Saturday, June 4, 2016

Hexadecimal

  1. I had to study the hexadecimal system in 1966 in College and after so I knew more about it then when I learned Fortran and Cobol which were the main computer languages taught in college then in California. Later I taught myself the Basic language which the early home computers were based upon. Then came MS Dos and then finally Windows 95 through Microsoft for all the IBM clones made then in the 1980s and after. However, MSDOS and Windows were actually complete operating systems which included a lot of various kinds of programming. My wife asked me what the hexadecimal system was even though she learned Fortran in College too in graduate school while getting her master's degree in Business. However, I guess she hadn't also studied what hexidecimals were at the time so I showed her a lot of this stuff and she was very interested after seeing "The Martian" with me tonight again with Matt Damon on HBO.  The Martian uses this as an initial communications system when the original one had broken down to help bring back an astronaut.

    If you haven't seen  "The Martian" you should because it is technically relatively accurate regarding science and technology. Of course it is all Hollywooded up but it's really great when they get all or most of the science right in a movie.

  1.  

     begin quote from:

  1. What is hexadecimal? - Definition from WhatIs.com

    http://whatis.techtarget.com/definition/hexadecimal
    Hexadecimal describes a base-16 number system. That is, it describes a numbering system containing 16 sequential numbers as base units (including 0) before ...


Part of the Mathematics glossary:
Hexadecimal describes a base-16 number system. That is, it describes a numbering system containing 16 sequential numbers as base units (including 0) before adding a new position for the next number. (Note that we're using "16" here as a decimal number to explain a number that would be "10" in hexadecimal.) The hexadecimal numbers are 0-9 and then use the letters A-F. We show the equivalence of binary, decimal, and hexadecimal numbers in the table below.
Hexadecimal is a convenient way to express binary numbers in modern computers in which a byte is almost always defined as containing eight binary digits. When showing the contents of computer storage (for example, when getting a core dump of storage in order to debug a new computer program or when expressing a string of text characters or a string of binary values in coding a program or HTML page), one hexadecimal digit can represent the arrangement of four binary digits. Two hexadecimal digits can represent eight binary digits, or a byte.
BinaryDecimalHexadecimal
000
111
1022
1133
10044
10155
11066
11177
100088
100199
101010A
101111B
110012C
110113D
111014E
111115F
100001610
100011711
etcetcetc
  1.  

    • From Yahoo Answers

      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... Full answer

      What is the hexadecimal number system used for and who invented it?

      3 answers

      Answers

      Relevance
      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.

      A common use of hexadecimal numerals is found in HTML and CSS. They use hexadecimal notation (hex triplets) to specify colours on web pages; there is just the # symbol, not a separate symbol for "hexadecimal". Twenty-four-bit color is represented in the format #RRGGBB, where RR specifies the value of the Red component of the color, GG the Green component and BB the Blue component. For example, a shade of red that is (238,9,63) in decimal is coded as #EE093F. This syntax is borrowed from the X Window System.

      Example of conversion from hexadecimal triplet to decimal triplet: Hexadecimal triplet: FFCF4B

      Separate the triplets: FF CF 4B

      Convert each hexadecimal value to a decimal representation:

      FF = 15*16 + 15*1 = 255
      CF = 12*16 + 15*1 = 207
      4B = 4*16 + 11*1 = 75
      Hexadecimal triplet FFCF4B = Decimal triplet 255,207,75

      Hexadecimal is used also in more generic computing, as the most commonly found form of expressing a guaranteeably human-readable string representation of a byte. All the possible values of a byte (256 values) can be represented using 2 digits in hexadecimal notation. Some people assume that using 8-bit "ASCII" to represent the value of a byte should work, but this has a number of problems, firstly there are a number of unprintable control characters, secondly ASCII itself stops at 7 bits with the remainder being system specific extentions and finally even assuming all characters in the machines set were displayable as something neither users nor input methods are generally prepared to handle 256 unique characters.

      In URLs, all characters can be coded hexadecimally, even those not normally permitted. Each 2-digit (1 byte) hexadecimal sequence is preceded by a percent sign and refers to a specific ASCII character code. For example, in the URL [1], the ASCII character code for a space (" ") is 20.

      Knuth, Donald. (1969).

  1.   begin quote from:

    Hexadecimal - Wikipedia, the free encyclopedia

    en.wikipedia.org/wiki/Hexadecimal
    In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen ...
  2. Hexadecimal numeral system - Simple English...

    simple.wikipedia.org/wiki/Hexadecimal_numeral_system...
    The hexadecimal numeral system, also known as just hex, is a numeral system made up of 16 symbols (base 16). The standard numeral system is called ...
  3. Hexadecimal number system | Alternate number bases...

    www.khanacademy.org/math/pre-algebra/applying-math-r...
    Hexadecimal number system (base 16) ... Binary and hexadecimal number systems. Introduction to ... Converting from decimal to hexadecimal representation.
  4. Hexadecimals - Math is Fun

    www.mathsisfun.com/hexadecimals.html
    There are 16 Hexadecimal digits. They are the same as the decimal digits up to 9 , but then there are the letters A, B, C, D, E and F in place of the decimal ...
  5. What is hexadecimal? - Definition from WhatIs.com

    http://whatis.techtarget.com/definition/hexadecimal
    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 09 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 2AF316, depending on the choice of notation.
    As an example, the hexadecimal number 2AF316 can be converted to an equivalent decimal representation. Observe that 2AF316 is equal to a sum of (200016 + A0016 + F016 + 316), 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.

    Contents

    Representation

    Written representation

    Using 0–9 and A–F

    0hex = 0dec = 0oct
    0 0 0 0
    1hex = 1dec = 1oct 0 0 0 1
    2hex = 2dec = 2oct 0 0 1 0
    3hex = 3dec = 3oct 0 0 1 1

    4hex = 4dec = 4oct 0 1 0 0
    5hex = 5dec = 5oct 0 1 0   1
    6hex = 6dec = 6oct 0 1 1 0
    7hex = 7dec = 7oct 0 1 1 1

    8hex = 8dec = 10oct 1 0 0 0
    9hex = 9dec = 11oct 1 0 0 1
    Ahex = 10dec = 12oct 1 0 1 0
    Bhex = 11dec = 13oct 1 0 1 1

    Chex = 12dec = 14oct 1 1 0 0
    Dhex = 13dec = 15oct 1 1 0 1
    Ehex = 14dec = 16oct 1 1 1 0
    Fhex = 15dec = 17oct 1 1 1 1
    In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give the base explicitly: 15910 is decimal 159; 15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h.
    In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:
    • In URIs (including URLs), character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the space (blank) character, ASCII code point 20 in hex, 32 in decimal.
    • In XML and XHTML, characters can be expressed as hexadecimal numeric character references using the notation &#xcode;, where the x denotes that code is a hex code point (of 1- to 6-digits) assigned to the character in the Unicode standard. Thus 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 102310 on ten fingers. 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.
    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 −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).

    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 (410). This example converts 11112 to base ten. Since each position in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from the right:
    • 00012 = 110
    • 00102 = 210
    • 01002 = 410
    • 10002 = 810
    Therefore:
    11112 = 810 + 410 + 210 + 110
      = 1510
    With little practice, mapping 11112 to F16 in one step becomes easy: see table in Written representation. The advantage of using hexadecimal rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hexadecimal, it is trivial to regard the binary string as 4-digit groups and map each to a single hexadecimal digit.
    This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
    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 hihi−1...h2h1 be the hexadecimal digits representing the number.
    1. i ← 1
    2. hi ← d mod 16
    3. d ← (d − hi) / 16
    4. If d = 0 (return series hi) else increment i and go to step 2
    "16" may be replaced with any other base that may be desired.
    The following is a JavaScript implementation of the above algorithm for converting any number to a hexadecimal in String representation. Its purpose is to illustrate the above algorithm. To work with data seriously, however, it is much more advisable to work with bitwise operators.
    function toHex(d) {
      var r = d % 16;
      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

    A hexadecimal multiplication table
    It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value and then performing multiplication and addition to get the final representation. That is, to convert the number B3AD to decimal one can split the hexadecimal number into its digits: B (1110), 3 (310), A (1010) and D (1310), and then get the final result by multiplying each decimal representation by 16p, where p is the corresponding hex digit position, counting from right to left, beginning with 0. In this case we have B3AD = (11 × 163) + (3 × 162) + (10 × 161) + (13 × 160), which is 45997 base 10.

    Tools for conversion

    Most modern computer systems with graphical user interfaces provide a built-in calculator utility, capable of performing conversions between various radices, in general including hexadecimal.
    In Microsoft Windows, the Calculator utility can be set to Scientific mode (called Programmer mode in some versions), which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 (octal) and 2 (binary), the bases most commonly used by programmers. In Scientific Mode, the on-screen numeric keypad includes the hexadecimal digits A through F, which are active when "Hex" is selected. In hex mode, however, the Windows Calculator supports only integers.

    Real numbers

    Rational numbers

    As with other numeral systems, the hexadecimal system can be used to represent rational numbers, although repeating expansions are common since sixteen (10hex) 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 (42), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no cyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix; thus, when using hexadecimal notation, all fractions with denominators that are not a power of two result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than decimal for representing rational numbers since a larger proportion lie outside its range of finite representation.
    All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hexadecimal number with a finite number of digits 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:
    Name Abbreviation Number base
    Binary bin 2
    Octal oct 8
    Decimal dec 10
    Hexadecimal hex 16

    See also

    References


  6. "PDF The Unicode Standard, Version 7" (PDF).

  • "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.

  • "Modula-2 - Vocabulary and representation". Modula -2. Retrieved 1 November 2015.

  • The VHDL MINI-REFERENCE: VHDL IDENTIFIERS, NUMBERS, STRINGS, AND EXPRESSIONS

  • "*read-base* variable in Common Lisp".

  • "*print-base* 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 0-201-13448-9. The source code of the book in TeX (and a required set of macros CTAN.org) is available online on CTAN.

  • 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.

  • This somewhat odd sequence was from the next six sequential numeric keyboard codes in the LGP-30's 6-bit character code. LGP-30 PROGRAMMING MANUAL

  • "ILLIAC Programming" (PDF). University of Illinois via Bitsavers. pp. 3–2. Retrieved 18 December 2014.

  • http://pubs.opengroup.org/onlinepubs/9699919799/functions/printf.html

  • Knuth, Donald. (1969). The Art of Computer Programming, Volume 2. ISBN 0-201-03802-1. (Chapter 17.)

  • A.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 296-366. See pages 317 and 322.

  • Schwartzman, S. (1994). The Words of Mathematics: an etymological dictionary of mathematical terms used in English. ISBN 0-88385-511-9.

  • "Base 4^2 Hexadecimal Symbol Proposal".

  • "Intuitor Hex Headquarters".

  • "A proposal for addition of the six Hexadecimal digits (A-F) to Unicode".



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