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{{Short description|Encoding for data, using 0s and 1s}}
{{Short description |Encoded data represented in binary notation}}
{{For|the binary form of computer software|Machine code}}
{{For|the binary form of computer software|Machine code}}
[[Image:Wikipedia in binary.gif|thumb|The word 'Wikipedia' represented in [[ASCII]] binary code, made up of 9 bytes (72 bits).]]
[[Image:Wikipedia in binary.gif|thumb |The ASCII-encoded letters of "Wikipedia" represented as binary codes.]]
A '''binary code''' represents [[plain text|text]], [[instruction set|computer processor instructions]], or any other [[data]] using a two-symbol system. The two-symbol system used is often "0" and "1" from the [[binary number|binary number system]]. The binary code assigns a pattern of binary digits, also known as [[bit]]s, to each character, instruction, etc. For example, a binary [[string (computer science)|string]] of eight bits (which is also called a byte) can represent any of 256 possible values and can, therefore, represent a wide variety of different items.
[[Image:Binary to Hexadecimal or Decimal.jpg|thumb |Values represented in binary, hex and decimal]]
A '''binary code''' is the value of a [[data encoding |data-encoding]] convention represented in a [[Binary number |binary]] notation that usually is a sequence of 0s and 1s, sometimes called a ''[[bit]] string''. For example, [[ASCII]] is an 8-bit text encoding that in addition to the [[human readable]] form (letters) can be represented as binary. ''Binary code'' can also refer to the [[mass noun]] ''code'' that is not human readable in nature such as [[machine code]] and [[bytecode]].


In computing and telecommunications, binary codes are used for various methods of [[encoding]] data, such as [[character string]]s, into bit strings. Those methods may use fixed-width or [[variable-length code|variable-width]] strings. In a fixed-width binary code, each letter, digit, or other character is represented by a bit string of the same length; that bit string, interpreted as a [[binary number]], is usually displayed in code tables in [[octal]], [[decimal]] or [[hexadecimal]] notation. There are many [[character sets]] and many [[character encoding]]s for them.
Even though all modern computer data is binary in nature, and therefore can be represented as binary, other [[numerical base]]s may be used. [[Power of 2]] bases (including [[hexadecimal |hex]] and [[octal]]) are sometimes considered binary code since their power-of-2 nature makes them inherently linked to binary. [[Decimal]] is, of course, a commonly used representation. For example, ASCII characters are often represented as either decimal or hex. Some types of data such as [[image]] data is sometimes represented as hex, but rarely as decimal.
[[File:Binary to Hexadecimal or Decimal.jpg|thumb|Binary to Hexadecimal or Decimal]]


A [[bit string]], interpreted as a binary number, can be [[binary number#Decimal|translated into a decimal number]]. For example, the [[letter case|lower case]] ''a'', if represented by the bit string <code>01100001</code> (as it is in the standard [[ASCII]] code), can also be represented as the decimal number 97.
==History==
 
==History of binary codes==
{{further|Binary number#History}}
{{further|Binary number#History}}
  {{Disputed section|date=April 2015}}
  {{Disputed section|date=April 2015}}
Line 27: Line 25:
[[George Boole]] published a paper in 1847 called 'The Mathematical Analysis of Logic' that describes an algebraic system of logic, now known as [[Boolean algebra (logic)|Boolean algebra]]. Boole's system was based on binary, a yes-no, on-off approach that consisted of the three most basic operations: AND, OR, and NOT.<ref name="Boolean operations">{{Cite web|url=http://www.kerryr.net/pioneers/boolean.htm|title=What's So Logical About Boolean Algebra?|website=www.kerryr.net}}</ref> This system was not put into use until a graduate student from [[Massachusetts Institute of Technology]], [[Claude Shannon]], noticed that the Boolean algebra he learned was similar to an electric circuit. In 1937, Shannon wrote his master's thesis, ''[[A Symbolic Analysis of Relay and Switching Circuits]]'', which implemented his findings. Shannon's thesis became a starting point for the use of the binary code in practical applications such as computers, electric circuits, and more.<ref name="Claude Shannon">{{Cite web|url=http://www.kerryr.net/pioneers/shannon.htm|title=Claude Shannon (1916 - 2001)|website=www.kerryr.net}}</ref>
[[George Boole]] published a paper in 1847 called 'The Mathematical Analysis of Logic' that describes an algebraic system of logic, now known as [[Boolean algebra (logic)|Boolean algebra]]. Boole's system was based on binary, a yes-no, on-off approach that consisted of the three most basic operations: AND, OR, and NOT.<ref name="Boolean operations">{{Cite web|url=http://www.kerryr.net/pioneers/boolean.htm|title=What's So Logical About Boolean Algebra?|website=www.kerryr.net}}</ref> This system was not put into use until a graduate student from [[Massachusetts Institute of Technology]], [[Claude Shannon]], noticed that the Boolean algebra he learned was similar to an electric circuit. In 1937, Shannon wrote his master's thesis, ''[[A Symbolic Analysis of Relay and Switching Circuits]]'', which implemented his findings. Shannon's thesis became a starting point for the use of the binary code in practical applications such as computers, electric circuits, and more.<ref name="Claude Shannon">{{Cite web|url=http://www.kerryr.net/pioneers/shannon.htm|title=Claude Shannon (1916 - 2001)|website=www.kerryr.net}}</ref>


==Other forms of binary code==
===Timeline===
{{Main|List of binary codes}}
* 1875: [[Émile Baudot]] "Addition of binary strings in his ciphering system," which, eventually, led to the ASCII of today.
* 1884: The [[Linotype machine]] where the matrices are sorted to their corresponding channels after use by a binary-coded slide rail.
* 1932: [[C. E. Wynn-Williams]] "Scale of Two" counter<ref name="Glaser">{{Harvnb|Glaser|1971}}</ref>
* 1937: [[Alan Turing]] electro-mechanical binary multiplier
* 1937: [[George Stibitz]] [[Excess three code|"excess three" code]] in the [[George Stibitz#Computer|Complex Computer]]<ref name="Glaser"/>
* 1937: [[Atanasoff–Berry Computer]]<ref name="Glaser"/>
* 1938: [[Konrad Zuse]] [[Z1 (computer)|Z1]]
 
==Rendering==
{{original research|section|date=March 2015}}
{{original research|section|date=March 2015}}
[[File:Bagua-name-earlier.svg|thumb|Daoist Bagua]]
[[File:Bagua-name-earlier.svg|thumb|Daoist Bagua]]
The bit string is not the only type of binary code: in fact, a binary system in general, is any system that allows only two choices such as a switch in an electronic system or a simple true or false test.
A binary code can be rendered using any two distinguishable indications. In addition to the bit string, other notable ways to render a binary code are described below.
 
===Braille===
[[Braille]] is a type of binary code that is widely used by the blind to read and write by touch, named for its creator, Louis Braille. This system consists of grids of six dots each, three per column, in which each dot has two states: raised or not raised. The different combinations of raised and flattened dots are capable of representing all letters, numbers, and punctuation signs.
 
===Bagua {{anchor|BaGua}}===
The ''[[bagua]]'' are diagrams used in ''[[feng shui]],'' [[Taoist]] [[cosmology]] and ''[[I Ching]]'' studies. The ''ba gua'' consists of 8 trigrams; ''bā'' meaning 8 and ''guà'' meaning divination figure. The same word is used for the 64 guà (hexagrams). Each figure combines three lines (''yáo'') that are either broken ([[Yin and yang|''yin'']]) or unbroken (''yang''). The relationships between the trigrams are represented in two arrangements, the primordial, "Earlier Heaven"  or "Fuxi" ''bagua'', and the manifested, "Later Heaven", or "King Wen" ''bagua''.<ref name='wilhelm'>{{cite book |last=Wilhelm |first=Richard |author-link=Richard Wilhelm (sinologist) |others=trans. by [[Cary F. Baynes]], foreword by [[C. G. Jung]], preface to 3rd ed. by [[Hellmut Wilhelm]] (1967) |title=The I Ching or Book of Changes |publisher=Princeton University Press |year=1950 |location=Princeton, NJ |url=https://books.google.com/books?id=bbU9AAAAIAAJ&pg=PA266 |isbn=978-0-691-09750-3 |pages=266, 269}}</ref> (See also, the [[King Wen sequence]] of the 64 hexagrams).


===Ifá, Ilm Al-Raml and Geomancy{{anchor|Ifá}}===
; Braille: [[Braille]] is a binary code that is widely used to enable the blind to read and write by touch. The system consists of grids of six dots each, three per column, in which each dot is either raised or flat (not raised). The different combinations of raised and flat dots encode information such as letters, numbers, and punctuation.
The [[Ifá]]/Ifé system of divination in African religions, such as of [[Yoruba people|Yoruba]], [[Igbo people|Igbo]], and [[Ewe people|Ewe]], consists of an elaborate traditional ceremony producing 256 oracles made up by 16 symbols with 256 = 16 x 16. An initiated priest, or [[Babalawo]], who had memorized oracles, would request sacrifice from consulting clients and make prayers. Then, divination nuts or a pair of chains are used to produce random binary numbers,<ref>{{Cite book |last=Olupona |first=Jacob K. |url=https://www.worldcat.org/oclc/839396781 |title=African Religions: A Very Short Introduction |publisher=[[Oxford University Press]] |year=2014 |isbn=978-0-19-979058-6 |location=Oxford |pages=45 |oclc=839396781}}</ref> which are drawn with sandy material on an "Opun" figured wooden tray representing the totality of fate.


Through the spread of [[Islamic]] culture, Ifé/Ifá was assimilated as the "Science of Sand" (ilm al-raml), which then spread further and became "Science of Reading the Signs on the Ground" ([[Geomancy]]) in Europe.
; Bagua {{anchor|BaGua}}: The ''[[bagua]]'' is a set of diagrams used in ''[[feng shui]],'' [[Taoist]] [[cosmology]] and ''[[I Ching]]'' studies. The ''ba gua'' consists of 8 trigrams, each a combination of three lines (''yáo'') that are either broken ([[Yin and yang|''yin'']]) or unbroken (''yang'').<ref name='wilhelm'>{{cite book |last=Wilhelm |first=Richard |author-link=Richard Wilhelm (sinologist) |others=trans. by [[Cary F. Baynes]], foreword by [[C. G. Jung]], preface to 3rd ed. by [[Hellmut Wilhelm]] (1967) |title=The I Ching or Book of Changes |publisher=Princeton University Press |year=1950 |location=Princeton, NJ |url=https://books.google.com/books?id=bbU9AAAAIAAJ&pg=PA266 |isbn=978-0-691-09750-3 |pages=266, 269}}</ref>


This was thought to be another possible route from which computer science was inspired,<ref>{{Cite web|last=Eglash|first=Ron|date=June 2007|title=The fractals at the heart of African designs|url=https://www.ted.com/talks/ron_eglash_the_fractals_at_the_heart_of_african_designs/up-next#t-13472|url-status=live|access-date=2021-04-15|website=www.ted.com|archive-url=https://web.archive.org/web/20210727161435/https://www.ted.com/talks/ron_eglash_the_fractals_at_the_heart_of_african_designs/up-next |archive-date=2021-07-27 }}</ref> as Geomancy arrived at Europe at an earlier stage (about 12th Century, described by [[Hugo of Santalla|Hugh of Santalla]]) than [[I Ching]] (17th Century, described by [[Gottfried Wilhelm Leibniz]]).
; Ifá{{anchor|Ifá}}: The [[Ifá]]/Ifé system of divination in African religions, such as of [[Yoruba people |Yoruba]], [[Igbo people |Igbo]], and [[Ewe people |Ewe]], consists of an elaborate traditional ceremony producing 256 oracles made up by 16 symbols with 256 = 16 x 16. A priest, or [[Babalawo]], requests sacrifice from consulting clients and makes prayers. Then, divination [[nut (fruit)|nuts]] or a pair of [[chain]]s are used to produce random binary numbers,<ref>{{Cite book |last=Olupona |first=Jacob K. |title=African Religions: A Very Short Introduction |publisher=[[Oxford University Press]] |year=2014 |isbn=978-0-19-979058-6 |location=Oxford |pages=45 |oclc=839396781}}</ref> which are drawn with sandy material on an "Opun" figured wooden tray representing the totality of fate.<ref>{{Cite web|last=Eglash|first=Ron|date=June 2007|title=The fractals at the heart of African designs|url=https://www.ted.com/talks/ron_eglash_the_fractals_at_the_heart_of_african_designs/up-next#t-13472|url-status=live|access-date=2021-04-15|website=www.ted.com|archive-url=https://web.archive.org/web/20210727161435/https://www.ted.com/talks/ron_eglash_the_fractals_at_the_heart_of_african_designs/up-next |archive-date=2021-07-27 }}</ref>


==Coding systems==
==Encoding==
[[File:2D Binary Index.svg|thumb|An example of a recursive [[binary space partitioning]] [[quadtree]] for a 2D index]]
[[File:2D Binary Index.svg|thumb|An example of a recursive [[binary space partitioning]] [[quadtree]] for a 2D index]]
Innumerable encoding systems exists. Some notable examples are described here.


===ASCII code===
; ASCII: The [[American Standard Code for Information Interchange]] (ASCII) character encoding, is a 7-bit convention for representing (normal/printing) characters and [[Control character |control]] operations. Each printing and control character is assigned a number from 0 to 127. For example, "a" is represented by decimal code 97 which is rendered as bit string <code>1100001</code>.
The [[American Standard Code for Information Interchange]] (ASCII), uses a 7-bit binary code to represent text and other characters within computers, communications equipment, and other devices. Each letter or symbol is assigned a number from 0 to 127. For example, lowercase "a" is represented by <code>1100001</code> as a bit string (which is decimal 97).
 
===Binary-coded decimal===
[[Binary-coded decimal]] (BCD) is a binary encoded representation of integer values that uses a 4-bit [[nibble]] to encode decimal digits. Four binary bits can encode up to 16 distinct values; but, in BCD-encoded numbers, only ten values in each nibble are legal, and encode the decimal digits zero, through nine. The remaining six values are illegal and may cause either a machine exception or unspecified behavior, depending on the computer implementation of BCD arithmetic.
 
BCD arithmetic is sometimes preferred to floating-point numeric formats in commercial and financial applications where the complex rounding behaviors of floating-point numbers is inappropriate.<ref name="Cowlishaw_GDA">{{cite web |first=Mike F. |last=Cowlishaw |author-link=Mike F. Cowlishaw |title=General Decimal Arithmetic |orig-year=1981, 2008 |publisher=IBM |date=2015 |url=http://speleotrove.com/decimal/<!-- http://www2.hursley.ibm.com/decimal/ --> |access-date=2016-01-02}}</ref>
 
==Early uses of binary codes==
* 1875: [[Émile Baudot]] "Addition of binary strings in his ciphering system," which, eventually, led to the ASCII of today.
* 1884: The [[Linotype machine]] where the matrices are sorted to their corresponding channels after use by a binary-coded slide rail.
* 1932: [[C. E. Wynn-Williams]] "Scale of Two" counter<ref name="Glaser">{{Harvnb|Glaser|1971}}</ref>
* 1937: [[Alan Turing]] electro-mechanical binary multiplier
* 1937: [[George Stibitz]] [[Excess three code|"excess three" code]] in the [[George Stibitz#Computer|Complex Computer]]<ref name="Glaser"/>
* 1937: [[Atanasoff–Berry Computer]]<ref name="Glaser"/>
* 1938: [[Konrad Zuse]] [[Z1 (computer)|Z1]]
 
==Current uses of binary==
Most modern computers use binary encoding for instructions and data. [[CD]]s, [[DVD]]s, and [[Blu-ray Disc]]s represent sound and video digitally in binary form. Telephone calls are carried digitally on long-distance and mobile phone networks using [[pulse-code modulation]], and on [[voice over IP]] networks.


==Weight of binary codes==
; Binary-coded decimal: [[Binary-coded decimal]] (BCD) is an encoding of integer values that consists of a 4-bit [[nibble]] for each decimal digit. As a decimal digit is only 1 of 10 values (0 to 9) but 4 bits can encode up to 16 values, and BCD element is invalid for a value greater than 9.<ref name="Cowlishaw_GDA">{{cite web |first=Mike F. |last=Cowlishaw |author-link=Mike F. Cowlishaw |title=General Decimal Arithmetic |orig-year=1981, 2008 |publisher=IBM |date=2015 |url=http://speleotrove.com/decimal/<!-- http://www2.hursley.ibm.com/decimal/ --> |access-date=2016-01-02}}</ref>
The weight of a binary code, as defined in the table of [[constant-weight code]]s,<ref>[http://www.research.att.com/~njas/codes/Andw/ Table of Constant Weight Binary Codes]</ref> is the [[Hamming weight]] of the binary words coding for the represented words or sequences.


==See also==
==See also==
* [[Binary number]]
* {{Annotated link |Binary file}}
* [[List of binary codes]]
* {{Annotated link |Bit array}}
* [[Binary file]]
* {{Annotated link |Constant-weight code}}
* [[Unicode]]
* {{Annotated link |Gray code}}
* [[Gray code]]
* {{Annotated link |List of binary codes}}
* {{Annotated link |Unicode}}


==References==
==References==
Line 88: Line 70:
* [https://web.archive.org/web/20170423014446/http://www.eng.tau.ac.il/~litsyn/tableand/ Table of Nonlinear Binary Codes]. Maintained by Simon Litsyn, E. M. Rains, and N. J. A. Sloane. Updated until 1999.
* [https://web.archive.org/web/20170423014446/http://www.eng.tau.ac.il/~litsyn/tableand/ Table of Nonlinear Binary Codes]. Maintained by Simon Litsyn, E. M. Rains, and N. J. A. Sloane. Updated until 1999.
* {{Cite book | last = Glaser | first = Anton | title = History of Binary and other Nondecimal Numeration | publisher = Tomash | year = 1971 | chapter = Chapter VII Applications to Computers | isbn = 978-0-938228-00-4}} cites some pre-ENIAC milestones.
* {{Cite book | last = Glaser | first = Anton | title = History of Binary and other Nondecimal Numeration | publisher = Tomash | year = 1971 | chapter = Chapter VII Applications to Computers | isbn = 978-0-938228-00-4}} cites some pre-ENIAC milestones.
* [https://www.amazon.it/01010011-01100101-01100111-01110010-01110100/dp/B0CHD5Q8W1/ref=sr_1_1?qid=1694173977&refinements=p_27:%22Luigi+Usai%22&s=books&sr=1-1&ufe=app_do:amzn1.fos.9d4f9b77-768c-4a4e-94ad-33674c20ab35 First book in the world fully written in binary code]: (<abbr>IT</abbr>) Luigi Usai, ''01010011 01100101 01100111 01110010 01100101 01110100 01101001'', Independently published, 2023, <nowiki>ISBN 979-8-8604-3980-1</nowiki>. <small>URL consulted September 8, 2023</small>.
* [https://www.amazon.it/01010011-01100101-01100111-01110010-01110100/dp/B0CHD5Q8W1/ref=sr_1_1?qid=1694173977&refinements=p_27:%22Luigi+Usai%22&s=books&sr=1-1&ufe=app_do:amzn1.fos.9d4f9b77-768c-4a4e-94ad-33674c20ab35 First book in the world fully written in binary code]: (<abbr>IT</abbr>) Luigi Usai, ''01010011 01100101 01100111 01110010 01100101 01110100 01101001'', Independently published, 2023, {{ISBN|979-8-8604-3980-1}}. <small>URL consulted September 8, 2023</small>.


[[Category:Computer data]]
[[Category:Computer data]]

Latest revision as of 01:28, 18 December 2025

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File:Wikipedia in binary.gif
The ASCII-encoded letters of "Wikipedia" represented as binary codes.
File:Binary to Hexadecimal or Decimal.jpg
Values represented in binary, hex and decimal

A binary code is the value of a data-encoding convention represented in a binary notation that usually is a sequence of 0s and 1s, sometimes called a bit string. For example, ASCII is an 8-bit text encoding that in addition to the human readable form (letters) can be represented as binary. Binary code can also refer to the mass noun code that is not human readable in nature such as machine code and bytecode.

Even though all modern computer data is binary in nature, and therefore can be represented as binary, other numerical bases may be used. Power of 2 bases (including hex and octal) are sometimes considered binary code since their power-of-2 nature makes them inherently linked to binary. Decimal is, of course, a commonly used representation. For example, ASCII characters are often represented as either decimal or hex. Some types of data such as image data is sometimes represented as hex, but rarely as decimal.

History

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File:Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg
Gottfried Leibniz

Invention

The modern binary number system, the basis for binary code, is an invention by Gottfried Leibniz in 1689 and appears in his article Explication de l'Arithmétique Binaire (English: Explanation of the Binary Arithmetic) which uses only the characters 1 and 0, and some remarks on its usefulness. Leibniz's system uses 0 and 1, like the modern binary numeral system. Binary numerals were central to Leibniz's intellectual and theological ideas. He believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing.[1][2] In Leibniz's view, binary numbers represented a fundamental form of creation, reflecting the simplicity and unity of the divine.[2] Leibniz was also attempting to find a way to translate logical reasoning into pure mathematics. He viewed the binary system as a means of simplifying complex logical and mathematical processes, believing that it could be used to express all concepts of arithmetic and logic.[2]

Previous Ideas

Leibniz explained in his work that he encountered the I Ching by Fu Xi[2] that dates from the 9th century BC in China,[3] through French Jesuit Joachim Bouvet and noted with fascination how its hexagrams correspond to the binary numbers from 0 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical visual binary mathematics he admired.[4][5] Leibniz saw the hexagrams as an affirmation of the universality of his own religious belief.[5] After Leibniz ideas were ignored, the book had confirmed his theory that life could be simplified or reduced down to a series of straightforward propositions. He created a system consisting of rows of zeros and ones. During this time period, Leibniz had not yet found a use for this system.[6] The binary system of the I Ching is based on the duality of yin and yang.[7] Slit drums with binary tones are used to encode messages across Africa and Asia.[7] The Indian scholar Pingala (around 5th–2nd centuries BC) developed a binary system for describing prosody in his Chandashutram.[8][9]

Mangareva people in French Polynesia were using a hybrid binary-decimal system before 1450.[10] In the 11th century, scholar and philosopher Shao Yong developed a method for arranging the hexagrams which corresponds, albeit unintentionally, to the sequence 0 to 63, as represented in binary, with yin as 0, yang as 1 and the least significant bit on top. The ordering is also the lexicographical order on sextuples of elements chosen from a two-element set.[11]

File:George Boole color.jpg
George Boole

In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as scarcely visible variations in the font in any random text.[12] Importantly for the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only; as by Bells, by Trumpets, by Lights and Torches, by the report of Muskets, and any instruments of like nature".[12]

Boolean Logical System

George Boole published a paper in 1847 called 'The Mathematical Analysis of Logic' that describes an algebraic system of logic, now known as Boolean algebra. Boole's system was based on binary, a yes-no, on-off approach that consisted of the three most basic operations: AND, OR, and NOT.[13] This system was not put into use until a graduate student from Massachusetts Institute of Technology, Claude Shannon, noticed that the Boolean algebra he learned was similar to an electric circuit. In 1937, Shannon wrote his master's thesis, A Symbolic Analysis of Relay and Switching Circuits, which implemented his findings. Shannon's thesis became a starting point for the use of the binary code in practical applications such as computers, electric circuits, and more.[14]

Timeline

Rendering

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File:Bagua-name-earlier.svg
Daoist Bagua

A binary code can be rendered using any two distinguishable indications. In addition to the bit string, other notable ways to render a binary code are described below.

Braille
Braille is a binary code that is widely used to enable the blind to read and write by touch. The system consists of grids of six dots each, three per column, in which each dot is either raised or flat (not raised). The different combinations of raised and flat dots encode information such as letters, numbers, and punctuation.
Bagua Script error: No such module "anchor".
The bagua is a set of diagrams used in feng shui, Taoist cosmology and I Ching studies. The ba gua consists of 8 trigrams, each a combination of three lines (yáo) that are either broken (yin) or unbroken (yang).[16]
IfáScript error: No such module "anchor".
The Ifá/Ifé system of divination in African religions, such as of Yoruba, Igbo, and Ewe, consists of an elaborate traditional ceremony producing 256 oracles made up by 16 symbols with 256 = 16 x 16. A priest, or Babalawo, requests sacrifice from consulting clients and makes prayers. Then, divination nuts or a pair of chains are used to produce random binary numbers,[17] which are drawn with sandy material on an "Opun" figured wooden tray representing the totality of fate.[18]

Encoding

File:2D Binary Index.svg
An example of a recursive binary space partitioning quadtree for a 2D index

Innumerable encoding systems exists. Some notable examples are described here.

ASCII
The American Standard Code for Information Interchange (ASCII) character encoding, is a 7-bit convention for representing (normal/printing) characters and control operations. Each printing and control character is assigned a number from 0 to 127. For example, "a" is represented by decimal code 97 which is rendered as bit string 1100001.
Binary-coded decimal
Binary-coded decimal (BCD) is an encoding of integer values that consists of a 4-bit nibble for each decimal digit. As a decimal digit is only 1 of 10 values (0 to 9) but 4 bits can encode up to 16 values, and BCD element is invalid for a value greater than 9.[19]

See also

References

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  1. ↑ Script error: No such module "citation/CS1".
  2. ↑ a b c d Leibniz G., Explication de l'Arithmétique Binaire, Die Mathematische Schriften, ed. C. Gerhardt, Berlin 1879, vol.7, p.223; Engl. transl.[1]
  3. ↑ Script error: No such module "citation/CS1".
  4. ↑ Script error: No such module "citation/CS1".
  5. ↑ a b Script error: No such module "citation/CS1".
  6. ↑ Script error: No such module "citation/CS1".
  7. ↑ a b Script error: No such module "citation/CS1".
  8. ↑ Script error: No such module "citation/CS1".
  9. ↑ W. S. Anglin and J. Lambek, The Heritage of Thales, Springer, 1995, Template:ISBN
  10. ↑ Script error: No such module "Citation/CS1".
  11. ↑ Script error: No such module "Citation/CS1".
  12. ↑ a b Script error: No such module "citation/CS1".
  13. ↑ Script error: No such module "citation/CS1".
  14. ↑ Script error: No such module "citation/CS1".
  15. ↑ a b c Script error: No such module "Footnotes".
  16. ↑ Script error: No such module "citation/CS1".
  17. ↑ Script error: No such module "citation/CS1".
  18. ↑ Script error: No such module "citation/CS1".
  19. ↑ Script error: No such module "citation/CS1".

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External links

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