Hexadecimal describes a base-16 number
system. That is, it describes a numbering
system containing 16 sequential numbers as base units (including 0) before ...
Hexadecimal
From Wikipedia, the free encyclopedia
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,
B,
C,
D,
E,
F (or alternatively
a,
b,
c,
d,
e,
f)
to represent values ten to fifteen. Hexadecimal numerals are widely
used by computer system designers and programmers. Several different
notations are used to represent hexadecimal constants in computing
languages; the prefix "0x" is widespread due to its use in
Unix and
C
(and related operating systems and languages). Alternatively, some
authors denote hexadecimal values using a suffix or subscript. For
example, one could write 0x2AF3 or 2AF3
16, depending on the choice of notation.
As an example, the hexadecimal number 2AF3
16 can be converted to an equivalent decimal representation. Observe that 2AF3
16 is equal to a sum of (2000
16 + A00
16 + F0
16 + 3
16), by decomposing the numeral into a series of
place value terms. Converting each term to decimal, one can further write:
Each hexadecimal digit represents four binary digits (
bits), and the primary use of hexadecimal notation is a human-friendly representation of
binary-coded values in computing and digital electronics. One hexadecimal digit represents a
nibble, which is half of an
octet or byte (8 bits). For example,
byte
values can range from 0 to 255 (decimal), but may be more conveniently
represented as two hexadecimal digits in the range 00 to FF. Hexadecimal
is also commonly used to represent computer
memory addresses.
Representation
Written representation
Using 0–9 and A–F
| 0hex |
= |
0dec |
= |
0oct |
|
0 |
0 |
0 |
0 |
| 1hex |
= |
1dec |
= |
1oct |
0 |
0 |
0 |
1 |
| 2hex |
= |
2dec |
= |
2oct |
0 |
0 |
1 |
0 |
| 3hex |
= |
3dec |
= |
3oct |
0 |
0 |
1 |
1 |
|
| 4hex |
= |
4dec |
= |
4oct |
0 |
1 |
0 |
0 |
| 5hex |
= |
5dec |
= |
5oct |
0 |
1 |
0 |
1 |
| 6hex |
= |
6dec |
= |
6oct |
0 |
1 |
1 |
0 |
| 7hex |
= |
7dec |
= |
7oct |
0 |
1 |
1 |
1 |
|
| 8hex |
= |
8dec |
= |
10oct |
1 |
0 |
0 |
0 |
| 9hex |
= |
9dec |
= |
11oct |
1 |
0 |
0 |
1 |
| Ahex |
= |
10dec |
= |
12oct |
1 |
0 |
1 |
0 |
| Bhex |
= |
11dec |
= |
13oct |
1 |
0 |
1 |
1 |
|
| Chex |
= |
12dec |
= |
14oct |
1 |
1 |
0 |
0 |
| Dhex |
= |
13dec |
= |
15oct |
1 |
1 |
0 |
1 |
| Ehex |
= |
14dec |
= |
16oct |
1 |
1 |
1 |
0 |
| Fhex |
= |
15dec |
= |
17oct |
1 |
1 |
1 |
1 |
In contexts where the
base
is not clear, hexadecimal numbers can be ambiguous and confused with
numbers expressed in other bases. There are several conventions for
expressing values unambiguously. A numerical subscript (itself written
in decimal) can give the base explicitly: 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 6-digits) assigned to the character in the Unicode standard. Thus ’ represents the right single quotation mark (’), Unicode code point number 2019 in hex, 8217 (thus ’ in decimal).[1]
- 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 and CSS and X Window can be expressed with six hexadecimal digits (two each for the red, green and blue components, in that order) prefixed with
#: white, for example, is represented #FFFFFF .[2] CSS allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange: ).
- *nix (Unix and related) shells, AT&T assembly language and likewise the C programming language, which was designed for Unix (and the syntactic descendants of C – including C++, C#, D, Java, JavaScript, Python and Windows PowerShell) use the prefix
0x for numeric constants represented in hex: 0x5A3. Character and string constants may express character codes in hexadecimal with the prefix \x followed by two hex digits: '\x1B' represents the Esc control character; "\x1B[0m\x1B[25;1H" is a string containing 11 characters (plus a trailing NUL to mark the end of the string) with two embedded Esc characters.[3] To output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used.
- In MIME (e-mail extensions) quoted-printable encoding, characters that cannot be represented as literal ASCII characters are represented by their codes as two hexadecimal digits (in ASCII) prefixed by an equal to sign
=, as in Espa=F1a
to send "España" (Spain). (Hexadecimal F1, equal to decimal 241, is the
code number for the lower case n with tilde in the ISO/IEC 8859-1
character set.)
- In Intel-derived assembly languages and Modula-2,[4] hexadecimal is denoted with a suffixed H or h:
FFh or 05A3H.
Some implementations require a leading zero when the first hexadecimal
digit character is not a decimal digit, so one would write 0FFh instead of FFh
- Other assembly languages (6502, Motorola), Pascal, Delphi, some versions of BASIC (Commodore), Game Maker Language, Godot and Forth use
$ as a prefix: $5A3.
- Some assembly languages (Microchip) use the notation
H'ABCD' (for ABCD16).
- Ada and VHDL enclose hexadecimal numerals in based "numeric quotes":
16#5A3#. For bit vector constants VHDL uses the notation x"5A3".[5]
- Verilog represents hexadecimal constants in the form
8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant.
- The Smalltalk language uses the prefix
16r: 16r5A3
- PostScript and the Bourne shell and its derivatives denote hex with prefix
16#: 16#5A3. For PostScript, binary data (such as image pixels) can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC...
- In early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong.
- Common Lisp uses the prefixes
#x and #16r. Setting the variables *read-base*[6] and *print-base*[7]
to 16 can also used to switch the reader and printer of a Common Lisp
system to Hexadecimal number representation for reading and printing
numbers. Thus Hexadecimal numbers can be represented without the #x or
#16r prefix code, when the input or output base has been changed to 16.
- MSX BASIC,[8] QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with
&H: &H5A3
- BBC BASIC and Locomotive BASIC use
& for hex.[9]
- TI-89 and 92 series uses a
0h prefix: 0h5A3
- ALGOL 68 uses the prefix
16r to denote hexadecimal numbers: 16r5a3. Binary, quaternary (base-4) and octal numbers can be specified similarly.
- The most common format for hexadecimal on IBM mainframes (zSeries) and midrange computers (IBM System i) running the traditional OS's (zOS, zVSE, zVM, TPF, IBM i) is
X'5A3', and is used in Assembler, PL/I, COBOL, JCL,
scripts, commands and other places. This format was common on other
(and now obsolete) IBM systems as well. Occasionally quotation marks
were used instead of apostrophes.
- Donald Knuth introduced the use of a particular typeface to represent a particular radix in his book The TeXbook.[10] Hexadecimal representations are written there in a typewriter typeface: 5A3
- Any IPv6 address can be written as eight groups of four hexadecimal digits, 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.
Early written representations
Bruce Alan Martin's hexadecimal notation proposal
[11]
The use of the letters
A through
F to represent the digits above 9 was not universal in the early history of computers.
- During the 1950s, some installations favored using the digits 0 through 5 with a macron to denote the values 10–15 as 0, 1, 2, 3, 4 and 5.
- Bendix G-15 computers used the letters U through Z.
- The Librascope LGP-30 used the letters F, G, J, K, Q and W.[12]
- The ILLIAC I computer used the letters K, S, N, J, F and L.[13]
- Bruce Alan Martin of Brookhaven National Laboratory considered the choice of A–F "ridiculous" and in a 1968 letter to the editor of the CACM proposed an entirely new set of symbols based on the bit locations, which did not gain much, if any, acceptance.[11]
- Soviet programmable calculators Б3-34 and similar used the symbols "−", "L", "C", "Г", "E", " " (space) on their displays.
- Seven-segment display decoder chips used various schemes for outputting values above nine:
- The Texas Instruments 7446/7/8/9 and 74246/7/8/9 use truncated versions of "2", "3", "4", "5" and "6" for digits A–E. Digit F (1111 binary) was blank.
- The National Semiconductor MM74C912 displayed "o" for A and B, "−" for C, D and E, and blank for F.
- The CD4511 just displays blanks.
Verbal and digital representations
There are no traditional numerals to represent the quantities from ten to fifteen — letters are used as a substitute — and most
European languages lack non-decimal names for the numerals above ten. Even though English has names for several non-decimal powers (
pair for the first
binary power,
score for the first
vigesimal power,
dozen,
gross and
great gross for the first three
duodecimal
powers), no English name describes the hexadecimal powers (decimal 16,
256, 4096, 65536, ... ). Some people read hexadecimal numbers digit by
digit like a phone number:
4DA is "four-dee-ay". However, the letter
A sounds like "eight", and
D can easily be mistaken for the "-ty" suffix: Is it
4D or
forty? Other people avoid confusion by using the
NATO phonetic alphabet:
4DA is "four-delta-alfa", the
Joint Army/Navy Phonetic Alphabet ("four-dog-able"), or a similar ad hoc system.
Hexadecimal finger-counting scheme.
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. 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.
However, some
[who?] prefer instead to use the hexadecimal notation to express the exact bit patterns used in the
processor, so a sequence of hexadecimal digits may represent a
signed or even a
floating point value. This way, the negative number −42
10 can be written as FFFF FFD6 in a 32-bit
CPU register (in
two's-complement), as C228 0000 in a 32-bit
FPU register or C045 0000 0000 0000 in a 64-bit FPU register (in the
IEEE floating-point standard).
Hexadecimal exponential notation
Just as decimal numbers can be represented in
exponential notation, so too can hexadecimal. By convention, the letter
P (or
p, for "power") represents
times two raised to the power of, whereas
E (or
e) serves a similar purpose in decimal as part of the
E notation. The number after the
P is
decimal and represents the
binary exponent.
Usually the number is normalised so that the leading hexadecimal digit is 1 (unless the value is exactly 0).
Example: 1.3DEp42 represents
1.3DE16 × 242.
Using the
%a or
%A conversion specifiers, this notation can be produced by implementations of the
printf family of functions following the
C99 specification and (
Single Unix Specification)
IEEE Std 1003.1 POSIX standard.
[14] Hexadecimal exponential notation is required by the
IEEE 754-2008 binary floating-point 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 (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:
- 00012 = 110
- 00102 = 210
- 01002 = 410
- 10002 = 810
Therefore:
| 11112 |
= 810 + 410 + 210 + 110 |
| |
= 1510 |
With little practice, mapping 1111
2 to F
16 in one step becomes easy: see table in
Written representation.
The advantage of using hexadecimal rather than decimal increases
rapidly with the size of the number. When the number becomes large,
conversion to decimal is very tedious. However, when mapping to
hexadecimal, it is trivial to regard the binary string as 4-digit groups
and map each to a single hexadecimal digit.
This example shows the conversion of a binary number to decimal,
mapping each digit to the decimal value, and adding the results.
| 010111101011010100102 |
= 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:
| 010111101011010100102 |
= |
0101 |
1110 |
1011 |
0101 |
00102 |
| |
= |
5 |
E |
B |
5 |
216 |
| |
= |
5EB5216 |
The conversion from hexadecimal to binary is equally direct.
The
octal
system can also be useful as a tool for people who need to deal
directly with binary computer data. Octal represents data as three bits
per character, rather than four.
Division-remainder in source base
As with all bases there is a simple
algorithm
for converting a representation of a number to hexadecimal by doing
integer division and remainder operations in the source base. In theory,
this is possible from any base, but for most humans only decimal and
for most computers only binary (which can be converted by far more
efficient methods) can be easily handled with this method.
Let d be the number to represent in hexadecimal, and the series h
ih
i−1...h
2h
1 be the hexadecimal digits representing the number.
- i ← 1
- hi ← d mod 16
- d ← (d − hi) / 16
- If d = 0 (return series hi) else increment i and go to step 2
"16" may be replaced with any other base that may be desired.
The following is a
JavaScript
implementation of the above algorithm for converting any number to a
hexadecimal in String representation. Its purpose is to illustrate the
above algorithm. To work with data seriously, however, it is much more
advisable to work with
bitwise operators.
function toHex(d) {
var r = d % 16;
var result;
if (d - r == 0)
result = toChar(r);
else
result = toHex( (d - r)/16 ) + toChar(r);
return result;
}
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 × 163) + (3 × 162) + (10 × 161) + (13 × 160), which is 45997 base 10.
Tools for conversion
Most modern computer systems with
graphical user interfaces
provide a built-in calculator utility, capable of performing
conversions between various radices, in general including hexadecimal.
In
Microsoft Windows, the
Calculator
utility can be set to Scientific mode (called Programmer mode in some
versions), which allows conversions between radix 16 (hexadecimal), 10
(decimal), 8 (
octal) and 2 (
binary), the bases most commonly used by programmers. In Scientific Mode, the on-screen
numeric keypad
includes the hexadecimal digits A through F, which are active when
"Hex" is selected. In hex mode, however, the Windows Calculator supports
only integers.
Real numbers
Rational numbers
As with other numeral systems, the hexadecimal system can be used to represent
rational numbers, although
repeating expansions are common since sixteen (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: 2; b − 1 = 15: 3, 5; b + 1 = 17: 1116 |
| Fraction |
Prime factors |
Positional representation |
Positional representation |
Prime factors |
Fraction |
| 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) |
1.414213562373095048... |
1.6A09E667F3BCD... |
| √3 (the length of the diagonal of a unit cube) |
1.732050807568877293... |
1.BB67AE8584CAA... |
| √5 (the length of the diagonal of a 1×2 rectangle) |
2.236067977499789696... |
2.3C6EF372FE95... |
| φ (phi, the golden ratio = (1+√5)/2) |
1.618033988749894848... |
1.9E3779B97F4A... |
| π (pi, the ratio of circumference to diameter of a circle) |
3.141592653589793238462643
383279502884197169399375105... |
3.243F6A8885A308D313198A2E0
3707344A4093822299F31D008... |
| e (the base of the natural logarithm) |
2.718281828459045235... |
2.B7E151628AED2A6B... |
| τ (the Thue–Morse constant) |
0.412454033640107597... |
0.6996 9669 9669 6996... |
γ (the limiting difference between the
harmonic series and the natural logarithm) |
0.577215664901532860... |
0.93C467E37DB0C7A4D1B... |
Powers
Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.
| 2x |
Value |
| 20 |
1 |
| 21 |
2 |
| 22 |
4 |
| 23 |
8 |
| 24 |
10hex |
| 25 |
20hex |
| 26 |
40hex |
| 27 |
80hex |
| 28 |
100hex |
| 29 |
200hex |
| 2A (210dec) |
400hex |
| 2B (211dec) |
800hex |
| 2C (212dec) |
1000hex |
| 2D (213dec) |
2000hex |
| 2E (214dec) |
4000hex |
| 2F (215dec) |
8000hex |
| 210 (216dec) |
10000hex |
Cultural
Etymology
The word
hexadecimal is composed of
hexa-, derived from the
Greek έξ (hex) for
six, and
-decimal, derived from the
Latin for
tenth. Webster's Third New International online derives
hexadecimal as an alteration of the all-Latin
sexadecimal (which appears in the earlier Bendix documentation). The earliest date attested for
hexadecimal in Merriam-Webster Collegiate online is 1954, placing it safely in the category of
international scientific vocabulary (ISV). It is common in ISV to mix Greek and Latin
combining forms freely. The word
sexagesimal (for base 60) retains the Latin prefix.
Donald Knuth has pointed out that the etymologically correct term is
senidenary (or possibly,
sedenary), from the Latin term for
grouped by 16. (The terms
binary,
ternary and
quaternary are from the same Latin construction, and the etymologically correct terms for
decimal and
octal arithmetic are
denary and
octonary, respectively.)
[15] 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".
[16][17] 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.
[18] The
etymologically proper
Greek term would be
hexadecadic (although in
Modern Greek,
deca-hexadic/
δεκαεξαδικός is more commonly used).
Use in Chinese culture
The traditional
Chinese units of weight were base-16. For example, one jīn (斤) in the old system equals sixteen
taels. The
suanpan (Chinese
abacus) could 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.
[19] Some proposals unify standard measures so that they are multiples of 16.
[20][21][22]
An example of unified standard measures is
hexadecimal time, which subdivides a day by 16 so that there are 16 "hexhours" in a day.
[22]
Key to number base notation
Simple key for notations used in article:
See also
References
Best Answer: In mathematics and computer science, base-16, hexadecimal, or simply hex, is a numeral system with a radix or base of 16 usually written using the symbols 0–9 and A–F or a–f. For example, the decimal numeral 79 whose binary representation is 01001111 can be written as 4F in hexadecimal (4 = 0100, F = 1111). The current hexadecimal system was first introduced to the computing world in 1963 by IBM. An earlier version, using the digits 0–9 and u–z, was used by the Bendix G-15 computer, introduced in 1956.
The use of the letters A through F to represent the digits above 9 was not universal in the early history of computers.Bruce Alan Martin's hexadecimal notation proposal[11]
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. Another system for counting up to FF16 (25510) is illustrated on the right.Hexadecimal finger-counting scheme.
It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value and then performing multiplication and addition to get the final representation. That is, to convert the number B3AD to decimal one can split the hexadecimal number into its digits: B (1110), 3 (310), A (1010) and D (1310), and then get the final result by multiplying each decimal representation by 16p, where p is the corresponding hex digit position, counting from right to left, beginning with 0. In this case we have B3AD = (11 × 163) + (3 × 162) + (10 × 161) + (13 × 160), which is 45997 base 10.A hexadecimal multiplication table
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