Editing Binary number

{{Short description|Number expressed in the base-2 numeral system}}
{{Use dmy dates|date=February 2021}}
{{Table Numeral Systems}}
A '''binary number''' is a [[number]] expressed in the '''[[Radix|base]]-2 [[numeral system]]''' or '''binary numeral system''', a method for representing [[number]]s that uses only two symbols for the [[natural number]]s: typically "0" ([[zero]]) and "1" ([[one]]). A ''binary number''  may also refer to a [[rational number]] that has a finite representation in the binary numeral system, that is, the quotient of an [[integer]] by a power of two.

The base-2 numeral system is a [[positional notation]] with a [[radix]] of [[2]]. Each digit is referred to as a [[bit]], or binary digit. Because of its straightforward implementation in [[digital electronic circuit]]ry using [[logic gate]]s, the binary system is used by almost all modern [[computer|computers and computer-based devices]], as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation.<ref>{{cite web |title=3.3. Binary and Its Advantages — CS160 Reader |url=https://computerscience.chemeketa.edu/cs160Reader/Binary/Binary.html |website=computerscience.chemeketa.edu |access-date=22 May 2024}}</ref>

{{-}}
{{Aligned table |class=wikitable |cols=2
|style=float:right; |rowstyle=text-align:right;
|row1header=y
| Decimal<br>number | Binary<br>number
|  0 |    0
|  1 |    1
|  2 |   10
|  3 |   11
|  4 |  100
|  5 |  101
|  6 |  110
|  7 |  111
|  8 | 1000
|  9 | 1001
| 10 | 1010
| 11 | 1011
| 12 | 1100
| 13 | 1101
| 14 | 1110
| 15 | 1111
}}

==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>

==Representation==
Any number can be represented by a sequence of [[bit]]s (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. Any of the following rows of symbols can be interpreted as the binary numeric value of 667:

{| style="text-align:center;"
| 1 || 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1 || 1
|-
| <nowiki>|</nowiki> || ― || <nowiki>|</nowiki> || ― || ― || <nowiki>|</nowiki> || <nowiki>|</nowiki> || ― || <nowiki>|</nowiki> || <nowiki>|</nowiki>
|-
| ☒ || ☐ || ☒ || ☐ || ☐ || ☒ || ☒ || ☐ || ☒ || ☒
|-
| y || n || y || n || n || y || y || n || y || y
|-
| T || F || T || F || F || T || T || F || T || T
|}

[[Image:Binary clock.svg|250px|thumbnail|right|A [[binary clock]] might use [[Light-emitting diode|LEDs]] to express binary values. In this clock, each column of LEDs shows a [[binary-coded decimal]] numeral of the traditional [[sexagesimal]] time.]]

The numeric value represented in each case depends on the value assigned to each symbol. In the earlier days of computing, switches, punched holes, and punched paper tapes were used to represent binary values.<ref>{{Cite web|url=https://www.bbc.com/bitesize/guides/zwsbwmn/revision/1|title=Introducing binary – Revision 1 – GCSE Computer Science|website=BBC|access-date=2019-06-26}}</ref> In a modern computer, the numeric values may be represented by two different [[voltage]]s; on a [[Magnetic field|magnetic]] [[Disk storage|disk]], [[Magnetic polarity|magnetic polarities]] may be used. A "positive", "[[yes and no|yes]]", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use.

In keeping with the customary representation of numerals using [[Arabic numerals]], binary numbers are commonly written using the symbols '''0''' and '''1'''. When written, binary numerals are often subscripted, prefixed, or suffixed to indicate their base, or [[radix]]. The following notations are equivalent:
* 100101 binary (explicit statement of format)
* 100101b (a suffix indicating binary format; also known as [[Intel convention]]<ref name="Kueveler-Schwoch_1996">{{cite book|title=Arbeitsbuch Informatik – eine praxisorientierte Einführung in die Datenverarbeitung mit Projektaufgabe|language=de|first1=Gerd|last1=Küveler|first2=Dietrich|last2=Schwoch|date=2013|orig-year=1996|publisher=Vieweg-Verlag, reprint: Springer-Verlag|isbn=978-3-528-04952-2|id=9783322929075|doi=10.1007/978-3-322-92907-5|url=https://books.google.com/books?id=b8-dBgAAQBAJ}}</ref><ref name="Kueveler-Schwoch_2007">{{cite book|title=Informatik für Ingenieure und Naturwissenschaftler: PC- und Mikrocomputertechnik, Rechnernetze|language=de|first1=Gerd|last1=Küveler|first2=Dietrich|last2=Schwoch|date=2007-10-04|publisher=Vieweg, reprint: Springer-Verlag|edition=5|volume=2|isbn=978-3834891914|id=9783834891914|url=https://books.google.com/books?id=xQbvPYxceY0C}}</ref>)
* 100101B (a suffix indicating binary format)
* bin 100101 (a prefix indicating binary format)
* 100101<sub>2</sub> (a subscript indicating base-2 (binary) notation)
* %100101 (a prefix indicating binary format; also known as [[Motorola convention]]<ref name="Kueveler-Schwoch_1996"/><ref name="Kueveler-Schwoch_2007"/>)
* 0b100101 (a prefix indicating binary format, common in programming languages)
* 6b100101 (a prefix indicating number of bits in binary format, common in programming languages)
* #b100101 (a prefix indicating binary format, common in Lisp programming languages)

When spoken, binary numerals are usually read digit-by-digit, to distinguish them from decimal numerals. For example, the binary numeral 100 is pronounced ''one zero zero'', rather than ''one hundred'', to make its binary nature explicit and for purposes of correctness. Since the binary numeral 100 represents the value four, it would be confusing to refer to the numeral as ''one hundred'' (a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct ''value''), but this does not make its binary nature explicit.

==Counting in binary==
Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Before examining binary counting, it is useful to briefly discuss the more familiar [[decimal]] counting system as a frame of reference.

===Decimal counting===
[[Decimal]] counting uses the ten symbols ''0'' through ''9''. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called the ''first digit''. When the available symbols for this position are exhausted, the least significant digit is reset to ''0'', and the next digit of higher significance (one position to the left) is incremented (''overflow''), and incremental substitution of the low-order digit resumes. This method of reset and overflow is repeated for each digit of significance. Counting progresses as follows:

:000, 001, 002, ... 007, 008, 009, (rightmost digit is reset to zero, and the digit to its left is incremented)
:0'''1'''0, 011, 012, ...
:&nbsp;&nbsp;&nbsp;...
:090, 091, 092, ... 097, 098, 099, (rightmost two digits are reset to zeroes, and next digit is incremented)
:'''1'''00, 101, 102, ...

===Binary counting===
[[File:Binary counter.gif|thumb|This counter shows how to count in binary from numbers zero through thirty-one.]]
[[File:Binary_guess_number_trick_SMIL.svg|thumb|upright|link={{filepath:binary_guess_number_trick_SMIL.svg}}|A party trick to guess a number from which cards it is printed on uses the bits of the binary representation of the number. In the SVG file, click a card to toggle it]]
Binary counting follows the exact same procedure, and again the incremental substitution begins with the least significant binary digit, or ''bit'' (the rightmost one, also called the ''first bit''), except that only the two symbols ''0'' and ''1'' are available. Thus, after a bit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next bit to the left:

:0000,
:000'''1''', (rightmost bit starts over, and the next bit is incremented)
:00'''1'''0, 0011, (rightmost two bits start over, and the next bit is incremented)
:0'''1'''00, 0101, 0110, 0111, (rightmost three bits start over, and the next bit is incremented)
:'''1'''000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ...

In the binary system, each bit represents an increasing power of 2, with the rightmost bit representing 2<sup>0</sup>, the next representing 2<sup>1</sup>, then 2<sup>2</sup>, and so on. The value of a binary number is the sum of the powers of 2 represented by each "1" bit. For example, the binary number 100101 is converted to decimal form as follows:

:100101<sub>2</sub> = [ ( '''1''' ) × 2<sup>5</sup> ] + [ ( '''0''' ) × 2<sup>4</sup> ] + [ ( '''0''' ) × 2<sup>3</sup> ] + [ ( '''1''' ) × 2<sup>2</sup> ] + [ ( '''0''' ) × 2<sup>1</sup> ] + [ ( '''1''' ) × 2<sup>0</sup> ]

:100101<sub>2</sub> = [ '''1''' × 32 ] + [ '''0''' × 16 ] + [ '''0''' × 8 ] + [ '''1''' × 4 ] + [ '''0''' × 2 ] + [ '''1''' × 1 ]

:'''100101<sub>2</sub> = 37<sub>10</sub>'''

==Fractions==

[[Fractions]] in binary arithmetic [[Repeating decimal|terminate]] only if the [[denominator]] is a [[power of 2]]. As a result, 1/10 does not have a finite binary representation ('''10''' has prime factors '''2''' and '''5'''). This causes 10 × 1/10 not to precisely equal 1 in binary [[floating-point arithmetic]]. As an example, to interpret the binary expression for 1/3 = .010101..., this means: 1/3 = 0 × '''2<sup>−1</sup>''' + 1 × '''2<sup>−2</sup>''' + 0 × '''2<sup>−3</sup>''' + 1 × '''2<sup>−4</sup>''' + ... = 0.3125 + ... An exact value cannot be found with a sum of a finite number of inverse powers of two, the zeros and ones in the binary representation of 1/3 alternate forever.

{| class="wikitable"
|-
! Fraction
! [[Base 10|Decimal]]
! Binary
! Fractional approximation
|-
| 1/1
| 1{{pad|0.25em}}or{{pad|0.25em}}0.999...
| 1{{pad|0.25em}}or{{pad|0.25em}}0.{{Overline|1}}
| 1/2 + 1/4 + 1/8...
|-
| 1/2
| 0.5{{pad|0.25em}}or{{pad|0.25em}}0.4999...
| 0.1{{pad|0.25em}}or{{pad|0.25em}}0.0{{Overline|1}}
| 1/4 + 1/8 + 1/16 . . .
|-
| 1/3
| 0.333...
| 0.{{Overline|01}}
| 1/4 + 1/16 + 1/64 . . .
|-
| 1/4
| 0.25{{pad|0.25em}}or{{pad|0.25em}}0.24999...
| 0.01{{pad|0.25em}}or{{pad|0.25em}}0.00{{Overline|1}}
| 1/8 + 1/16 + 1/32 . . .
|-
| 1/5
| 0.2{{pad|0.25em}}or{{pad|0.25em}}0.1999...
| 0.{{Overline|0011}}
| 1/8 + 1/16 + 1/128 . . .
|-
| 1/6
| 0.1666...
| 0.0{{Overline|01}}
| 1/8 + 1/32 + 1/128 . . .
|-
| 1/7
| 0.142857142857...
| 0.{{Overline|001}}
| 1/8 + 1/64 + 1/512 . . .
|-
| 1/8
| 0.125{{pad|0.25em}}or{{pad|0.25em}}0.124999...
| 0.001{{pad|0.25em}}or{{pad|0.25em}}0.000{{Overline|1}}
| 1/16 + 1/32 + 1/64 . . .
|-
| 1/9
| 0.111...
| 0.{{Overline|000111}}
| 1/16 + 1/32 + 1/64 . . .
|-
| 1/10
| 0.1{{pad|0.25em}}or{{pad|0.25em}}0.0999...
| 0.0{{Overline|0011}}
| 1/16 + 1/32 + 1/256 . . .
|-
| 1/11
| 0.090909...
| 0.{{Overline|0001011101}}
| 1/16 + 1/64 + 1/128 . . .
|-
| 1/12
| 0.08333...
| 0.00{{Overline|01}}
| 1/16 + 1/64 + 1/256 . . .
|-
| 1/13
| 0.076923076923...
| 0.{{Overline|000100111011}}
| 1/16 + 1/128 + 1/256 . . .
|-
| 1/14
| 0.0714285714285...
| 0.0{{Overline|001}}
| 1/16 + 1/128 + 1/1024 . . .
|-
| 1/15
| 0.0666...
| 0.{{Overline|0001}}
| 1/16 + 1/256 . . .
|-
| 1/16
| 0.0625{{pad|0.25em}}or{{pad|0.25em}}0.0624999...
| 0.0001{{pad|0.25em}}or{{pad|0.25em}}0.0000{{Overline|1}}
| 1/32 + 1/64 + 1/128 . . .
|}

==Binary arithmetic==
[[Arithmetic]] in binary is much like arithmetic in other [[positional notation]] [[numeral system]]s. Addition, subtraction, multiplication, and division can be performed on binary numerals.

===Addition===
{{main | Adder (electronics)}}
[[Image:Half Adder.svg|thumbnail|200px|right|The [[circuit diagram]] for a binary [[Adder (electronics)|half adder]], which adds two bits together, producing sum and carry bits]]

The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying:

:0 + 0 → 0
:0 + 1 → 1
:1 + 0 → 1
:1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 2<sup>1</sup>)  )
Adding two "1" digits produces a digit "0", while 1 will have to be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:

:5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 10<sup>1</sup>)  )
:7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 10<sup>1</sup>)  )

This is known as ''carrying''. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:

   {{brown|1 1 1 1 1    (carried digits)}}
     0 1 1 0 1
 +   1 0 1 1 1
 -------------
 = 1 0 0 1 0 0 = 36

In this example, two numerals are being added together: 01101<sub>2</sub> (13<sub>10</sub>) and 10111<sub>2</sub> (23<sub>10</sub>). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 10<sub>2</sub>. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 10<sub>2</sub> again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 11<sub>2</sub>. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 100100<sub>2</sub> (36<sub>10</sub>).

When computers must add two numbers, the rule that:
x [[Exclusive or|xor]] y = (x + y) [[Modulo operation|mod]] 2
for any two bits x and y allows for very fast calculation, as well.

====Long carry method====
A simplification for many binary addition problems is the "long carry method" or "Brookhouse Method of Binary Addition". This method is particularly useful when one of the numbers contains a long stretch of ones. It is based on the simple premise that under the binary system, when given a stretch of digits composed entirely of {{varserif|n}} ones (where {{varserif|n}} is any integer length), adding 1 will result in the number 1 followed by a string of {{varserif|n}} zeros. That concept follows, logically, just as in the decimal system, where adding 1 to a string of {{varserif|n}} 9s will result in the number 1 followed by a string of {{varserif|n}} 0s:

      Binary                        Decimal
     1 1 1 1 1     likewise        9 9 9 9 9
  +          1                  +          1
   ———————————                   ———————————
   1 0 0 0 0 0                   1 0 0 0 0 0

Such long strings are quite common in the binary system.  From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations. In the following example, two numerals are being added together: 1 1 1 0 1 1 1 1 1 0<sub>2</sub> (958<sub>10</sub>) and 1 0 1 0 1 1 0 0 1 1<sub>2</sub> (691<sub>10</sub>), using the traditional carry method on the left, and the long carry method on the right:

 Traditional Carry Method                       Long Carry Method
                                 vs.
   {{brown|1 1 1   1 1 1 1 1      (carried digits)   1 ←     1 ←}}            carry the 1 until it is one digit past the "string" below
     1 1 1 0 1 1 1 1 1 0                       <s>1 1 1</s> 0 <s>1 1 1 1 1</s> 0  cross out the "string",
 +   1 0 1 0 1 1 0 0 1 1                   +   1 0 <s>1</s> 0 1 1 0 0 <s>1</s> 1  and cross out the digit that was added to it
 ———————————————————————                    ——————————————————————
 = 1 1 0 0 1 1 1 0 0 0 1                     1 1 0 0 1 1 1 0 0 0 1

The top row shows the carry bits used. Instead of the standard carry from one column to the next, the lowest-ordered "1" with a "1" in the corresponding place value beneath it may be added and a "1" may be carried to one digit past the end of the series.  The "used" numbers must be crossed off, since they are already added. Other long strings may likewise be cancelled using the same technique. Then, simply add together any remaining digits normally. Proceeding in this manner gives the final answer of 1 1 0 0 1 1 1 0 0 0 1<sub>2</sub> (1649<sub>10</sub>).  In our simple example using small numbers, the traditional carry method required eight carry operations, yet the long carry method required only two, representing a substantial reduction of effort.

====Addition table====

{| class="wikitable" style="text-align:center"
|-
! style="width:1.5em" |
! style="width:1.5em" | 0
! style="width:1.5em" | 1
|-
! 0
| 0
| 1
|-
! 1
| 1
| 10
|}

The binary addition table is similar to, but not the same as, the [[Logical disjunction#Truth table|truth table]] of the [[logical disjunction]] operation <math>\lor</math>. The difference is that <math>1 \lor 1 = 1</math>, while <math>1+1=10</math>.

=== Subtraction ===
{{further|signed number representations|two's complement}}

[[Subtraction]] works in much the same way:

:0 − 0 → 0
:0 − 1 → 1, borrow 1
:1 − 0 → 1
:1 − 1 → 0
Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column. This is known as ''borrowing''. The principle is the same as for carrying. When the result of a subtraction is less than 0, the least possible value of a digit, the procedure is to "borrow" the deficit divided by the radix (that is, 10/10) from the left, subtracting it from the next positional value.

     *   * * *   (starred columns are borrowed from)
   1 1 0 1 1 1 0
 −     1 0 1 1 1
 ----------------
 = 1 0 1 0 1 1 1

   *             (starred columns are borrowed from)
   1 0 1 1 1 1 1
 –   1 0 1 0 1 1
 ----------------
 = 0 1 1 0 1 0 0

Subtracting a positive number is equivalent to ''adding'' a [[negative number]] of equal [[absolute value]]. Computers use [[signed number representations]] to handle negative numbers—most commonly the [[two's complement]] notation. Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation, subtraction can be summarized by the following formula:

: {{math|1=A − B = A + not B + 1}}

===Multiplication===<!-- This section is linked from [[Binary-coded decimal]] -->
[[Multiplication]] in binary is similar to its decimal counterpart. Two numbers {{varserif|A}} and {{varserif|B}} can be multiplied by partial products: for each digit in {{varserif|B}}, the product of that digit in {{varserif|A}} is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in {{varserif|B}} that was used. The sum of all these partial products gives the final result.

Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:
* If the digit in {{varserif|B}} is 0, the partial product is also 0
* If the digit in {{varserif|B}} is 1, the partial product is equal to {{varserif|A}}

For example, the binary numbers 1011 and 1010 are multiplied as follows:

            1 0 1 1   ({{varserif|A}})
          × 1 0 1 0   ({{varserif|B}})
          ---------
            0 0 0 0   ← to the rightmost 'zero' in {{varserif|B}}
    +     1 0 1 1     ← to the next 'one' in {{varserif|B}}
    +   0 0 0 0
    + 1 0 1 1
    ---------------
    = 1 1 0 1 1 1 0

Binary numbers can also be multiplied with bits after a [[binary point]]:

                1 0 1 . 1 0 1     {{varserif|A}} (5.625 in decimal)
              × 1 1 0 . 0 1       {{varserif|B}} (6.25 in decimal)
              -------------------
                    1 . 0 1 1 0 1   ← to a 'one' in {{varserif|B}}
      +           0 0 . 0 0 0 0     ← to a 'zero' in {{varserif|B}}
      +         0 0 0 . 0 0 0
      +       1 0 1 1 . 0 1
      +     1 0 1 1 0 . 1
      ---------------------------
      =   1 0 0 0 1 1 . 0 0 1 0 1 (35.15625 in decimal)

See also [[Booth's multiplication algorithm]].

====Multiplication table====
{| class="wikitable" style="text-align:center"
|-
! style="width:1.5em" |
! style="width:1.5em" | 0
! style="width:1.5em" | 1
|-
! 0
| 0
| 0
|-
! 1
| 0
| 1
|}

The binary multiplication table is the same as the [[Logical conjunction#Truth table|truth table]] of the [[logical conjunction]] operation <math>\land</math>.

===Division===
{{See also|Division algorithm#Integer division (unsigned) with remainder}}

[[Long division]] in binary is again similar to its decimal counterpart.

In the example below, the [[divisor]] is 101<sub>2</sub>, or 5 in decimal, while the [[Division (mathematics)|dividend]] is 11011<sub>2</sub>, or 27 in decimal. The procedure is the same as that of decimal [[long division]]; here, the divisor 101<sub>2</sub> goes into the first three digits 110<sub>2</sub> of the dividend one time, so a "1" is written on the top line. This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit (a "1") is included to obtain a new three-digit sequence:

               1
         ___________
 1 0 1   ) 1 1 0 1 1
         − 1 0 1
           -----
           0 0 1

The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted:

              1 0 1
        ___________
 1 0 1  ) 1 1 0 1 1
        − 1 0 1
          -----
              1 1 1
          −   1 0 1
              -----
              0 1 0

Thus, the [[quotient]] of 11011<sub>2</sub> divided by 101<sub>2</sub> is 101<sub>2</sub>, as shown on the top line, while the remainder, shown on the bottom line, is 10<sub>2</sub>. In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2.

Aside from long division, one can also devise the procedure so as to allow for over-subtracting from the partial remainder at each iteration, thereby leading to alternative methods which are less systematic, but more flexible as a result.

===Square root===
The process of [[Methods of computing square roots#Binary numeral system (base 2)|taking a binary square root]] digit by digit is essentially the same as for a decimal square root but much simpler, due to the binary nature.  First group the digits in pairs, using a leading 0 if necessary so there are an even number of digits.  Now at each step, consider the answer so far, extended with the digits 01.  If this can be subtracted from the current remainder, do so.  Then extend the remainder with the next pair of digits.  If you subtracted, the next digit of the answer is 1, otherwise it's 0.

                              1                          1  1                       1  1  0                   1  1  0  1
  -------------             -------------              -------------              -------------             -------------
 √ 10 10 10 01             √ 10 10 10 01              √ 10 10 10 01              √ 10 10 10 01             √ 10 10 10 01
                            - 1                        - 1                        - 1                       - 1         
 Answer so far is 0,        ----                       ----                       ----                      ----        
 extended by 01 is 001,       1 10                       1 10                       1 10                      1 10
 this CAN be subtracted                                - 1 01                     - 1 01                    - 1 01
 from first pair 10,       Answer so far is 1,         -------                    -------                   -------
 so first digit of         extended by 01 is 101,           1 10                       1 10 01                   1 10 01
 answer is 1.              this CAN be subtracted                                                              - 1 10 01
                           from remainder 110, so     Answer so far is 11,       Answer so far is 110,         ----------
                           next answer digit is 1.    extended by 01 is 1101,    extended by 01 is 11001,              0
                                                      this is TOO BIG to         this CAN be subtracted
                                                      subtract from remainder    from remainder 11001, so           Done!
                                                      110, so next digit of      next digit of answer is 1.
                                                      answer is 0.

==Bitwise operations==
{{Main|Bitwise operation}}
Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using [[logical connective|Boolean logical operators]]. When a string of binary symbols is manipulated in this way, it is called a [[bitwise operation]]; the logical operators [[Logical conjunction|AND]], [[Logical disjunction|OR]], and [[Exclusive disjunction|XOR]] may be performed on corresponding bits in two binary numerals provided as input. The logical [[Negation|NOT]] operation may be performed on individual bits in a single binary numeral provided as input. Sometimes, such operations may be used as arithmetic short-cuts, and may have other computational benefits as well.  For example, an [[arithmetic shift]] left of a binary number is the equivalent of multiplication by a (positive, integral) power of 2.

==Conversion to and from other numeral systems==

===Decimal to binary===
[[File:Decimal to Binary Conversion.gif|alt=|frame|Conversion of (357)<sub>10</sub> to binary notation results in (101100101)]]
To convert from a base-10 [[Integer (computer science)|integer]] to its base-2 (binary) equivalent, the number is [[division by two|divided by two]]. The remainder is the [[least-significant bit]]. The quotient is again divided by two; its remainder becomes the next least significant bit. This process repeats until a quotient of one is reached. The sequence of remainders (including the final quotient of one) forms the binary value, as each remainder must be either zero or one when dividing by two. For example, (357)<sub>10</sub> is expressed as (101100101)<sub>2.</sub><ref>{{Cite web|url=https://www.chalkstreet.com/aptipedia/knowledgebase/base-system/|title=Base System|access-date=31 August 2016|archive-date=23 October 2017|archive-url=https://web.archive.org/web/20171023010521/https://www.chalkstreet.com/aptipedia/knowledgebase/base-system/|url-status=usurped}}</ref>

=== Binary to decimal ===
Conversion from base-2 to base-10 simply inverts the preceding algorithm. The bits of the binary number are used one by one, starting with the most significant (leftmost) bit. Beginning with the value 0, the prior value is doubled, and the next bit is then added to produce the next value. This can be organized in a multi-column table. For example, to convert 10010101101<sub>2</sub> to decimal:

{| style= "border: 1px solid #a2a9b1; border-spacing: 3px; background-color: #f8f9fa; color: black; margin: 0.5em 0 0.5em 1em; padding: 0.2em; line-height: 1.5em; width:22em"
!Prior value
! style="text-align:left" | × 2 +
!Next bit
!= Next value
|-
|align="right"|0 ||× 2 +|| '''1''' || = 1
|-
|align="right"|1 ||× 2 +|| '''0''' || = 2
|-
|align="right"|2 ||× 2 +|| '''0''' || = 4
|-
|align="right"|4 ||× 2 +|| '''1''' || = 9
|-
|align="right"|9 ||× 2 +|| '''0''' || = 18
|-
|align="right"|18 ||× 2 +|| '''1''' || = 37
|-
|align="right"|37 ||× 2 +|| '''0''' || = 74
|-
|align="right"|74 ||× 2 +|| '''1''' || = 149
|-
|align="right"|149 ||× 2 +|| '''1''' || = 299
|-
|align="right"|299 ||× 2 +|| '''0''' || = 598
|-
|align="right"|598 ||× 2 +|| '''1''' || = '''1197'''
|}

The result is 1197<sub>10</sub>. The first Prior Value of 0 is simply an initial decimal value. This method is an application of the [[Horner scheme]].

{|
! Binary&nbsp;
| 1 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1 || 0 || 1 ||
|-
! Decimal&nbsp;
| 1×2<sup>10</sup> + || 0×2<sup>9</sup> + || 0×2<sup>8</sup> + || 1×2<sup>7</sup> + || 0×2<sup>6</sup> + || 1×2<sup>5</sup> + || 0×2<sup>4</sup> + || 1×2<sup>3</sup> + || 1×2<sup>2</sup> + || 0×2<sup>1</sup> + || 1×2<sup>0</sup> = || 1197
|}

The fractional parts of a number are converted with similar methods. They are again based on the equivalence of shifting with doubling or halving.

In a fractional binary number such as 0.11010110101<sub>2</sub>, the first digit is <math display="inline">\frac{1}{2} </math>, the second <math display="inline"> (\frac{1}{2})^2 = \frac{1}{4} </math>, etc. So if there is a 1 in the first place after the decimal, then the number is at least <math display="inline"> \frac{1}{2} </math>, and vice versa. Double that number is at least 1. This suggests the algorithm: Repeatedly double the number to be converted, record if the result is at least 1, and then throw away the integer part.

For example, <math display="inline"> (\frac{1}{3})_{10} </math>, in binary, is:

{| class="wikitable"
!Converting!!Result
|-
|<math display="inline"> \frac{1}{3} </math> || 0.
|-
|<math display="inline"> \frac{1}{3} \times 2 = \frac{2}{3} < 1 </math> || 0.0
|-
|<math display="inline"> \frac{2}{3} \times 2 = 1\frac{1}{3} \ge 1</math> || 0.01
|-
|<math display="inline"> \frac{1}{3} \times 2 = \frac{2}{3} < 1 </math> || 0.010
|-
|<math display="inline"> \frac{2}{3} \times 2 = 1\frac{1}{3} \ge 1 </math> || 0.0101
|}

Thus the repeating decimal fraction 0.{{overline|3}}... is equivalent to the repeating binary fraction 0.{{overline|01}}... .

Or for example, 0.1<sub>10</sub>, in binary, is:

{| class="wikitable"
! Converting                !! Result
|-
|          '''0.1'''        || 0.
|-
|0.1 × 2 = '''0.2''' < 1    || 0.0
|-
|0.2 × 2 = '''0.4''' < 1    || 0.00
|-
|0.4 × 2 = '''0.8''' < 1    || 0.000
|-
|0.8 × 2 = '''1.6''' ≥ 1 || 0.0001
|-
|0.6 × 2 = '''1.2''' ≥ 1 || 0.00011
|-
|0.2 × 2 = '''0.4''' < 1    || 0.000110
|-
|0.4 × 2 = '''0.8''' < 1    || 0.0001100
|-
|0.8 × 2 = '''1.6''' ≥ 1 || 0.00011001
|-
|0.6 × 2 = '''1.2''' ≥ 1 || 0.000110011
|-
|0.2 × 2 = '''0.4''' < 1    || 0.0001100110
|}

This is also a repeating binary fraction 0.0{{overline|0011}}... .  It may come as a surprise that terminating decimal fractions can have repeating expansions in binary. It is for this reason that many are surprised to discover that 1/10 + ... + 1/10 (addition of 10 numbers) differs from 1 in binary [[floating-point arithmetic]]. In fact, the only binary fractions with terminating expansions are of the form of an integer divided by a power of 2, which 1/10 is not.

The final conversion is from binary to decimal fractions. The only difficulty arises with repeating fractions, but otherwise the method is to shift the fraction to an integer, convert it as above, and then divide by the appropriate power of two in the decimal base. For example:

<math display="block">\begin{align}
x & = & 1100&.1\overline{01110}\ldots \\
x\times 2^6 & = & 1100101110&.\overline{01110}\ldots \\
x\times 2 & = & 11001&.\overline{01110}\ldots \\
x\times(2^6-2) & = & 1100010101 \\
x & = & 1100010101/111110 \\
x & = & (789/62)_{10}
\end{align}</math>

Another way of converting from binary to decimal, often quicker for a person familiar with [[hexadecimal]], is to do so indirectly—first converting (<math>x</math> in binary) into (<math>x</math> in hexadecimal) and then converting (<math>x</math> in hexadecimal) into (<math>x</math> in decimal).

For very large numbers, these simple methods are inefficient because they perform a large number of multiplications or divisions where one operand is very large. A simple divide-and-conquer algorithm is more effective asymptotically: given a binary number, it is divided by 10<sup>''k''</sup>, where ''k'' is chosen so that the quotient roughly equals the remainder; then each of these pieces is converted to decimal and the two are [[Concatenation|concatenated]]. Given a decimal number, it can be split into two pieces of about the same size, each of which is converted to binary, whereupon the first converted piece is multiplied by 10<sup>''k''</sup> and added to the second converted piece, where ''k'' is the number of decimal digits in the second, least-significant piece before conversion.

===Hexadecimal===
{{Main|Hexadecimal}}
{{Hexadecimal table}}
Binary may be converted to and from hexadecimal more easily. This is because the [[radix]] of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 2<sup>4</sup>, so it takes four digits of binary to represent one digit of hexadecimal, as shown in the adjacent table.

To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits:

:3A<sub>16</sub> = 0011 1010<sub>2</sub>
:E7<sub>16</sub> = 1110 0111<sub>2</sub>

To convert a binary number into its hexadecimal equivalent, divide it into groups of four bits. If the number of bits isn't a multiple of four, simply insert extra '''0''' bits at the left (called [[Padding (cryptography)#Bit padding|padding]]). For example:

:1010010<sub>2</sub> = 0101 0010 grouped with padding = 52<sub>16</sub>
:11011101<sub>2</sub> = 1101 1101 grouped = DD<sub>16</sub>

To convert a hexadecimal number into its decimal equivalent, multiply the decimal equivalent of each hexadecimal digit by the corresponding power of 16 and add the resulting values:

:C0E7<sub>16</sub> = (12 × 16<sup>3</sup>) + (0 × 16<sup>2</sup>) + (14 × 16<sup>1</sup>) + (7 × 16<sup>0</sup>) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,383<sub>10</sub>

===Octal===
{{Main|Octal}}
Binary is also easily converted to the [[octal]] numeral system, since octal uses a radix of 8, which is a [[power of two]] (namely, 2<sup>3</sup>, so it takes exactly three binary digits to represent an octal digit). The correspondence between octal and binary numerals is the same as for the first eight digits of [[hexadecimal]] in the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is equivalent to octal 7, and so forth.

{| class="wikitable"  style="text-align:center"
!Octal!!Binary
|-
| 0 || 000
|-
| 1 || 001
|-
| 2 || 010
|-
| 3 || 011
|-
| 4 || 100
|-
| 5 || 101
|-
| 6 || 110
|-
| 7 || 111
|}

Converting from octal to binary proceeds in the same fashion as it does for [[hexadecimal]]:

:65<sub>8</sub> = 110 101<sub>2</sub>
:17<sub>8</sub> = 001 111<sub>2</sub>

And from binary to octal:

:101100<sub>2</sub> = 101 100<sub>2</sub> grouped = 54<sub>8</sub>
:10011<sub>2</sub> = 010 011<sub>2</sub> grouped with padding = 23<sub>8</sub>

And from octal to decimal:

:65<sub>8</sub> = (6 × 8<sup>1</sup>) + (5 × 8<sup>0</sup>) = (6 × 8) + (5 × 1) = 53<sub>10</sub>
:127<sub>8</sub> = (1 × 8<sup>2</sup>) + (2 × 8<sup>1</sup>) + (7 × 8<sup>0</sup>) = (1 × 64) + (2 × 8) + (7 × 1) = 87<sub>10</sub>

==Representing real numbers==
<!-- This section is linked from [[Chaitin's constant]] -->
Non-integers can be represented by using negative powers, which are set off from the other digits by means of a [[radix point]] (called a [[decimal point]] in the decimal system). For example, the binary number 11.01<sub>2</sub> means:

{|
|'''1''' × 2<sup>1</sup>  || (1 × 2 = '''2''')           || plus
|-
|'''1''' × 2<sup>0</sup>  || (1 × 1 = '''1''')           || plus
|-
|'''0''' × 2<sup>−1</sup> || (0 × {{frac|2}} = '''0''')  || plus
|-
|'''1''' × 2<sup>−2</sup> || (1 × {{frac|4}} = '''0.25''')
|}

For a total of 3.25 decimal.

All [[dyadic fraction|dyadic rational numbers]] <math>\frac{p}{2^a}</math> have a ''terminating'' binary numeral—the binary representation has a finite number of terms after the radix point. Other [[rational numbers]] have binary representation, but instead of terminating, they ''recur'', with a finite sequence of digits repeating indefinitely. For instance

<math display="block">\frac{1_{10}}{3_{10}} = \frac{1_2}{11_2} = 0.01010101\overline{01}\ldots\,_2 </math>
<math display="block">\frac{12_{10}}{17_{10}} = \frac{1100_2}{10001_2} = 0.10110100 10110100\overline{10110100}\ldots\,_2 </math>

The phenomenon that the binary representation of any rational is either terminating or recurring also occurs in other radix-based numeral systems. See, for instance, the explanation in [[decimal]]. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that [[0.111... = 1 (binary)|0.111111...]] is the sum of the [[geometric series]] 2<sup>−1</sup> + 2<sup>−2</sup> + 2<sup>−3</sup> + ... which is 1.

Binary numerals that neither terminate nor recur represent [[irrational number]]s. For instance,
* 0.10100100010000100000100... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational
* 1.0110101000001001111001100110011111110... is the binary representation of <math>\sqrt{2}</math>, the [[square root of 2]], another irrational. It has no discernible pattern.

==See also==
{{Portal|Mathematics}}
* [[ASCII]]
* [[Balanced ternary]]
* [[Bitwise operation]]
* [[Binary code]]
* [[Binary-coded decimal]]
* [[Finger binary]]
* [[Gray code]]
* [[IEEE 754]]
* [[Linear-feedback shift register]]
* [[Offset binary]]
* [[Quibinary]]
* [[Reduction of summands]]
* [[Redundant binary representation]]
* [[Repeating decimal]]
* [[Two's complement]]
* [[Unicode]]

==References==
{{Reflist}}

==External links==
{{Wikibooks|Fractals/Mathematics/binary}}
<!--=========================================================

  There are MANY web pages devoted to binary numbers. This section should not be a repository of such web pages.
  Before adding another external link here, make sure it provides a unique resource beyond what is already here.
  We do not need yet another explanation of the binary number system.

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-->
{{Commons category|Binary numeral system}}
* [http://www.cut-the-knot.org/do_you_know/BinaryHistory.shtml Binary System] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/blue/frac_conv.shtml Conversion of Fractions] at [[cut-the-knot]]
* Sir Francis Bacon's BiLiteral Cypher system {{Webarchive|url=https://web.archive.org/web/20160923014940/http://www.baconlinks.com/docs/BILITERAL.doc |date=23 September 2016 }}, predates binary number system.

{{Authority control}}

[[Category:Binary arithmetic]]
[[Category:Computer arithmetic]]
[[Category:Elementary arithmetic]]
[[Category:Gottfried Wilhelm Leibniz]]
[[Category:Power-of-two numeral systems]]