x01 as in the case of Tiger. The two variants are otherwise identical.
Unicode blockA Unicode block is one of several contiguous ranges of numeric character codes (code points) of the Unicode character set that are defined by the Unicode Consortium for administrative and documentation purposes. Typically, proposals such as the addition of new glyphs are discussed and evaluated by considering the relevant block or blocks as a whole.
Each block is generally, but not always, meant to include all the glyphs used by one or more specific languages, or in some general application area such as mathematics, surveying, decorative typesetting, social forums, etc..
Unicode blocks are identified by unique names, which use only ASCII characters and are usually descriptive of the nature of the symbols, in English; such as "Tibetan" or "Supplemental Arrows-A". (When comparing block names, one is supposed to equate uppercase with lowercase letters, and ignore any whitespace, hyphens, and underbars; so the last name is equivalent to "supplemental_arrows__a" and "SUPPLEMENTALARROWSA".)
Blocks are pairwise disjoint, that is, they do not overlap. The starting code point and the size (number of code points) of each block are always multiples of 16; therefore, in the hexadecimal notation, the starting (smallest) point is U+xxx0 and the ending (largest) point is U+yyyF, where 'xxx and yyy are three or more hexadecimal digits. (These constraints are intended to simplify the display of glyphs in Unicode Consortium documents, as tables with 16 columns labeled with the last hexadecimal digit of the code point.) The size of a block may range from the minimum of 16 to a maximum of 65,536 code points.
Every assigned code point has a glyph property called "Block", whose value is a character string naming the unique block that owns that point. However, a block may also contain unassigned code points, usually reserved for future additions of characters that "logically" should belong to that block. Code points not belonging to any of the named blocks, e.g. in the unassigned planes 3–13, have the value block="No_block".Each Unicode point also has a property called "General Category", that attempts to describes the role of the corresponding symbol in the languages or applications for whose sake it was included in the system. Examples of General Categories are "Lu" (meaning upper-case letter), "Nd" (decimal digit), "Pi" (open-quote punctuation), and "Mn" (non-spacing mark, i.e. a diacritic for the preceding glyph). This division is completely independent of code blocks: the code points with a given General Category generally span many blocks, and do not have to be consecutive, not even within each block.In descriptions of the Unicode system, a block may be subdivided into more specific subgroups, such as the "Chess symbols" in the block "Miscellaneous symbols". Those subgroups are not "blocks" in the technical sense used by the Unicode consortium, and are named only for the convenience of users.
Unicode 12.1 defines 300 blocks:
163 in plane 0, the Basic Multilingual Plane (BMP)
127 in plane 1, the Supplementary Multilingual Plane (SMP)
6 in plane 2, the Supplementary Ideographic Plane (SIP)
2 in plane 14 (E in hexadecimal), the Supplementary Special-purpose Plane (SSP)
One each in planes 15 (Fhex) and 16 (10hex), called Supplementary Private Use Area-A and -B
Unicode inputUnicode input is the insertion of a specific Unicode character on a computer by a user; it is a common way to input characters not directly supported by a physical keyboard. Unicode characters can be produced either by selecting them from a display or by typing a certain sequence of keys on a physical keyboard. In addition, a character produced by one of these methods in one web page or document can be copied into another. Unicode is similar to ASCII but provides many more options and encodes many more signs.A Unicode input system needs to provide a large repertoire of characters, ideally all valid Unicode code points. This is different from a keyboard layout which defines keys and their combinations only for a limited number of characters appropriate for a certain locale.
Web colorsWeb colors are colors used in displaying web pages on the World Wide Web, and the methods for describing and specifying those colors. Colors may be specified as an RGB triplet or in hexadecimal format (a hex triplet) or according to their common English names in some cases. A color tool or other graphics software is often used to generate color values. In some uses, hexadecimal color codes are specified with notation using a leading number sign (#). A color is specified according to the intensity of its red, green and blue components, each represented by eight bits. Thus, there are 24 bits used to specify a web color within the sRGB gamut, and 16,777,216 colors that may be so specified.
Colors outside the sRGB gamut can be specified in Cascading Style Sheets by making one or more of the red, green and blue components negative or greater than 100%, so the color space is theoretically an unbounded extrapolation of sRGB similar to scRGB. Specifying a non-sRGB color this way requires the RGB() function call; it is impossible with the hexadecimal syntax (and thus impossible in legacy HTML documents that do not use CSS).
The first versions of Mosaic and Netscape Navigator used the X11 color names as the basis for their color lists, as both started as X Window System applications.
Web colors have an unambiguous colorimetric definition, sRGB, which relates the chromaticities of a particular phosphor set, a given transfer curve, adaptive whitepoint, and viewing conditions. These have been chosen to be similar to many real-world monitors and viewing conditions, in order to allow rendering to be fairly close to the specified values even without color management. User agents vary in the fidelity with which they represent the specified colors. More advanced user agents use color management to provide better color fidelity; this is particularly important for Web-to-print applications.
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