Numeral systems 

Hindu–Arabic numeral system 
East Asian 
Alphabetic 
Former 
Positional systems by base 
Nonstandard positional numeral systems 
List of numeral systems 
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 humanfriendly representation of binarycoded 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 would be expressed in hexadecimal as 2AF3_{16}. In programming, a number of notations are used to support hexadecimal representation, usually involving a prefix or suffix. The prefix 10,9950x
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 4bit values and represented by two hexadecimal digits.
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Transcription
 [Voiceover] We're all familiar with the base 10 number system, were often called the decimal number system, where we have 10 digits Zero, one, two, three, four, five, six, seven, eight, nine. Now, we started to see that we can have alternate number system. We can have a base two number system, or it's the binary number system, where instead of 10 digits you only have two digits. Each place, instead of being a power of ten is going to be a power of two. Now you can imagine that we can keep extending this. We can extend to base three, four, five, six, seven, eight, nine, or we could even go above 10. What I want to show you in this video is a fairly, heavily used number system that is larger than, or that has more digits than base 10, and that base is 16. Base 16, often called the hexadecimal. Hexadecimal number system. As you can imagine, instead of only having 10 digits, it is going to have 16. What are those digits going to be? As we'll see, instead of the place is being powers of two or powers of ten, there will be powers of 16. Let's see, we can reuse the existing 10 digits from the decimal number system. We can reuse zero, one, two, three, four, five, six, seven, eight, nine, but then we're going to need to have six more digits. The convention is to use the first six letters. A, B, C, D, E, and F. You might say this is crazy. These are letters, not numbers, but remember these are just arbitrary squiggles of ink on a piece of paper. These are just arbitrary symbols that we're grown to associate with things. You're grown to associate this symbol right over here with eight thing, with the word eight which you associate with when you see that many objects. If you're thinking in hexadecimal, this isn't the letter A that makes you want to say "ah", or the letter B that makes you want to say "bababababa". This is, literally, this represents if you had 10 things laying around. You would say, "I have A things over there." If you have 11, you'd say, "I have B things over there." 12, C things. 13, instead of saying, "I have 13 things there", "I have D things there." Instead of saying, "I have 14", you could say, "I can have E things there." Instead of saying, "I have 15", you could say, "I have F things there." Now, how does that help? Well, let's see if we can represent the same number 231, or 231 in decimal. If we can represent that same number in hexadecimal. What I'll do is I'll give you what the number is, and then I'll show you how we convert it. I'll show you the place value, and I'll show you how we convert it. 231 in hexadecimal. 231 in hexadecimal is the number E seven. E seven. Once again, you're like, "This looks crazy. "This is like I'm playing like battleship or something." What's E seven? This is a number and I would say yes. This is a number. Now remember, base 16. What are these place values represent? This first place represents 16 to the zero power or still represents the ones place. This is the ones place. This is seven ones. Now, what is this place here represents? Well, in base 10, that was 10 to the first power. In base two, that was two to the first power. On base 16, this is going to be, I'll leave those there, in base 16, this is going to be 16 to the first power. This is literally, well let me write out the word, this is literally sixteens. This is E sixteens plus seven ones. Let me write that down. This is E sixteens plus seven ones. That's what this number represents. Now, if we want to start rewriting this or reconceptualizing it in our decimal number system, what is E sixteens? Well, the E if we think in decimal, E is 14. E is 14. This is really, we can really think of this if you want to think them decimals. This is 14 sixteens. It's 14 sixteens. Well, that's just the same thing as 14 times 16. 14 times 16 is equal to 224. Maybe I should do that in same color. This thing right over here is going to be 224. 14 sixteens, 14 times 16 is 224 plus seven ones. Well, 224 plus 7 is going to be give you 231. Hopefully, you can appreciate it. You can represent the same quantity in any of these different number systems. In any number that you can represent in decimal, you can also represent that number in binary, or in hexadecimal, or in base three, or in base 60, or in base 31, whatever you want to do. You might have noticed the pattern. The more symbols that we have, so in base 16, you have 16 symbols, the less place values we need to represent the same quantity. One way to think about it is each of the places are containing more information. This is one of 16 characters. While this over here is only one of two characters. This is one of ten characters. The more symbols that you have, the more digits that you could put in each place, the less places that you need to represent a given quantity. Another way to think about it is when you have a high base, like base 16, as you take powers of 16, the next place right over here would be 16 squared, which, of course, is two hundred and, wait a minute, 256. You're clearly going to be able to represent bigger numbers faster, I guess you could say, or with less digits. It's just an interesting thing to observe. But hopefully, you're going to kick out of, as much of a kick out of base 16 as I do, and it's actually useful. This actually is used if you look at most web pages. If you look at the actual code for there, or I guess you could say the formatting line, the HTML for the webpage, when they specify colors, they tend to specify in hexadecimal. That's because they're specifying the colors, the intensity of the red, the green, or the blue, between zero and 255. Two digits of hexadecimal are perfect for that, because if you think about it, what is F F? What would this be if you rewrite it in the decimal number system, and I encourage you after this video is done, I encourage you to do that to figure that out on your own. If you really want to do something fun, let me give you another one. Try to figure out what A F three is. Again, this isn't very specialized. I just wanted to give you another interesting thing to work on.
Contents
Representation
Written representation
Using 0–9 and A–F
0_{hex}  =  0_{dec}  =  0_{oct}  0  0  0  0  
1_{hex}  =  1_{dec}  =  1_{oct}  0  0  0  1  
2_{hex}  =  2_{dec}  =  2_{oct}  0  0  1  0  
3_{hex}  =  3_{dec}  =  3_{oct}  0  0  1  1  
4_{hex}  =  4_{dec}  =  4_{oct}  0  1  0  0  
5_{hex}  =  5_{dec}  =  5_{oct}  0  1  0  1  
6_{hex}  =  6_{dec}  =  6_{oct}  0  1  1  0  
7_{hex}  =  7_{dec}  =  7_{oct}  0  1  1  1  
8_{hex}  =  8_{dec}  =  10_{oct}  1  0  0  0  
9_{hex}  =  9_{dec}  =  11_{oct}  1  0  0  1  
A_{hex}  =  10_{dec}  =  12_{oct}  1  0  1  0  
B_{hex}  =  11_{dec}  =  13_{oct}  1  0  1  1  
C_{hex}  =  12_{dec}  =  14_{oct}  1  1  0  0  
D_{hex}  =  13_{dec}  =  15_{oct}  1  1  0  1  
E_{hex}  =  14_{dec}  =  16_{oct}  1  1  1  0  
F_{hex}  =  15_{dec}  =  17_{oct}  1  1  1  1 
In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give the base explicitly: 159_{10} is decimal 159; 159_{16} is hexadecimal 159, which is equal to 345_{10}. Some authors prefer a text subscript, such as 159_{decimal} and 159_{hex}, or 159_{d} and 159_{h}.
In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:
 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
ode;
, where the x denotes that code is a hex code point (of 1 to 6digits) assigned to the character in the Unicode standard. Thus’
represents the right single quotation mark (’), Unicode code point number 2019 in hex, 8217 (thus’
in decimal).^{[1]}  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 3hexdigit 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 (email extensions) quotedprintable 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 inEspa=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 88591 character set.)  In Intelderived assembly languages and Modula2,^{[4]} hexadecimal is denoted with a suffixed H or h:
FFh
or05A3H
. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write0FFh
instead ofFFh
 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 ABCD_{16}). 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 notationx"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 *readbase*^{[6]} and *printbase*^{[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]}  TI89 and 92 series uses a
0h
prefix:0h5A3
 ALGOL 68 uses the prefix
16r
to denote hexadecimal numbers:16r5a3
. Binary, quaternary (base4) 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 thirtytwo hexadecimal digits, often in unequal hyphenseparated groupings, for example
{3F2504E04F8941D39A0C0305E82C3301}
.
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^{[which?]} 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 G15 (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 LGP30 (1956) used the letters F, G, J, K, Q and W for the values 10 to 15.^{[15]}^{[12]}
 The Honeywell Datamatic D1000 (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 Bibibinary 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 Б334 (1980) and similar used the symbols "−", "L", "C", "Г", "E", " " (space) for the values 10 to 15 on their displays.^{[citation needed]}
 Sevensegment 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 nondecimal names for the numerals above ten. Even though English has names for several nondecimal powers (pair for the first binary power, score for the first vigesimal power, dozen, gross and great gross for the first three duodecimal powers), no English name describes the hexadecimal powers (decimal 16, 256, 4096, 65536, ... ). Some people read hexadecimal numbers digit by digit like a phone number, or using the NATO phonetic alphabet, the Joint Army/Navy Phonetic Alphabet, or a similar ad hoc system.
Systems of counting on digits have been devised for both binary and hexadecimal. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 1023_{10} on ten fingers.^{[18]} Another system for counting up to FF_{16} (255_{10}) is illustrated on the right.
Signs
The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −42_{10} and so on.
Hexadecimal can also be used to express the exact bit patterns used in the processor, so a sequence of hexadecimal digits may represent a signed or even a floating point value. This way, the negative number −42_{10} can be written as FFFF FFD6 in a 32bit CPU register (in two'scomplement), as C228 0000 in a 32bit FPU register or C045 0000 0000 0000 in a 64bit FPU register (in the IEEE floatingpoint standard).
Hexadecimal exponential notation
Just as decimal numbers can be represented in exponential notation, so too can hexadecimal numbers. By convention, the letter P (or p, for "power") represents times two raised to the power of, whereas E (or e) serves a similar purpose in decimal as part of the E notation. The number after the P is decimal and represents the binary exponent.
Usually the number is normalised so that the leading hexadecimal digit is 1 (unless the value is exactly 0).
Example: 1.3DEp42 represents 1.3DE_{16} × 2^{42}.
Hexadecimal exponential notation is required by the IEEE 7542008 binary floatingpoint standard. This notation can be used for floatingpoint 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 (4_{10}). This example converts 1111_{2} to base ten. Since each position in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from the right:
 0001_{2} = 1_{10}
 0010_{2} = 2_{10}
 0100_{2} = 4_{10}
 1000_{2} = 8_{10}
Therefore:
1111_{2}  = 8_{10} + 4_{10} + 2_{10} + 1_{10} 
= 15_{10} 
With little practice, mapping 1111_{2} to F_{16} in one step becomes easy: see table in Written representation. The advantage of using hexadecimal rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hexadecimal, it is trivial to regard the binary string as 4digit groups and map each to a single hexadecimal digit.
This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
(01011110101101010010)_{2}  = 262144_{10} + 65536_{10} + 32768_{10} + 16384_{10} + 8192_{10} + 2048_{10} + 512_{10} + 256_{10} + 64_{10} + 16_{10} + 2_{10} 
= 387922_{10} 
Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently, and converted directly:
(01011110101101010010)_{2}  =  0101_{ }  1110_{ }  1011_{ }  0101_{ }  0010_{2} 
=  5  E  B  5  2_{16}  
=  5EB52_{16} 
The conversion from hexadecimal to binary is equally direct.
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 2_{16} = 11 32 23 11 02_{4}.
The octal (base 8) system can also be converted with relative ease, although not quite as trivially as with bases 2 and 4. Each octal digit corresponds to three binary digits, rather than four. Therefore we can convert between octal and hexadecimal via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four.
Divisionremainder 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 h_{i}h_{i−1}...h_{2}h_{1} be the hexadecimal digits representing the number.
 i ← 1
 h_{i} ← d mod 16
 d ← (d − h_{i}) / 16
 If d = 0 (return series h_{i}) 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) {
return toChar(r);
}
return toHex( (d  r)/16 ) + toChar(r);
}
function toChar(n) {
const alpha = "0123456789ABCDEF";
return alpha.charAt(n);
}
Addition and multiplication
It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value and then performing multiplication and addition to get the final representation. That is, to convert the number B3AD to decimal one can split the hexadecimal number into its digits: B (11_{10}), 3 (3_{10}), A (10_{10}) and D (13_{10}), and then get the final result by multiplying each decimal representation by 16^{p}, where p is the corresponding hex digit position, counting from right to left, beginning with 0. In this case we have B3AD = (11 × 16^{3}) + (3 × 16^{2}) + (10 × 16^{1}) + (13 × 16^{0}), which is 45997 base 10.
Tools for conversion
Most modern computer systems with graphical user interfaces provide a builtin 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 onscreen 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 (10_{hex}) has only a single prime factor (two):
1/2  =  0.8 
1/3  =  0.5 
1/4  =  0.4 
1/5  =  0.3 
1/6  =  0.2A 
1/7  =  0.249 
1/8  =  0.2 
1/9  =  0.1C7 
1/A  =  0.19 
1/B  =  0.1745D 
1/C  =  0.15 
1/D  =  0.13B 
1/E  =  0.1249 
1/F  =  0.1 
1/10  =  0.1 
1/11  =  0.0F 
where an overline denotes a recurring pattern.
For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two for binary or dividing one by sixteen for hexadecimal, both of these fractions are written as 0.1
. Because the radix 16 is a perfect square (4^{2}), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no cyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix; thus, when using hexadecimal notation, all fractions with denominators that are not a power of two result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than decimal for representing rational numbers since a larger proportion lie outside its range of finite representation.
All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hexadecimal number with a finite number of digits 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 0.199999999999... in hexadecimal. However, hexadecimal is more efficient than bases 12 and 60 for representing fractions with powers of two in the denominator (e.g., decimal one sixteenth is 0.1 in hexadecimal, 0.09 in duodecimal, 0;3,45 in sexagesimal and 0.0625 in decimal).
n  Decimal Prime factors of base, b = 10: 2, 5; b − 1 = 9: 3; b + 1 = 11: 11 
Hexadecimal Prime factors of base, b = 16_{10} = 10: 2; b − 1 = 15_{10} = F: 3, 5; b + 1 = 17_{10} = 11: 11  

Fraction  Prime factors  Positional representation  Positional representation  Prime factors  Fraction(1/n)  
2  1/2  2  0.5  0.8  2  1/2 
3  1/3  3  0.3333... = 0.3  0.5555... = 0.5  3  1/3 
4  1/4  2  0.25  0.4  2  1/4 
5  1/5  5  0.2  0.3  5  1/5 
6  1/6  2, 3  0.16  0.2A  2, 3  1/6 
7  1/7  7  0.142857  0.249  7  1/7 
8  1/8  2  0.125  0.2  2  1/8 
9  1/9  3  0.1  0.1C7  3  1/9 
10  1/10  2, 5  0.1  0.19  2, 5  1/A 
11  1/11  11  0.09  0.1745D  B  1/B 
12  1/12  2, 3  0.083  0.15  2, 3  1/C 
13  1/13  13  0.076923  0.13B  D  1/D 
14  1/14  2, 7  0.0714285  0.1249  2, 7  1/E 
15  1/15  3, 5  0.06  0.1  3, 5  1/F 
16  1/16  2  0.0625  0.1  2  1/10 
17  1/17  17  0.0588235294117647  0.0F  11  1/11 
18  1/18  2, 3  0.05  0.0E38  2, 3  1/12 
19  1/19  19  0.052631578947368421  0.0D79435E5  13  1/13 
20  1/20  2, 5  0.05  0.0C  2, 5  1/14 
21  1/21  3, 7  0.047619  0.0C3  3, 7  1/15 
22  1/22  2, 11  0.045  0.0BA2E8  2, B  1/16 
23  1/23  23  0.0434782608695652173913  0.0B21642C859  17  1/17 
24  1/24  2, 3  0.0416  0.0A  2, 3  1/18 
25  1/25  5  0.04  0.0A3D7  5  1/19 
26  1/26  2, 13  0.0384615  0.09D8  2, D  1/1A 
27  1/27  3  0.037  0.097B425ED  3  1/1B 
28  1/28  2, 7  0.03571428  0.0924  2, 7  1/1C 
29  1/29  29  0.0344827586206896551724137931  0.08D3DCB  1D  1/1D 
30  1/30  2, 3, 5  0.03  0.08  2, 3, 5  1/1E 
31  1/31  31  0.032258064516129  0.08421  1F  1/1F 
32  1/32  2  0.03125  0.08  2  1/20 
33  1/33  3, 11  0.03  0.07C1F  3, B  1/21 
34  1/34  2, 17  0.02941176470588235  0.078  2, 11  1/22 
35  1/35  5, 7  0.0285714  0.075  5, 7  1/23 
36  1/36  2, 3  0.027  0.071C  2, 3  1/24 
Irrational numbers
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)  213562373095048... 1.414  1.6A09E667F3BCD... 
√3 (the length of the diagonal of a unit cube)  050807568877293... 1.732  1.BB67AE8584CAA... 
√5 (the length of the diagonal of a 1×2 rectangle)  067977499789696... 2.236  2.3C6EF372FE95... 
φ (phi, the golden ratio = (1+√5)/2)  033988749894848... 1.618  1.9E3779B97F4A... 
π (pi, the ratio of circumference to diameter of a circle)  592653589793238462643 3.141 279502884197169399375105... 383 
3.243F6A8885A308D313198A2E0 3707344A4093822299F31D008... 
e (the base of the natural logarithm)  281828459045235... 2.718  2.B7E151628AED2A6B... 
τ (the Thue–Morse constant)  454033640107597... 0.412  0.6996 9669 9669 6996... 
γ (the limiting difference between the harmonic series and the natural logarithm) 
215664901532860... 0.577  0.93C467E37DB0C7A4D1B... 
Powers
Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.
2^{x}  Value  Value (Decimal) 

2^{0}  1  1 
2^{1}  2  2 
2^{2}  4  4 
2^{3}  8  8 
2^{4}  10_{hex}  16_{dec} 
2^{5}  20_{hex}  32_{dec} 
2^{6}  40_{hex}  64_{dec} 
2^{7}  80_{hex}  128_{dec} 
2^{8}  100_{hex}  256_{dec} 
2^{9}  200_{hex}  512_{dec} 
2^{A} (2^{10dec})  400_{hex}  1024_{dec} 
2^{B} (2^{11dec})  800_{hex}  2048_{dec} 
2^{C} (2^{12dec})  1000_{hex}  4096_{dec} 
2^{D} (2^{13dec})  2000_{hex}  8192_{dec} 
2^{E} (2^{14dec})  4000_{hex}  16,384_{dec} 
2^{F} (2^{15dec})  8000_{hex}  32,768_{dec} 
2^{10} (2^{16dec})  10000_{hex}  65,536_{dec} 
Cultural
Etymology
The word hexadecimal is composed of hexa, derived from the Greek ἕξ (hex) for six, and decimal, derived from the Latin for tenth. Webster's Third New International online derives hexadecimal as an alteration of the allLatin sexadecimal (which appears in the earlier Bendix documentation). The earliest date attested for hexadecimal in MerriamWebster Collegiate online is 1954, placing it safely in the category of international scientific vocabulary (ISV). It is common in ISV to mix Greek and Latin combining forms freely. The word sexagesimal (for base 60) retains the Latin prefix. Donald Knuth has pointed out that the etymologically correct term is senidenary (or possibly, sedenary), from the Latin term for grouped by 16. (The terms binary, ternary and quaternary are from the same Latin construction, and the etymologically correct terms for decimal and octal arithmetic are denary and octonary, respectively.)^{[22]} Alfred B. Taylor used senidenary in his mid1800s work on alternative number bases, although he rejected base 16 because of its "incommodious number of digits".^{[23]}^{[24]} Schwartzman notes that the expected form from usual Latin phrasing would be sexadecimal, but computer hackers would be tempted to shorten that word to sex.^{[25]} The etymologically proper Greek term would be hexadecadic / ἑξαδεκαδικός / hexadekadikós (although in Modern Greek, decahexadic / δεκαεξαδικός / dekaexadikos is more commonly used).
In hexadecimal, numbers with nondecreasing digits are called plaindrones, those with nonincreasing digits are called nialpdromes, those with descending digits are called katadromes, and those with ascending digits are called metadromes.^{[26]}^{[27]}
Use in Chinese culture
The traditional Chinese units of measurement were base16. 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 4bit 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:
 Most programming languages already have facilities to parse ASCIIencoded hexadecimal
 Being exactly half a byte, 4bits is easier to process than the 5 or 6 bits of Base32 and Base64 respectively
 The symbols 09 and AF 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 halfbyte (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 4bit value from the original data will be encoded as an 8bit 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 Base16encoded form. Most modern programming languages directly include support for formatting and parsing Base16encoded numbers.
See also
 Base32, Base64 (content encoding schemes)
 Hexadecimal time
 IBM hexadecimal floating point
 Hex editor
 Hex dump
 Bailey–Borwein–Plouffe formula (BBP)
References
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 ^ "Hexadecimal web colors explained".
 ^ 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.  ^ "Modula2  Vocabulary and representation". Modula 2. Retrieved 1 November 2015.
 ^ The VHDL MINIREFERENCE: VHDL IDENTIFIERS, NUMBERS, STRINGS, AND EXPRESSIONS
 ^ "*readbase* variable in Common Lisp".
 ^ "*printbase* variable in Common Lisp".
 ^ MSX is Coming — Part 2: Inside MSX Compute!, issue 56, January 1985, p. 52
 ^ 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.  ^ Donald E. Knuth. The TeXbook (Computers and Typesetting, Volume A). Reading, Massachusetts: Addison–Wesley, 1984. ISBN 0201134489. The source code of the book in TeX Archived 20070927 at the Wayback Machine. (and a required set of macros CTAN.org) is available online on CTAN.
 ^ ^{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.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} Savard, John J. G. (2018) [2005]. "Computer Arithmetic". quadibloc. The Early Days of Hexadecimal. Archived from the original on 20180716. Retrieved 20180716.
 ^ "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 20170601. Retrieved 20170601.
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 G15 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.
 ^ Gill, S.; Neagher, R. E.; Muller, D. E.; Nash, J. P.; Robertson, J. E.; Shapin, T.; Whesler, D. J. (19560901). 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 20170531. Retrieved 20141218.
 ^ ROYAL PRECISION Electronic Computer LGP  30 PROGRAMMING MANUAL. Port Chester, New York: Royal McBee Corporation. April 1957. Archived from the original on 20170531. Retrieved 20170531. (NB. This somewhat odd sequence was from the next six sequential numeric keyboard codes in the LGP30's 6bit character code.)
 ^ NEC Parametron Digital Computer Type NEAC1103 (PDF). Tokyo, Japan: Nippon Electric Company Ltd. 1960. Cat. No. 3405C. Archived (PDF) from the original on 20170531. Retrieved 20170531.
 ^ BCDtoSevenSegment Decoders/Drivers: SN54246/SN54247/SN54LS247, SN54LS248 SN74246/SN74247/SN74LS247/SN74LS248 (PDF), Texas Instruments, March 1988 [March 1974], SDLS083, archived (PDF) from the original on 20170329, retrieved 20170330,
[…] 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. […]
 ^ Clarke, Arthur; Pohl, Frederik (2008). The Last Theorem. Ballantine. p. 91. ISBN 9780007289981.
 ^ "ISO/IEC 9899:1999  Programming languages  C". Iso.org. 20111208. Retrieved 20140408.
 ^ "Rationale for International Standard  Programming Languages  C" (PDF). 5.10. April 2003. pp. 52, 153–154, 159. Archived (PDF) from the original on 20160606. Retrieved 20101017.
 ^ 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 20160621. Retrieved 20160621.
 ^ Knuth, Donald. (1969). The Art of Computer Programming, Volume 2. ISBN 0201038021. (Chapter 17.)
 ^ Alfred B. Taylor, Report on Weights and Measures, Pharmaceutical Association, 8th Annual Session, Boston, Sept. 15, 1859. See pages and 33 and 41.
 ^ Alfred B. Taylor, "Octonary numeration and its application to a system of weights and measures", Proc Amer. Phil. Soc. Vol XXIV, Philadelphia, 1887; pages 296366. See pages 317 and 322.
 ^ Schwartzman, S. (1994). The Words of Mathematics: an etymological dictionary of mathematical terms used in English. ISBN 0883855119.
 ^ Gardner, Martin (1984). Martin Gardner's Sixth Book of Mathematical Diversions from "Scientific American". University of Chicago Press. p. 105. ISBN 9780226282503.
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 ^ "Intuitor Hex Headquarters". Intuitor. Retrieved October 28, 2018.
 ^ Niemietz, Ricardo Cancho (October 21, 2003). "A proposal for addition of the six Hexadecimal digits (AF) to Unicode". Retrieved October 28, 2018.
 ^ ^{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.