Editing Binary number (section)

==History==
The modern  binary number system was studied in Europe in the 16th and 17th centuries by [[Thomas Harriot]], and [[Gottfried Leibniz]]. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Europe and India.

===Egypt===
{{See also|Ancient Egyptian mathematics}}
[[File:Oudjat.SVG|thumb|240px|left|Arithmetic values thought to have been represented by parts of the [[Eye of Horus]]]]
The scribes of ancient Egypt used two different systems for their fractions, [[Egyptian fraction]]s (not related to the binary number system) and [[Eye of Horus|Horus-Eye]] fractions (so called because many historians of mathematics believe that the symbols used for this system could be arranged to form the eye of [[Horus]], although this has been disputed).<ref>{{citation|title=The Oxford Handbook of the History of Mathematics|editor1-first=Eleanor|editor1-last=Robson|editor1-link=Eleanor Robson|editor2-first=Jacqueline|editor2-last=Stedall|editor2-link=Jackie Stedall|publisher=Oxford University Press|year=2009|isbn=9780199213122|page=790|url=https://books.google.com/books?id=xZMSDAAAQBAJ&pg=PA790|contribution=Myth No. 2: the Horus eye fractions}}</ref> Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a [[hekat]] is expressed as a sum of the binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, and 1/64. Early forms of this system can be found in documents from the [[Fifth Dynasty of Egypt]], approximately 2400 BC, and its fully developed hieroglyphic form dates to the [[Nineteenth Dynasty of Egypt]], approximately 1200 BC.<ref>{{citation|title=Numerical Notation: A Comparative History|first=Stephen|last=Chrisomalis|publisher=Cambridge University Press|year=2010|isbn=9780521878180|pages=42–43|url=https://books.google.com/books?id=ux--OWgWvBQC&pg=PA42}}.</ref>

The method used for [[ancient Egyptian multiplication]] is also closely related to binary numbers. In this method, multiplying one number by a second is performed by a sequence of steps in which a value (initially the first of the two numbers) is either doubled or has the first number added back into it; the order in which these steps are to be performed is given by the binary representation of the second number. This method can be seen in use, for instance, in the [[Rhind Mathematical Papyrus]], which dates to around 1650 BC.<ref>{{citation|title=How Mathematics Happened: The First 50,000 Years|first=Peter Strom|last=Rudman|publisher=Prometheus Books|year=2007|isbn=9781615921768|pages=135–136|url=https://books.google.com/books?id=BtcQq4RUfkUC&pg=PA135}}.</ref>

===China===
[[File:Bagua-name-earlier.svg|thumb|160px|Daoist Bagua]]
The ''[[I Ching]]'' dates from the 9th century BC in China.<ref name="HackerMoore2002">{{cite book|author1=Edward Hacker|author2=Steve Moore|author3=Lorraine Patsco|title=I Ching: An Annotated Bibliography|url=https://books.google.com/books?id=S5hLpfFiMCQC&pg=PR13|year=2002|publisher=Routledge|isbn=978-0-415-93969-0|page=13}}</ref> The binary notation in the ''I Ching'' is used to interpret its [[quaternary numeral system|quaternary]] [[I Ching divination|divination]] technique.<ref name=redmond-hon/>

It is based on taoistic duality of [[yin and yang]].<ref name="scientific">{{cite book|author1=Jonathan Shectman|title=Groundbreaking Scientific Experiments, Inventions, and Discoveries of the 18th Century|url=https://books.google.com/books?id=SsbChdIiflsC&pg=PA29|year=2003|publisher=Greenwood Publishing|isbn=978-0-313-32015-6|page=29}}</ref> [[Ba gua|Eight trigrams (Bagua)]] and a set of [[Hexagram (I Ching)|64 hexagrams ("sixty-four" gua)]], analogous to the three-bit and six-bit binary numerals, were in use at least as early as the [[Zhou dynasty]] of ancient China.<ref name="HackerMoore2002"/>

The [[Song dynasty]] scholar [[Shao Yong]] (1011–1077) rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically.<ref name=redmond-hon>{{cite book|last1=Redmond|first1=Geoffrey|last2=Hon|first2=Tze-Ki|title=Teaching the I Ching|date=2014|publisher=Oxford University Press|isbn=978-0-19-976681-9|page=227}}</ref> Viewing the [[least significant bit]] on top of single hexagrams in Shao Yong's square<ref name="Marshall">
{{cite web
|url= http://www.biroco.com/yijing/sequence.htm
|title= Yijing hexagram sequences: The Shao Yong square (Fuxi sequence)
|last= Marshall
|first= Steve
|date=
|website=
|publisher=
|access-date=2022-09-15
|quote="You could say [the Fuxi binary sequence] is a more sensible way of rendering hexagram as binary numbers ... The reasoning, if any, that informs [the King Wen] sequence is unknown."
}}
</ref>
and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63.
<ref name="Shao Yong’s ”Xiantian Tu'‘">{{cite book|last1=Zhonglian|first1=Shi|last2=Wenzhao|first2=Li|last3=Poser|first3=Hans|title=Leibniz' Binary System and Shao Yong's "Xiantian Tu" in :Das Neueste über China: G.W. Leibnizens Novissima Sinica von 1697 : Internationales Symposium, Berlin 4. bis 7. Oktober 1997|date=2000|
publisher=Franz Steiner Verlag|location=Stuttgart|isbn=3515074481|pages=165–170|url=https://books.google.com/books?id=DkIpP2SsGlIC&pg=PA165|ref=ID3515074481}}</ref>
=== Classical antiquity ===
[[Etruscan civilization|Etruscans]] divided the outer edge of [[Haruspex|divination livers]] into sixteen parts, each inscribed with the name of a divinity and its region of the sky. Each liver region produced a binary reading which was combined into a final binary for divination.<ref>{{Cite journal |last=Collins |first=Derek |date=2008 |title=Mapping the Entrails: The Practice of Greek Hepatoscopy |url=https://www.jstor.org/stable/27566714 |journal=The American Journal of Philology |volume=129 |issue=3 |pages=319–345 |jstor=27566714 |issn=0002-9475}}</ref> 

Divination at Ancient Greek [[Dodona]] oracle worked by drawing from separate jars, questions tablets and "yes" and "no" pellets. The result was then combined to make a final prophecy.<ref>{{Cite book |last=Johnston |first=Sarah Iles |title=Ancient Greek divination |date=2008 |publisher=Wiley-Blackwell |isbn=978-1-4051-1573-5 |edition=1. publ |series=Blackwell ancient religions |location=Malden, Mass.}}</ref>

===India===
The Indian scholar [[Pingala]] (c. 2nd century BC) developed a binary system for describing [[prosody (poetry)|prosody]].<ref>{{Cite book|last1=Sanchez|first1=Julio|last2=Canton|first2=Maria P.|title=Microcontroller programming: the microchip PIC|year=2007|publisher=CRC Press|location=Boca Raton, Florida|isbn=978-0-8493-7189-9|page=37}}</ref><ref>W. S. Anglin and J. Lambek, ''The Heritage of Thales'', Springer, 1995, {{ISBN|0-387-94544-X}}</ref> He described meters in the form of short and long syllables (the latter equal in length to two short syllables).<ref>[http://www.sju.edu/~rhall/Rhythms/Poets/arcadia.pdf Math for Poets and Drummers] {{Webarchive|url=https://web.archive.org/web/20120616225617/http://www.sju.edu/~rhall/Rhythms/Poets/arcadia.pdf |date=16 June 2012 }} (pdf, 145KB)</ref> They were known as ''laghu'' (light) and ''guru'' (heavy) syllables.

Pingala's Hindu classic titled [[Chandaḥśāstra]] (8.23) describes the formation of a matrix in order to give a unique value to each meter. "Chandaḥśāstra" literally translates to ''science of meters'' in Sanskrit. The binary representations in Pingala's system increases towards the right, and not to the left like in the binary numbers of the modern [[positional notation]].<ref>{{Cite book|title=The mathematics of harmony: from Euclid to contemporary mathematics and computer science|first1=Alexey|last1=Stakhov|author1-link=Alexey Stakhov|first2=Scott Anthony|last2=Olsen|isbn=978-981-277-582-5|year=2009|publisher=World Scientific |url=https://books.google.com/books?id=K6fac9RxXREC}}</ref> In Pingala's system, the numbers start from number one, and not zero. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of [[place value]]s.<ref>B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50</ref>

=== Africa ===
The [[Ifá]] is an African divination system''.'' Similar to the ''I Ching'', but has up to 256 binary signs,<ref>{{Cite book |last=Landry |first=Timothy R. |title=Vodún: secrecy and the search for divine power |date=2019 |publisher=University of Pennsylvania Press |isbn=978-0-8122-5074-9 |edition=1st |series=Contemporary ethnography |location=Philadelphia |pages=25}}</ref> unlike the ''I Ching'' which has 64. The Ifá originated in 15th century West Africa among [[Yoruba people]]. In 2008, [[UNESCO]] added Ifá to its list of the "[[Masterpieces of the Oral and Intangible Heritage of Humanity]]".{{sfn|Landry|2019|p=154}}<ref>{{Cite web |title=Ifa Divination System |url=https://ich.unesco.org/en/RL/ifa-divination-system-00146 |access-date=5 July 2017}}</ref>

===Other cultures===
The residents of the island of [[Mangareva]] in [[French Polynesia]] were using a hybrid binary-[[decimal]] system before 1450.<ref>{{Cite journal|last1=Bender|first1=Andrea|last2=Beller|first2=Sieghard|title=Mangarevan invention of binary steps for easier calculation|journal=Proceedings of the National Academy of Sciences|volume=111|issue=4|date=16 December 2013|doi=10.1073/pnas.1309160110|pages=1322–1327|pmid=24344278|pmc=3910603|doi-access=free}}</ref> [[Slit drum]]s with binary tones are used to encode messages across Africa and Asia.<ref name="scientific"/>
Sets of binary combinations similar to the ''I Ching'' have also been used in traditional African divination systems, such as [[Ifá]] among others, as well as in [[Middle Ages|medieval]] Western [[geomancy]]. The majority of [[Australian Aboriginal languages|Indigenous Australian languages]] use a base-2 system.<ref>{{Cite journal |last1=Bowern |first1=Claire |last2=Zentz |first2=Jason |date=2012 |title=Diversity in the Numeral Systems of Australian Languages |url=https://www.jstor.org/stable/23621076 |journal=Anthropological Linguistics |volume=54 |issue=2 |pages=133–160 |jstor=23621076 |issn=0003-5483}}</ref>

===Western predecessors to Leibniz===
In the late 13th century [[Ramon Llull]] had the ambition to account for all wisdom in every branch of human knowledge of the time. For that purpose he developed a general method or "Ars generalis" based on binary combinations of a number of simple basic principles or categories, for which he has been considered a predecessor of computing science and artificial intelligence.<ref>(see Bonner 2007 [http://lullianarts.net/] {{Webarchive|url=https://web.archive.org/web/20140403194204/http://lullianarts.net/|date=3 April 2014}}, Fidora et al. 2011 [https://www.iiia.csic.es/es/publications/ramon-llull-ars-magna-artificial-intelligence/] {{Webarchive|url=https://web.archive.org/web/20190408011909/https://www.iiia.csic.es/es/publications/ramon-llull-ars-magna-artificial-intelligence/|date=8 April 2019}})</ref>

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.<ref name="Bacon1605" /> 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".<ref name="Bacon1605">{{Cite web
|last=Bacon
|first=Francis
|author-link=Francis Bacon
|title=The Advancement of Learning
|url=http://home.hiwaay.net/~paul/bacon/advancement/book6ch1.html
|year=1605
|volume=6
|location=London
|pages=Chapter 1
}}
</ref> (See [[Bacon's cipher]].)

In 1617, [[John Napier]] described a system he called [[location arithmetic]] for doing binary calculations using a non-positional representation by letters.
[[Thomas Harriot]] investigated several positional numbering systems, including binary, but did not publish his results; they were found later among his papers.<ref>{{cite journal|last=Shirley|first=John W.|title=Binary numeration before Leibniz|journal=American Journal of Physics|volume=19|year=1951|issue=8|pages=452–454|doi=10.1119/1.1933042|bibcode=1951AmJPh..19..452S}}</ref>
Possibly the first publication of the system in Europe was by [[Juan Caramuel y Lobkowitz]], in 1700.<ref>{{cite journal|last=Ineichen|first=R.|title=Leibniz, Caramuel, Harriot und das Dualsystem|language=de|journal=Mitteilungen der deutschen Mathematiker-Vereinigung|volume=16|year=2008|issue=1|pages=12–15|doi=10.1515/dmvm-2008-0009|s2cid=179000299|url=http://page.math.tu-berlin.de/~mdmv/archive/16/mdmv-16-1-12-ineichen.pdf}}</ref>

===Leibniz===
[[File:Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg|thumb|upright|Gottfried Leibniz]]
Leibniz wrote in excess of a hundred manuscripts on binary, most of them remaining unpublished.<ref name=":0">{{Citation |last=Strickland |first=Lloyd |title=Leibniz on Number Systems |date=2020 |work=Handbook of the History and Philosophy of Mathematical Practice |pages=1–31 |editor-last=Sriraman |editor-first=Bharath |url=https://link.springer.com/referenceworkentry/10.1007/978-3-030-19071-2_90-1 |access-date=2024-08-20 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-19071-2_90-1 |isbn=978-3-030-19071-2}}</ref> Before his first dedicated work in 1679, numerous manuscripts feature early attempts to explore binary concepts, including tables of numbers and basic calculations, often scribbled in the margins of works unrelated to mathematics.<ref name=":0" />

His first known work on binary, ''“On the Binary Progression"'', in 1679, Leibniz introduced conversion between decimal and binary, along with algorithms for performing basic arithmetic operations such as addition, subtraction, multiplication, and division using binary numbers. He also developed a form of binary algebra to calculate the square of a six-digit number and to extract square roots.<ref name=":0" />

His most well known work appears in his article ''Explication de l'Arithmétique Binaire'' (published in 1703).
The full title of Leibniz's article is translated into English as the ''"Explanation of Binary Arithmetic, which uses only the characters 1 and 0, with some remarks on its usefulness, and on the light it throws on the ancient Chinese figures of [[Fu Xi]]"''.<ref name="lnz">Leibniz G., Explication de l'Arithmétique Binaire, Die Mathematische Schriften, ed. C. Gerhardt, Berlin 1879, vol.7, p.223; Engl. transl.[https://www.leibniz-translations.com/binary]</ref> Leibniz's system uses 0 and 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows:<ref name="lnz" />
: 0 0 0 1 &nbsp; numerical value 2<sup>0</sup>
: 0 0 1 0 &nbsp; numerical value 2<sup>1</sup>
: 0 1 0 0 &nbsp; numerical value 2<sup>2</sup>
: 1 0 0 0 &nbsp; numerical value 2<sup>3</sup>

While corresponding with the Jesuit priest [[Joachim Bouvet]] in 1700, who had made himself an expert on the ''I Ching'' while a missionary in China, Leibniz explained his binary notation, and Bouvet demonstrated in his 1701 letters that the ''I Ching'' was an independent, parallel invention of binary notation.
Leibniz & Bouvet concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical [[mathematics]] he admired.<ref>[https://gwern.net/doc/cs/1980-swiderski.pdf#page=8 "Bouvet and Leibniz: A Scholarly Correspondence"], Swiderski 1980</ref> Of this parallel invention, Leibniz wrote in his "Explanation Of Binary Arithmetic" that "this restitution of their meaning, after such a great interval of time, will seem all the more curious."<ref>[https://www.leibniz-translations.com/binary.htm Leibniz]: "The Chinese lost the meaning of the Cova or Lineations of Fuxi, perhaps more than a thousand years ago, and they have written commentaries on the subject in which they have sought I know not what far out meanings, so that their true explanation now has to come from Europeans. Here is how: It was scarcely more than two years ago that I sent to Reverend Father Bouvet,<sup>3</sup> the celebrated French Jesuit who lives in Peking, my method of counting by 0 and 1, and nothing more was required to make him recognize that this was the key to the figures of Fuxi. Writing to me on 14 November 1701, he sent me this philosophical prince's grand figure, which goes up to 64, and leaves no further room to doubt the truth of our interpretation, such that it can be said that this Father has deciphered the enigma of Fuxi, with the help of what I had communicated to him. And as these figures are perhaps the most ancient monument of [GM VII, p227] science which exists in the world, this restitution of their meaning, after such a great interval of time, will seem all the more curious."</ref>

The relation was a central idea to his universal concept of a language or [[characteristica universalis]], a popular idea that would be followed closely by his successors such as [[Gottlob Frege]] and [[George Boole]] in forming [[Propositional Calculus|modern symbolic logic]].<ref>{{Cite book
|last=Aiton
|first=Eric J.
|title=Leibniz: A Biography
|year=1985
|publisher=Taylor & Francis
|isbn=0-85274-470-6
|pages=245–8
}}</ref>
Leibniz was first introduced to the ''[[I Ching]]'' through his contact with the French Jesuit [[Joachim Bouvet]], who visited China in 1685 as a missionary. Leibniz saw the ''I Ching'' hexagrams as an affirmation of the [[Universality (philosophy)|universality]] of his own religious beliefs as a Christian.<ref name="smith">{{cite book|author1=J.E.H. Smith|title=Leibniz: What Kind of Rationalist?: What Kind of Rationalist?|url=https://books.google.com/books?id=Da_oP3sJs1oC&pg=PA4153|year=2008|publisher=Springer|isbn=978-1-4020-8668-7|page=415}}</ref> Binary numerals were central to Leibniz's theology. He believed that binary numbers were symbolic of the Christian idea of ''[[creatio ex nihilo]]'' or creation out of nothing.<ref name="lniz">{{cite book|author1=Yuen-Ting Lai|title=Leibniz, Mysticism and Religion|url=https://books.google.com/books?id=U9dOmVt81UAC&pg=PA149|year=1998|publisher=Springer|isbn=978-0-7923-5223-5|pages=149–150}}</ref>

{{quote|[A concept that] is not easy to impart to the pagans, is the creation ''ex nihilo'' through God's almighty power. Now one can say that nothing in the world can better present and demonstrate this power than the origin of numbers, as it is presented here through the simple and unadorned presentation of One and Zero or Nothing.|Leibniz's letter to the [[Rudolph Augustus, Duke of Brunswick-Lüneburg|Duke of Brunswick]] attached with the ''I Ching'' hexagrams<ref name="smith"/>}}

===Later developments===
[[File:George Boole color.jpg|thumb|left|160px|George Boole]]
In 1854, British mathematician [[George Boole]] published a landmark paper detailing an [[algebra]]ic system of [[logic]] that would become known as [[Boolean algebra (logic)|Boolean algebra]]. His logical calculus was to become instrumental in the design of digital electronic circuitry.<ref>{{cite book |last=Boole |first=George |orig-year=1854 |url=https://www.gutenberg.org/ebooks/15114 |title=An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities |publisher=Cambridge University Press |edition=Macmillan, Dover Publications, reprinted with corrections [1958] |location=New York |year=2009 |isbn=978-1-108-00153-3}}</ref>

In 1937, [[Claude Shannon]] produced his master's thesis at [[MIT]] that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled ''[[A Symbolic Analysis of Relay and Switching Circuits]]'', Shannon's thesis essentially founded practical [[digital circuit]] design.<ref>{{cite thesis |title=A symbolic analysis of relay and switching circuits |last=Shannon |first=Claude Elwood |publisher=Massachusetts Institute of Technology |location=Cambridge |year=1940 |hdl=1721.1/11173 |type=Thesis }}</ref>

In November 1937, [[George Stibitz]], then working at [[Bell Labs]], completed a relay-based computer he dubbed the "Model K" (for "'''K'''itchen", where he had assembled it), which calculated using binary addition.<ref>{{cite web |url=http://www.invent.org/hall_of_fame/140.html |title=National Inventors Hall of Fame – George R. Stibitz |date=20 August 2008 |access-date=5 July 2010 |url-status=dead |archive-url=https://web.archive.org/web/20100709213530/http://www.invent.org/hall_of_fame/140.html |archive-date=9 July 2010}}</ref> Bell Labs authorized a full research program in late 1938 with Stibitz at the helm. Their Complex Number Computer, completed 8 January 1940, was able to calculate [[complex numbers]]. In a demonstration to the [[American Mathematical Society]] conference at [[Dartmouth College]] on 11 September 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a [[Teleprinter|teletype]]. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were [[John von Neumann]], [[John Mauchly]] and [[Norbert Wiener]], who wrote about it in his memoirs.<ref>{{cite web|url=http://stibitz.denison.edu/bio.html |title=George Stibitz : Bio |publisher=Math & Computer Science Department, Denison University |date=30 April 2004 |access-date=5 July 2010 }}</ref><ref>{{cite web|url=http://www.kerryr.net/pioneers/stibitz.htm |title=Pioneers – The people and ideas that made a difference – George Stibitz (1904–1995) |publisher=Kerry Redshaw |date=20 February 2006 |access-date=5 July 2010 }}</ref><ref>{{cite web|url=http://ei.cs.vt.edu/~history/Stibitz.html |title=George Robert Stibitz – Obituary |publisher=Computer History Association of California |date=6 February 1995 |access-date=5 July 2010}}</ref>

The [[Z1 (computer)|Z1 computer]], which was designed and built by [[Konrad Zuse]] between 1935 and 1938, used [[Boolean logic]] and binary [[Floating-point arithmetic|floating-point numbers]].<ref name="zuse">{{cite journal |title=Konrad Zuse's Legacy: The Architecture of the Z1 and Z3 |author-last=Rojas |author-first=Raúl |author-link=Raúl Rojas |journal=[[IEEE Annals of the History of Computing]] |volume=19 |number=2 |date=April–June 1997 |pages=5–16 |doi=10.1109/85.586067 |url=http://ed-thelen.org/comp-hist/Zuse_Z1_and_Z3.pdf |access-date=2022-07-03 |url-status=live |archive-url=https://web.archive.org/web/20220703082408/http://ed-thelen.org/comp-hist/Zuse_Z1_and_Z3.pdf |archive-date=2022-07-03}} (12 pages)</ref>