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# Hexadecimal ( Binary Arithmetic )

Rank: 100%     In and , hexadecimal (also base 16, or hex) is a positional with a , or base, of 16. It uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 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 values. Each hexadecimal digit represents four , also known as a , which is half a . 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 . For example, the decimal value would be expressed in hexadecimal as . 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 0x{{hexadecimal|10995|no}}.

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
In contexts where the 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 (including ), character codes are written as hexadecimal pairs prefixed with %: <nowiki></nowiki> where %20 is the space (blank) character, code point 20 in hex, 32 in decimal.
• In and , characters can be expressed as hexadecimal numeric character references using the notation &amp;#x''code'';, where the x denotes that code is a hex code point (of 1- to 6-digits) assigned to the character in the standard. Thus &amp;#x2019; represents the right single quotation mark (’), Unicode code point number 2019 in hex, 8217 (thus &amp;#8217; in decimal).
• In the standard, a character value is represented with U+ followed by the hex value, e.g. U+20AC is the (€).
• 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 as #FFFFFF. CSS also 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, , 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 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.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. To output an integer as hexadecimal with the function family, the format conversion code %X or %x is used.
• In (e-mail extensions) encoding, characters that cannot be represented as literal 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, 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, ), Pascal, , some versions of (), , 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 enclose hexadecimal numerals in based "numeric quotes": 16#5A3#. For bit vector constants uses the notation x"5A3".The VHDL MINI-REFERENCE: VHDL IDENTIFIERS, NUMBERS, STRINGS, AND EXPRESSIONS
• 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 language uses the prefix 16r: 16r5A3
• and the and its derivatives denote hex with prefix 16#: 16#5A3. For PostScript, binary data (such as image ) can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC...
• uses the prefixes #x and #16r. Setting the variables *read-base* and *print-base* 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 is Coming — Part 2: Inside MSX Compute!, issue 56, January 1985, p. 52 , and prefix hexadecimal numbers with &amp;H: &amp;H5A3
• and use &amp; for hex.BBC BASIC programs are not fully portable to (without modification) since the latter takes &amp; to prefix values. (Microsoft BASIC primarily uses &amp;O to prefix octal, and it uses &amp;H to prefix hexadecimal, but the ampersand alone yields a default interpretation as an octal prefix.
• 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 () and midrange computers (IBM System i) running the traditional OS's (zOS, zVSE, zVM, TPF, ) is X'5A3', and is used in Assembler, PL/I, , 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.
• introduced the use of a particular typeface to represent a particular radix in his book The TeXbook.Donald E. Knuth. The TeXbook (Computers and Typesetting, Volume A). Reading, Massachusetts: Addison–Wesley, 1984. . The source code of the book in TeX (and a required set of macros ftp://tug.ctan.org/pub/tex-archive/systems/knuth/lib/manmac.tex) is available online on . 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
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 to denote the values 10–15 as , , , , and .
• The SWAC (1950) and Bendix G-15 (1956) computers used the lowercase letters u, v, w, x, y and z for the values 10 to 15.
• The (1952) computer used the uppercase letters K, S, N, J, F and L for the values 10 to 15.
• The Librascope LGP-30 (1956) used the letters F, G, J, K, Q and W for the values 10 to 15.
• The Datamatic D-1000 (1957) used the lowercase letters b, c, d, e, f, and g whereas the  100 (1967) used the uppercase letters B, C, D, E, F and G for the values 10 to 15.
• The (1960) used the letters S, T, U, V, W and X for the values 10 to 15.
• The computer (1960) used the letters D, G, H, J, K (and possibly V) for values 10–15.
• The Pacific Data Systems 1020 (1964) used the letters L, C, A, S, M and D for the values 10 to 15.
• New numeric symbols and names were introduced in the notation by 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.
• 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.

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 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 power, , gross and for the first three 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.

(2019). 9780007289981, Ballantine.
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 or even a 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 . 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 .

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. Using the %a or %A conversion specifiers, this notation can be produced by implementations of the family of functions following the C99 specification and Single Unix Specification (IEEE Std 1003.1) standard.

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:
= 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.

 = 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:
00102
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 (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 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 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 — 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 16 p ( p being the corresponding hex digit position, counting from right to left, beginning with 0). In this case, we have that:

which is 45997 in base 10.

Tools for conversion
Most modern computer systems with graphical user interfaces provide a built-in calculator utility capable of performing conversions between the various radices, and in most cases would include the hexadecimal as well.

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 () and 2 (binary), the bases most commonly used by programmers. In Scientific Mode, the on-screen 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 arithmetic
Elementary operations such additions, subtractions, multiplications and divisions can be carried out indirectly through conversion to an alternate , such as the decimal system, since it's the most commonly adopted system, or the binary system, since 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 and the traditional subtraction algorithm.

Real numbers

Rational numbers
As with other numeral systems, the hexadecimal system can be used to represent , 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 (42), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a 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 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, and : 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 0.1 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.

21/2 0.50.8 1/2
31/3 0.3333... = 0.0.5555... = 0. 1/3
41/4 0.250.4 1/4
51/5 0.20. 1/5
61/6,0.10.2,1/6
71/770.0.71/7
81/8 0.1250.2 1/8
91/9 0.0. 1/9
101/10,0.10.1,1/A
111/11 0.0.B1/B
121/12,0.080.1,1/C
131/13130.0.D1/D
141/14, 70.00.1, 71/E
151/15,0.00.,1/F
161/16 0.06250.1 1/10
171/17170.0. 1/11
181/18,0.00.0,1/12
191/19190.0.131/13
201/20,0.050.0,1/14
211/21, 70.0., 71/15
221/22,0.00.0, B1/16
231/23230.0.171/17
241/24,0.0410.0,1/18
251/25 0.040. 1/19
261/26, 130.00.0, D1/1A
271/27 0.0. 1/1B
281/28, 70.030.0, 71/1C
291/29290.0.1D1/1D
301/30, ,0.00.0, ,1/1E
311/31310.0.1F1/1F
321/32 0.031250.08 1/20
331/33,0.0., B1/21
341/34, 170.00.0,1/22
351/35, 70.00., 71/23
361/36,0.020.0,1/24

Irrational numbers
The table below gives the expansions of some common irrational numbers in decimal and hexadecimal.
...1.6A09E667F3BCD...
...1.BB67AE8584CAA...
...2.3C6EF372FE95...
...1.9E3779B97F4A...

...
3.243F6A8885A308D313198A2E0
3707344A4093822299F31D008...
...2.B7E151628AED2A6B...
...0.6996 9669 9669 6996...
...0.93C467E37DB0C7A4D1B...

Powers
Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.
1
2
4
8
16dec
32dec
64dec
128dec
256dec
512dec
1024dec
2048dec
4096dec
8192dec
16,384dec
32,768dec
65,536dec

Cultural

Etymology

Use in Chinese culture
The traditional Chinese units of measurement were base-16. For example, one jīn (斤) in the old system equals sixteen . The (Chinese ) can be used to perform hexadecimal calculations such as additions and subtractions.

Primary numeral system
As with the 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. Some proposals unify standard measures so that they are multiples of 16.

An example of unified standard measures is , which subdivides a day by 16 so that there are 16 "hexhours" in a day.

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 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 ""), 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 , 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.

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