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In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9 and "A"–"F" (or "a"–"f") to represent values from ten to fifteen.
Software developers and system designers widely use hexadecimal numbers because they provide a convenient representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble). [1] For example, an 8-bit byte can have values ranging from 00000000 to 11111111 (0 to 255 decimal) in binary form, which can be written as 00 to FF in hexadecimal.
In mathematics, a subscript is typically used to specify the base. For example, the decimal value 12,389 would be expressed in hexadecimal as 306516. In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix 0x
is used in C, which would denote this value as 0x3065
.
Hexadecimal is used in the transfer encoding Base 16, in which each byte of the plain text is broken into two 4-bit values and represented by two hexadecimal digits.
In most current use cases, the letters A–F or a–f represent the values 10–15, while the numerals 0–9 are used to represent their decimal values.
There is no universal convention to use lowercase or uppercase, so each is prevalent or preferred in particular environments by community standards or convention; even mixed case is used. Some Seven-segment displays use mixed-case 'A b C d E F' to distinguish the digits A–F from one another and from 0–9.
There is some standardization of using spaces (rather than commas or another punctuation mark) to separate hex values in a long list. For instance, in the following hex dump, each 8-bit byte is a 2-digit hex number, with spaces between them, while the 32-bit offset at the start is an 8-digit hex number.
0000000057696b6970656469612c2074686520660000001072656520656e6379636c6f706564696100000020207468617420616e796f6e652063616e0000003020656469740a
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 equals 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h.
Donald Knuth introduced the use of a particular typeface to represent a particular radix in his book The TeXbook. [2] Hexadecimal representations are written there in a typewriter typeface: 5A3, C1F27ED
In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:
0x
to indicate a hex constant may have had origins in the IBM Stretch systems. It is derived from the 0
prefix already in use for octal constants. Byte values can be expressed 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 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.ode;
, for instance T
represents the character U+0054 (the uppercase letter "T"). If there is no x
the number is decimal (thus T
is the same character). [4] 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
. Some other implementations (such as NASM) allow C-style numbers (0x42
).$
as a prefix: $5A3
, $C1F27ED
.H'ABCD'
(for ABCD16). Similarly, Fortran 95 uses Z'ABCD'.16#5A3#
, 16#C1F27ED#
. For bit vector constants VHDL uses the notation x"5A3"
, x"C1F27ED"
. [6] 8'hFF
, where 8 is the number of bits in the value and FF is the hexadecimal constant.16r
: 16r5A3
16#
: 16#5A3
, 16#C1F27ED
.#x
and #16r
. Setting the variables *read-base* [7] and *print-base* [8] 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. [10] 0h
prefix: 0h5A3
, 0hC1F27ED
16r
to denote hexadecimal numbers: 16r5a3
, 16rC1F27ED
. Binary, quaternary (base-4), and octal numbers can be specified similarly.X'5A3'
or X'C1F27ED'
, 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.Sometimes the numbers are known to be Hex.
%
: <nowiki>http://www.example.com/name%20with%20spaces</nowiki>
where %20
is the code for the space (blank) character, ASCII code point 20 in hex, 32 in decimal.U+
followed by the hex value, e.g. U+00A1
is the inverted exclamation point (¡).#
: magenta, for example, is represented as #FF00FF
. [11] CSS also allows 3-hexdigit abbreviations with one hexdigit per component: #FA3
abbreviates #FFAA33
(a golden orange: ).=
: Espa=F1a
is "España" (F1hex is the code for ñ in the ISO/IEC 8859-1 character set). [12] )AA213FD51B3801043FBC
...:
). This, for example, is a valid IPv6 address: 2001:0db8:85a3:0000:0000:8a2e:0370:7334
or abbreviated by removing leading zeros as 2001:db8:85a3::8a2e:370:7334
(IPv4 addresses are usually written in decimal).3F2504E0-4F89-41D3-9A0C-0305E82C3301
.The use of the letters A through F to represent the digits above 9 was not universal in the early history of computers.
Since there were no traditional numerals to represent the quantities from ten to fifteen, alphabetic letters were re-employed as a substitute. Most European languages lack non-decimal-based words for some of the numerals eleven to fifteen. 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. In the wake of the adoption of hexadecimal among IBM System/360 programmers, Magnuson (1968) [23] suggested a pronunciation guide that gave short names to the letters of hexadecimal – for instance, "A" was pronounced "ann", B "bet", C "chris", etc. [23] Another naming-system was published online by Rogers (2007) [24] that tries to make the verbal representation distinguishable in any case, even when the actual number does not contain numbers A–F. Examples are listed in the tables below. Yet another naming system was elaborated by Babb (2015), based on a joke in Silicon Valley. [25]
Others have proposed using the verbal Morse Code conventions to express four-bit hexadecimal digits, with "dit" and "dah" representing zero and one, respectively, so that "0000" is voiced as "dit-dit-dit-dit" (....), dah-dit-dit-dah (-..-) voices the digit with a value of nine, and "dah-dah-dah-dah" (----) voices the hexadecimal digit for decimal 15.
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. [26] Another system for counting up to FF16 (25510) is illustrated on the right.
Number | Pronunciation |
---|---|
A | ann |
B | bet |
C | chris |
D | dot |
E | ernest |
F | frost |
1A | annteen |
A0 | annty |
5B | fifty-bet |
A01C | annty christeen |
1AD0 | annteen dotty |
3A7D | thirty-ann seventy-dot |
Number | Pronunciation |
---|---|
A | ten |
B | eleven |
C | twelve |
D | draze |
E | eptwin |
F | fim |
10 | tex |
11 | oneteek |
1F | fimteek |
50 | fiftek |
C0 | twelftek |
100 | hundrek |
1000 | thousek |
3E | thirtek-eptwin |
E1 | eptek-one |
C4A | twelve-hundrek-fourtek-ten |
1743 | one-thousek-seven- -hundrek-fourtek-three |
The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −4210, −B01D9 to represent −72136910 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. P notation uses the letter P (or p, for "power"), whereas E (or e) serves a similar purpose in decimal E notation. The number after the P is decimal and represents the binary exponent. Increasing the exponent by 1 multiplies by 2, not 16: 20p0 = 10p1 = 8p2 = 4p3 = 2p4 = 1p5. Usually, the number is normalized so that the hexadecimal digits start with 1. (zero is usually 0 with no P).
Example: 1.3DEp42 represents 1.3DE16 × 24210.
P notation is required by the IEEE 754-2008 binary floating-point standard and can be used for floating-point literals in the C99 edition of the C programming language. [27] Using the %a or %A conversion specifiers, this notation can be produced by implementations of the printf family of functions following the C99 specification [28] and Single Unix Specification (IEEE Std 1003.1) POSIX standard. [29]
Most computers manipulate binary data, but it is difficult for humans to work with a 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:
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. [30]
This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
(1001011100)2 | = 51210 + 6410 + 1610 + 810 + 410 |
= 60410 |
Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently and converted directly:
(1001011100)2 | = | 0010 | 0101 | 11002 | ||
= | 2 | 5 | C16 | |||
= | 25C16 |
The conversion from hexadecimal to binary is equally direct. [30]
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 2 5 C16 = 02 11 304.
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 hihi−1...h2h1 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.
functiontoHex(d){varr=d%16;if(d-r==0){returntoChar(r);}returntoHex((d-r)/16)+toChar(r);}functiontoChar(n){constalpha="0123456789ABCDEF";returnalpha.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 — before carrying out multiplication and addition to get the final representation. For example, 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 (p being the corresponding hex digit position, counting from right to left, beginning with 0). In this case, we have that:
B3AD = (11 × 163) + (3 × 162) + (10 × 161) + (13 × 160)
which is 45997 in base 10.
Many computer systems provide a calculator utility capable of performing conversions between the various radices frequently including hexadecimal.
In Microsoft Windows, the Calculator utility can be set to Programmer mode, which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 (octal), and 2 (binary), the bases most commonly used by programmers. In Programmer 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.
Elementary operations such as division can be carried out indirectly through conversion to an alternate numeral system, such as the commonly used decimal system or the binary system where each hex digit corresponds to four binary digits.
Alternatively, one can also perform elementary operations directly within the hex system itself — by relying on its addition/multiplication tables and its corresponding standard algorithms such as long division and the traditional subtraction algorithm.
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 lies 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 | Hexadecimal Prime factors of: base, b = 1610 = 10: 2; b − 1 = 1510 = F: 3, 5; b + 1 = 1710 = 11: 11 | ||||
---|---|---|---|---|---|---|
Reciprocal | Prime factors | Positional representation (decimal) | Positional representation (hexadecimal) | Prime factors | Reciprocal | |
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 |
37 | 1/37 | 37 | 0.027 | 0.06EB3E453 | 25 | 1/25 |
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.
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 |
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 such as additions and subtractions. [31]
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. [32] Some proposals unify standard measures so that they are multiples of 16. [33] [34] An early such proposal was put forward by John W. Nystrom in Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base, published in 1862. [35] Nystrom among other things suggested hexadecimal time, which subdivides a day by 16, so that there are 16 "hours" (or "10 tims", pronounced tontim) in a day. [36]
The word hexadecimal is first recorded in 1952. [37] It is macaronic in the sense that it combines Greek ἕξ (hex) "six" with Latinate -decimal. The all-Latin alternative sexadecimal (compare the word sexagesimal for base 60) is older, and sees at least occasional use from the late 19th century. [38] It is still in use in the 1950s in Bendix documentation. Schwartzman (1994) argues that use of sexadecimal may have been avoided because of its suggestive abbreviation to sex. [39] Many western languages since the 1960s have adopted terms equivalent in formation to hexadecimal (e.g. French hexadécimal, Italian esadecimale, Romanian hexazecimal, Serbian хексадецимални, etc.) but others have introduced terms which substitute native words for "sixteen" (e.g. Greek δεκαεξαδικός, Icelandic sextándakerfi, Russian шестнадцатеричной etc.)
Terminology and notation did not become settled until the end of the 1960s. In 1969, Donald Knuth argued that the etymologically correct term would be senidenary, or possibly sedenary, a Latinate term intended to convey "grouped by 16" modelled on binary, ternary, quaternary, etc. According to Knuth's argument, the correct terms for decimal and octal arithmetic would be denary and octonary, respectively. [40] 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". [41] [42]
The now-current notation using the letters A to F establishes itself as the de facto standard beginning in 1966, in the wake of the publication of the Fortran IV manual for IBM System/360, which (unlike earlier variants of Fortran) recognizes a standard for entering hexadecimal constants. [43] As noted above, alternative notations were used by NEC (1960) and The Pacific Data Systems 1020 (1964). The standard adopted by IBM seems to have become widely adopted by 1968, when Bruce Alan Martin in his letter to the editor of the CACM complains that
With the ridiculous choice of letters A, B, C, D, E, F as hexadecimal number symbols adding to already troublesome problems of distinguishing octal (or hex) numbers from decimal numbers (or variable names), the time is overripe for reconsideration of our number symbols. This should have been done before poor choices gelled into a de facto standard!
Martin's argument was that use of numerals 0 to 9 in nondecimal numbers "imply to us a base-ten place-value scheme": "Why not use entirely new symbols (and names) for the seven or fifteen nonzero digits needed in octal or hex. Even use of the letters A through P would be an improvement, but entirely new symbols could reflect the binary nature of the system". [19] He also argued that "re-using alphabetic letters for numerical digits represents a gigantic backward step from the invention of distinct, non-alphabetic glyphs for numerals sixteen centuries ago" (as Brahmi numerals, and later in a Hindu–Arabic numeral system), and that the recent ASCII standards (ASA X3.4-1963 and USAS X3.4-1968) "should have preserved six code table positions following the ten decimal digits -- rather than needlessly filling these with punctuation characters" (":;<=>?") that might have been placed elsewhere among the 128 available positions.
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 one of 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.
In computing and electronic systems, binary-coded decimal (BCD) is a class of binary encodings of decimal numbers where each digit is represented by a fixed number of bits, usually four or eight. Sometimes, special bit patterns are used for a sign or other indications.
In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are commonly represented in a computer as a group of binary digits (bits). The size of the grouping varies so the set of integer sizes available varies between different types of computers. Computer hardware nearly always provides a way to represent a processor register or memory address as an integer.
A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.
Octal is a numeral system with eight as the base.
A computer number format is the internal representation of numeric values in digital device hardware and software, such as in programmable computers and calculators. Numerical values are stored as groupings of bits, such as bytes and words. The encoding between numerical values and bit patterns is chosen for convenience of the operation of the computer; the encoding used by the computer's instruction set generally requires conversion for external use, such as for printing and display. Different types of processors may have different internal representations of numerical values and different conventions are used for integer and real numbers. Most calculations are carried out with number formats that fit into a processor register, but some software systems allow representation of arbitrarily large numbers using multiple words of memory.
In computer programming, Base64 is a group of binary-to-text encoding schemes that transforms binary data into a sequence of printable characters, limited to a set of 64 unique characters. More specifically, the source binary data is taken 6 bits at a time, then this group of 6 bits is mapped to one of 64 unique characters.
A binary code represents text, computer processor instructions, or any other data using a two-symbol system. The two-symbol system used is often "0" and "1" from the binary number system. The binary code assigns a pattern of binary digits, also known as bits, to each character, instruction, etc. For example, a binary string of eight bits can represent any of 256 possible values and can, therefore, represent a wide variety of different items.
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A binary number may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two.
A numerical digit or numeral is a single symbol used alone or in combinations, to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal digits.
Quaternary is a numeral system with four as its base. It uses the digits 0, 1, 2, and 3 to represent any real number. Conversion from binary is straightforward.
Dot-decimal notation is a presentation format for numerical data. It consists of a string of decimal numbers, using the full stop (dot) as a separation character.
Positional notation usually denotes the extension to any base of the Hindu–Arabic numeral system. More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred. In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.
Base32 is an encoding method based on the base-32 numeral system. It uses an alphabet of 32 digits, each of which represents a different combination of 5 bits (25). Since base32 is not very widely adopted, the question of notation—which characters to use to represent the 32 digits—is not as settled as in the case of more well-known numeral systems (such as hexadecimal), though RFCs and unofficial and de-facto standards exist. One way to represent Base32 numbers in human-readable form is using digits 0–9 followed by the twenty-two upper-case letters A–V. However, many other variations are used in different contexts. Historically, Baudot code could be considered a modified (stateful) base32 code.
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.
A hex editor is a 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 storage medium, 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.
Base36 is a binary-to-text encoding scheme that represents binary data in an ASCII string format by translating it into a radix-36 representation. The choice of 36 is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z.
Non-standard positional numeral systems here designates numeral systems that may loosely be described as positional systems, but that do not entirely comply with the following description of standard positional systems:
The octet is a unit of digital information in computing and telecommunications that consists of eight bits. The term is often used when the term byte might be ambiguous, as the byte has historically been used for storage units of a variety of sizes.
BCD, also called alphanumeric BCD, alphameric BCD, BCD Interchange Code, or BCDIC, is a family of representations of numerals, uppercase Latin letters, and some special and control characters as six-bit character codes.
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specifies the character sequence Esc [ 0 m Esc [ 2 5; 1 H. 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.