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 2AF316. 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.

## Representation

### Written representation

#### Using 0–9 and A–F

 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 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: 15910 is decimal 159; 15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h.

In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:

• In URIs (including URLs), character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the space (blank) character, ASCII code point 20 in hex, 32 in decimal.
• In XML and XHTML, characters can be expressed as hexadecimal numeric character references using the notation &#xcode;, 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 &#x2019; represents the right single quotation mark (’), Unicode code point number 2019 in hex, 8217 (thus &#8217; in decimal).[1]
• In the Unicode standard, a character value is represented with U+ followed by the hex value, e.g. U+20AC is the Euro sign (€).
• Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits (two each for the red, green and blue components, in that order) prefixed with #: white, for example, is represented #FFFFFF.[2] CSS allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange:     ).
• Unix (and related) shells, AT&T assembly language and likewise the C programming language (and its syntactic descendants such as C++, C#, D, Java, JavaScript, Python and Windows PowerShell) use the prefix 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.
• In MIME (e-mail extensions) quoted-printable encoding, characters that cannot be represented as literal ASCII characters are represented by their codes as two hexadecimal digits (in ASCII) prefixed by an equal to sign =, 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.)
• In Intel-derived assembly languages and Modula-2,[4] hexadecimal is denoted with a suffixed H or h: 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
• Other assembly languages (6502, Motorola), Pascal, Delphi, some versions of BASIC (Commodore), GameMaker Language, Godot and Forth use $as a prefix:$5A3.
• Some assembly languages (Microchip) use the notation H'ABCD' (for ABCD16). Similarly, Fortran 95 uses Z'ABCD'.
• Ada and VHDL enclose hexadecimal numerals in based "numeric quotes": 16#5A3#. For bit vector constants VHDL uses the notation x"5A3".[5]
• Verilog represents hexadecimal constants in the form 8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant.
• The Smalltalk language uses the prefix 16r: 16r5A3
• PostScript and the Bourne shell and its derivatives denote hex with prefix 16#: 16#5A3. For PostScript, binary data (such as image pixels) can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC...
• Common Lisp uses the prefixes #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.
• MSX BASIC,[8] QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H: &H5A3
• BBC BASIC and Locomotive BASIC use & for hex.[9]
• TI-89 and 92 series uses a 0h prefix: 0h5A3
• ALGOL 68 uses the prefix 16r to denote hexadecimal numbers: 16r5a3. Binary, quaternary (base-4) and octal numbers can be specified similarly.
• The most common format for hexadecimal on IBM mainframes (zSeries) and midrange computers (IBM System i) running the traditional OS's (zOS, zVSE, zVM, TPF, IBM i) is 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.
• Donald Knuth introduced the use of a particular typeface to represent a particular radix in his book The TeXbook.[10] Hexadecimal representations are written there in a typewriter typeface: 5A3
• Any IPv6 address can be written as eight groups of four hexadecimal digits (sometimes called hextets), where each group is separated by a colon (:). 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.
• Globally unique identifiers are written as thirty-two hexadecimal digits, often in unequal hyphen-separated groupings, for example {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.

### History of written representations

Bruce Alan Martin's hexadecimal notation proposal[11]

The use of the letters A through F to represent the digits above 9 was not universal in the early history of computers.

• During the 1950s, some installations favored using the digits 0 through 5 with an overline to denote the values 10–15 as 0, 1, 2, 3, 4 and 5.
• The SWAC (1950)[12] and Bendix G-15 (1956)[13][12] computers used the lowercase letters u, v, w, x, y and z for the values 10 to 15.
• The ILLIAC I (1952) computer used the uppercase letters K, S, N, J, F and L for the values 10 to 15.[14][12]
• The Librascope LGP-30 (1956) used the letters F, G, J, K, Q and W for the values 10 to 15.[15][12]
• The Honeywell Datamatic D-1000 (1957) used the lowercase letters b, c, d, e, f, and g whereas the Elbit 100 (1967) used the uppercase letters B, C, D, E, F and G for the values 10 to 15.[12]
• The Monrobot XI (1960) used the letters S, T, U, V, W and X for the values 10 to 15.[12]
• The NEC parametron computer NEAC 1103 (1960) used the letters D, G, H, J, K (and possibly V) for values 10–15.[16]
• The Pacific Data Systems 1020 (1964) used the letters L, C, A, S, M and D for the values 10 to 15.[12]
• New numeric symbols and names were introduced in the Bibi-binary notation by Boby Lapointe in 1968. This notation did not become very popular.
• Bruce Alan Martin of Brookhaven National Laboratory considered the choice of A–F "ridiculous". In a 1968 letter to the editor of the CACM, he proposed an entirely new set of symbols based on the bit locations, which did not gain much acceptance.[11]
• Soviet programmable calculators Б3-34 (1980) and similar used the symbols "−", "L", "C", "Г", "E", " " (space) for the values 10 to 15 on their displays.
• Seven-segment display decoder chips used various schemes for outputting values above nine. The Texas Instruments 7446/7447/7448/7449 and 74246/74247/74248/74249 use truncated versions of "2", "3", "4", "5" and "6" for the values 10 to 14. Value 15 (1111 binary) was blank.[17]

### Verbal and digital representations

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 102310 on ten fingers.[18] Another system for counting up to FF16 (25510) is illustrated on the right.

### Signs

The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −4210 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 −4210 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 1.3DE16 × 242.

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]

## Conversion

### Binary conversion

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 (410). This example converts 11112 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:

• 00012 = 110
• 00102 = 210
• 01002 = 410
• 10002 = 810

Therefore:

 11112 = 810 + 410 + 210 + 110 = 1510

With little practice, mapping 11112 to F16 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 = 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:

 (01011110101101010010)2 = 0101 1110 1011 0101 00102 = 5 E B 5 216 = 5EB5216

The conversion from hexadecimal to binary is equally direct.

### Other simple conversions

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 5 216 = 11 32 23 11 024.

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.

### 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. 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 hihi−1...h2h1 be the hexadecimal digits representing the number.

1. i ← 1
2. hi ← d mod 16
3. d ← (d − hi) / 16
4. If d = 0 (return series hi) 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;
if (d - r == 0) {
}
}

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 (1110), 3 (310), A (1010) and D (1310), and then get the final result by multiplying each decimal representation by 16p, where p is the corresponding hex digit position, counting from right to left, beginning with 0. In this case we have B3AD = (11 × 163) + (3 × 162) + (10 × 161) + (13 × 160), which is 45997 base 10.

### Tools for conversion

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.

## Real numbers

### Rational numbers

As with other numeral systems, the hexadecimal system can be used to represent rational numbers, although repeating expansions are common since sixteen (1016) 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 (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 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, 0.062510 (one sixteenth) is equivalent to 0.116, 0.0912, and 0;3,4560.

n Decimal
Prime factors of base, b = 10: 2, 5; b − 1 = 9: 3; b + 1 = 11: 11
Prime factors of base, b = 1610 = 10: 2; b − 1 = 1510 = F: 3, 5; b + 1 = 1710 = 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

### Irrational numbers

The table below gives the expansions of some common irrational numbers in decimal and hexadecimal.

Number Positional representation
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

Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.

2x Value Value (Decimal)
20 1 1
21 2 2
22 4 4
23 8 8
24 10hex 16dec
25 20hex 32dec
26 40hex 64dec
27 80hex 128dec
28 100hex 256dec
29 200hex 512dec
2A (210dec) 400hex 1024dec
2B (211dec) 800hex 2048dec
2C (212dec) 1000hex 4096dec
2D (213dec) 2000hex 8192dec
2E (214dec) 4000hex 16,384dec
2F (215dec) 8000hex 32,768dec
210 (216dec) 10000hex 65,536dec

## Cultural

### Etymology

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]

### Use in Chinese culture

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.

### Primary numeral system

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 (Transfer encoding)

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:

• Being exactly half a byte, 4-bits is easier to process than the 5 or 6 bits of Base32 and Base64 respectively
• The symbols 0-9 and A-F are universal in hexadecimal notation, so it is easily understood at a glance without needing to rely on a symbol lookup table
• Many CPU architectures have dedicated instructions that allow access to a half-byte (otherwise known as a "Nibble"), making it more efficient in hardware than Base32 and Base64

The main disadvantages of Base16 encoding are:

• Space efficiency is only 50%, since each 4-bit value from the original data will be encoded as an 8-bit byte. In contrast, Base32 and Base64 encodings have a space efficiency of 63% and 75% respectively.
• Possible added complexity of having to accept both uppercase and lowercase letters

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.

## References

1. ^ "The Unicode Standard, Version 7" (PDF). Unicode. Retrieved October 28, 2018.
2. ^
3. ^ The string "\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.
4. ^ "Modula-2 - Vocabulary and representation". Modula -2. Retrieved 1 November 2015.
5. ^
6. ^
7. ^
8. ^ MSX is Coming — Part 2: Inside MSX Compute!, issue 56, January 1985, p. 52
9. ^ BBC BASIC programs are not fully portable to Microsoft BASIC (without modification) since the latter takes & 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.
10. ^ Donald E. Knuth. The TeXbook (Computers and Typesetting, Volume A). Reading, Massachusetts: Addison–Wesley, 1984. ISBN 0-201-13448-9. The source code of the book in TeX Archived 2007-09-27 at the Wayback Machine (and a required set of macros CTAN.org) is available online on CTAN.
11. ^ a b Martin, Bruce Alan (October 1968). "Letters to the editor: On binary notation". Communications of the ACM. Associated Universities Inc. 11 (10): 658. doi:10.1145/364096.364107.
12. Savard, John J. G. (2018) [2005]. "Computer Arithmetic". quadibloc. The Early Days of Hexadecimal. Archived from the original on 2018-07-16. Retrieved 2018-07-16.
13. ^ "2.1.3 Sexadecimal notation". G15D Programmer's Reference Manual (PDF). Los Angeles, CA, USA: Bendix Computer, Division of Bendix Aviation Corporation. p. 4. Archived (PDF) from the original on 2017-06-01. Retrieved 2017-06-01. 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.
14. ^ Gill, S.; Neagher, R. E.; Muller, D. E.; Nash, J. P.; Robertson, J. E.; Shapin, T.; Whesler, D. J. (1956-09-01). Nash, J. P. (ed.). "ILLIAC Programming - A Guide to the Preparation of Problems For Solution by the University of Illinois Digital Computer" (PDF) (Fourth printing. Revised and corrected ed.). Urbana, Illinois, USA: Digital Computer Laboratory, Graduate College, University of Illinois. pp. 3–2. Archived (PDF) from the original on 2017-05-31. Retrieved 2014-12-18.
15. ^ ROYAL PRECISION Electronic Computer LGP - 30 PROGRAMMING MANUAL. Port Chester, New York: Royal McBee Corporation. April 1957. Archived from the original on 2017-05-31. Retrieved 2017-05-31. (NB. This somewhat odd sequence was from the next six sequential numeric keyboard codes in the LGP-30's 6-bit character code.)
16. ^ NEC Parametron Digital Computer Type NEAC-1103 (PDF). Tokyo, Japan: Nippon Electric Company Ltd. 1960. Cat. No. 3405-C. Archived (PDF) from the original on 2017-05-31. Retrieved 2017-05-31.
17. ^ BCD-to-Seven-Segment Decoders/Drivers: SN54246/SN54247/SN54LS247, SN54LS248 SN74246/SN74247/SN74LS247/SN74LS248 (PDF), Texas Instruments, March 1988 [March 1974], SDLS083, archived (PDF) from the original on 2017-03-29, retrieved 2017-03-30, […] 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. […]
18. ^ Clarke, Arthur; Pohl, Frederik (2008). The Last Theorem. Ballantine. p. 91. ISBN 978-0007289981.
19. ^ "ISO/IEC 9899:1999 - Programming languages - C". Iso.org. 2011-12-08. Retrieved 2014-04-08.
20. ^ "Rationale for International Standard - Programming Languages - C" (PDF). 5.10. April 2003. pp. 52, 153–154, 159. Archived (PDF) from the original on 2016-06-06. Retrieved 2010-10-17.
21. ^ The IEEE and The Open Group (2013) [2001]. "dprintf, fprintf, printf, snprintf, sprintf - print formatted output". The Open Group Base Specifications (Issue 7, IEEE Std 1003.1, 2013 ed.). Archived from the original on 2016-06-21. Retrieved 2016-06-21.
22. ^ Knuth, Donald. (1969). The Art of Computer Programming, Volume 2. ISBN 0-201-03802-1. (Chapter 17.)
23. ^ Alfred B. Taylor, Report on Weights and Measures, Pharmaceutical Association, 8th Annual Session, Boston, Sept. 15, 1859. See pages and 33 and 41.
24. ^ Alfred B. Taylor, "Octonary numeration and its application to a system of weights and measures", Proc Amer. Phil. Soc. Vol XXIV, Philadelphia, 1887; pages 296-366. See pages 317 and 322.
25. ^ Schwartzman, S. (1994). The Words of Mathematics: an etymological dictionary of mathematical terms used in English. ISBN 0-88385-511-9.
26. ^ Gardner, Martin (1984). Martin Gardner's Sixth Book of Mathematical Diversions from "Scientific American". University of Chicago Press. p. 105. ISBN 978-0226282503.
27. ^ Weisstein, Eric W. "Hexadecimal". Wolfram MathWorld. Retrieved October 27, 2018.
28. ^
29. ^ "Intuitor Hex Headquarters". Intuitor. Retrieved October 28, 2018.
30. ^ Niemietz, Ricardo Cancho (October 21, 2003). "A proposal for addition of the six Hexadecimal digits (A-F) to Unicode". Retrieved October 28, 2018.
31. ^ a b Nystrom, John William (1862). Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base. Philadelphia: Lippincott.
Binary number

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 Meter

Digital 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 Ones

EO, 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 dump

In 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 editor

A 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 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 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.

KEIS

KEIS is a stateful EBCDIC charset used in Hitachi mainframe systems. KEIS is an acronym for "Kanji processing Extended Information System".

A 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 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.

Nibble

In 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 reference

A 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.

Octal

The 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:

${\displaystyle \mathbf {74} _{10}=\mathbf {7} \times 10^{1}+\mathbf {4} \times 10^{0}}$

In the octal system each place is a power of eight. For example:

${\displaystyle \mathbf {112} _{8}=\mathbf {1} \times 8^{2}+\mathbf {1} \times 8^{1}+\mathbf {2} \times 8^{0}}$

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 system

Quaternary 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.

In 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.

Suanpan

The 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 block

A 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 input

Unicode 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 colors

Web 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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