TheInfoList

In mathematics and
digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons The electron is a subatomic particle In physica ...
, a binary number is a
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically "0" (
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

) and "1" (
one 1 (one, also called unit, and unity) is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can ...

). The base-2 numeral system is a
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications ope ...
with a
radix In a positional numeral system Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any of the (or ). More generally, a positional system is a numeral system in which the contribution ...

of 2. Each digit is referred to as a
bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algo ...
, or binary digit. Because of its straightforward implementation in
digital electronic circuit Digital electronics is a field of electronics Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter. It uses active devices to control elect ...
ry using
logic gate A logic gate is an idealized model of computation A model is an informative representation of an object, person or system. The term originally denoted the plan A plan is typically any diagram or list of steps with details of timing and resourc ...
s, the binary system is used by almost all modern , as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language.

# History

The modern binary number system was studied in Europe in the 16th and 17th centuries by
Thomas Harriot Thomas Harriot (; – 2 July 1621), also spelled Harriott, Hariot or Heriot, was an English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo ...
,
Juan Caramuel y Lobkowitz Juan Caramuel y Lobkowitz (Juan Caramuel de Lobkowitz, 23 May 1606 in Madrid Madrid (, ) is the capital and most-populous city of Spain , * gl, Reino de España, * oc, Reiaume d'Espanha, , , image_flag = Bandera de España.svg ...

, and
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, " ...
. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India. Leibniz was specifically inspired by the Chinese
I Ching The ''I Ching'' or ''Yi Jing'' (, ), usually translated as ''Book of Changes'' or ''Classic of Changes'', is an ancient Chinese divination Divination (from Latin ''divinare'', 'to foresee, to foretell, to predict, to prophesy') is the at ...
.

## Egypt

The scribes of ancient Egypt used two different systems for their fractions,
Egyptian fraction An Egyptian fraction is a finite sum of distinct unit fractionA unit fraction is a rational number written as a fraction where the numerator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, ...
s (not related to the binary number system) and 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 Horus or Her, Heru, Hor, Har in Ancient Egyptian, is one of the most significant ancient Egyptian deities Ancient Egyptian deities are the God (male deity), gods and goddesses worshipped in ancient Egypt. The beliefs and rituals surrounding ...

, although this has been disputed). Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a
hekat The hekat or heqat (transcribed ''HqA.t'') was an ancient Egyptian volume unit used to measure grain, bread, and beer. It equals 4.8 litre The litre (British English British English (BrE) is the standard dialect A standard ...
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 #REDIRECT Fifth Dynasty of Egypt #REDIRECT Fifth Dynasty of Egypt#REDIRECT Fifth Dynasty of Egypt The Fifth Dynasty of ancient Egypt Ancient Egypt was a civilization of Ancient history, ancient North Africa, concentrated along the lower r ...
, approximately 2400 BC, and its fully developed hieroglyphic form dates to the
Nineteenth Dynasty of Egypt The Nineteenth Dynasty of Egypt (notated Dynasty XIX), also known as the Ramessid dynasty, is classified as the second Dynasty of the Ancient Egyptian New Kingdom of Egypt, New Kingdom period, lasting from 1292 BC to 1189 BC. The 19th Dynasty and ...
, approximately 1200 BC. The method used for
ancient Egyptian multiplication In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum The British Museum, in the Bloomsbury Bloomsbury is a district in the West End of London The West End of London (commonly referred to as the West End ...

, which dates to around 1650 BC.

## China

The
I Ching The ''I Ching'' or ''Yi Jing'' (, ), usually translated as ''Book of Changes'' or ''Classic of Changes'', is an ancient Chinese divination Divination (from Latin ''divinare'', 'to foresee, to foretell, to predict, to prophesy') is the at ...
dates from the 9th century BC in China. The binary notation in the ''I Ching'' is used to interpret its
quaternary The Quaternary ( ) is the current and most recent of the three periods of the Cenozoic The Cenozoic ( ; ) is Earth's current geological era An era is a span of time defined for the purposes of chronology or historiography, as in the regnal ...

divination Divination (from Latin ''divinare'', 'to foresee, to foretell, to predict, to prophesy') is the attempt to gain insight into a question or situation by way of an occult The occult, in the broadest sense, is a category of supernatural ...
technique. It is based on taoistic duality of
yin and yang In Ancient Chinese philosophy Chinese philosophy originates in the Spring and Autumn period () and Warring States period (), during a period known as the "Hundred Schools of Thought", which was characterized by significan ...

. and a set of 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. The
Song Dynasty The Song dynasty (; ; 960–1279) was an imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song Emperor Taizu of Song (21 March 927 – 14 November 976), personal name Zhao Kua ...
scholar
Shao Yong Shao Yong (; 1011–1077), courtesy name A courtesy name (), also known as a style name, is a name bestowed upon one at adulthood in addition to one's given name. This practice is a tradition in the East Asian cultural sphere The ...

(1011–1077) rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Viewing the
least significant bit In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and softwa ...

on top of single hexagrams i
Shao Yong's square
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.

## India

The Indian scholar
Pingala Acharya Pingala ('; c. 3rd/2nd century BCE) was an ancient Indian poet and mathematician, and the author of the ' (also called ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody. The ' is a work of eight chapters in the late S ...
(c. 2nd century BC) developed a binary system for describing
prosody Prosody may refer to: * Sanskrit prosody, Prosody (Sanskrit), the study of poetic meters and verse in Sanskrit and one of the six Vedangas, or limbs of Vedic studies * Prosody (Greek), the theory and practice of Greek versification * Prosody (Lati ...
. He used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to
Morse code Morse code is a method used in telecommunication Telecommunication is the transmission of information by various types of technologies over wire A wire is a single usually cylindrical A cylinder (from Greek Greek may refer to: ...
.Binary Numbers in Ancient India
/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 Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications ope ...
. 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 Positional notation (or place-value notation, or positional numeral system) denotes usually the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
s.

## Other cultures

The residents of the island of
Mangareva Mangareva is the central and largest island of the Gambier Islands The Gambier Islands ( or ) are an archipelago in French Polynesia, located at the southeast terminus of the Tuamotu archipelago. They cover an area of , and are the remnant ...

in
French Polynesia )Territorial motto: ( en, "Great Tahiti of the Golden Haze") , anthem = "La Marseillaise "La Marseillaise" is the national anthem A national anthem is a Patriotism, patriotic musical composition symbolizing and evoking eulogies of the ...
were using a hybrid binary-
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
system before 1450.
Slit drum Slit may refer to: * Slit (protein) Slit is a protein family, family of secreted extracellular matrix proteins which play an important signalling role in the neural development of most bilaterians (animals with bilateral symmetry). While lower ...
s with binary tones are used to encode messages across Africa and Asia. Sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as
Ifá Ifá is a Yoruba people, Yoruba religion and system of divination. Its literary corpus is the ''Odu Ifá''. Orunmila is identified as the Grand Priest, as he is who revealed divinity and prophecy to the world. Babalawos or Iyanifas use either t ...
as well as in
medieval In the history of Europe The history of Europe concerns itself with the discovery and collection, the study, organization and presentation and the interpretation of past events and affairs of the people of Europe since the beginning of ...
Western
geomancy Geomancy (Greek language, Greek: γεωμαντεία, "earth divination") is a method of divination that interprets markings on the ground or the patterns formed by tossed handfuls of soil, rocks, or sand. The most prevalent form of divinator ...
.

## Western predecessors to Leibniz

In the late 13th century
Ramon Llull Ramon Llull (; c. 1232 – c. 1315/16) was a philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, ...

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. In 1605
Francis Bacon Francis Bacon, 1st Viscount St Alban, (; 22 January 1561 – 9 April 1626), also known as Lord Verulam, was an English philosopher and statesman who served as Attorney General for England and Wales, Attorney General and as Lord Chancellor of K ...

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. 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". (See
Bacon's cipher Bacon's cipher or the Baconian cipher is a method of steganographic Steganography ( ) is the practice of concealing a message within another message or a physical object. In computing/electronic contexts, a computer file, message, image, or video ...
.)
John Napier John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathema ...

in 1617 described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters.
Thomas Harriot Thomas Harriot (; – 2 July 1621), also spelled Harriott, Hariot or Heriot, was an English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo ...
investigated several positional numbering systems, including binary, but did not publish his results; they were found later among his papers. Possibly the first publication of the system in Europe was by
Juan Caramuel y Lobkowitz Juan Caramuel y Lobkowitz (Juan Caramuel de Lobkowitz, 23 May 1606 in Madrid Madrid (, ) is the capital and most-populous city of Spain , * gl, Reino de España, * oc, Reiaume d'Espanha, , , image_flag = Bandera de España.svg ...

, in 1700.

## Leibniz and the I Ching

Leibniz studied binary numbering in 1679; his 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 Fuxi or Fu Hsi (伏羲 ~ 伏犧 ~ 伏戲) is a culture hero A culture hero is a mythological Myth is a folklore genre Folklore is the expressive body of culture shared by a particular group of people; it encompasses the traditions commo ...

"''.Leibniz G., Explication de l'Arithmétique Binaire, Die Mathematische Schriften, ed. C. Gerhardt, Berlin 1879, vol.7, p.223; Engl. trans

/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: : 0 0 0 1   numerical value 20 : 0 0 1 0   numerical value 21 : 0 1 0 0   numerical value 22 : 1 0 0 0   numerical value 23 Leibniz interpreted the hexagrams of the I Ching as evidence of binary calculus. As a
Sinophile A Sinophile is a person who demonstrates a strong interest for Chinese culture Chinese culture () is one of the world's oldest cultures, originating thousands of years ago. The culture prevails across a large geographical region in ...
, Leibniz was aware of the I Ching, 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
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
he admired. The relation was a central idea to his universal concept of a language or
characteristica universalis The Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it ...
, a popular idea that would be followed closely by his successors such as
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...
and
George Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught Autodidacticism (also autodidactism) or self-education (also self-learning and self-teaching) is education Education is the process of facil ...

in forming modern symbolic logic. Leibniz was first introduced to the ''
I Ching The ''I Ching'' or ''Yi Jing'' (, ), usually translated as ''Book of Changes'' or ''Classic of Changes'', is an ancient Chinese divination Divination (from Latin ''divinare'', 'to foresee, to foretell, to predict, to prophesy') is the at ...
'' through his contact with the French Jesuit
Joachim Bouvet Joachim Bouvet (, courtesy name: 明远) (b. Le Mans, July 18, 1656 – June 28, 1730, Peking) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Re ...
, who visited China in 1685 as a missionary. Leibniz saw the ''I Ching'' hexagrams as an affirmation of the universality of his own religious beliefs as a Christian. Binary numerals were central to Leibniz's theology. He believed that binary numbers were symbolic of the Christian idea of ''
creatio ex nihilo (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republi ...
'' or creation out of nothing.

## Later developments

In 1854, British mathematician
George Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught Autodidacticism (also autodidactism) or self-education (also self-learning and self-teaching) is education Education is the process of facil ...

published a landmark paper detailing an
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

ic system of
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

that would become known as
Boolean algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. His logical calculus was to become instrumental in the design of digital electronic circuitry. In 1937,
Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbe ...
produced his master's thesis at
MIT Massachusetts Institute of Technology (MIT) is a private land-grant research university A research university is a university A university ( la, universitas, 'a whole') is an educational institution, institution of higher education, hi ...

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 In theoretical computer science Theoretical computer science (TCS) is a subset of general computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well ...
design. In November 1937,
George Stibitz George Robert Stibitz (April 30, 1904 – January 31, 1995) was a Bell Labs Nokia Bell Labs (formerly named Bell Labs Innovations (1996–2007), AT&T Bell Laboratories (1984–1996) and Bell Telephone Laboratories (1925–1984)) is an Ameri ...
, then working at
Bell Labs Nokia Bell Labs (formerly named Bell Labs Innovations (1996–2007), AT&T Bell Laboratories (1984–1996) and Bell Telephone Laboratories (1925–1984)) is an American industrial research and scientific development company A company, ab ...
, completed a relay-based computer he dubbed the "Model K" (for "Kitchen", where he had assembled it), which calculated using binary addition. 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 mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. In a demonstration to the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematics, mathematical research and scholarship, and serves the national and international community through its publicatio ...
conference at
Dartmouth College Dartmouth College ( ) is a private Private or privates may refer to: Music * "In Private "In Private" was the third single in a row to be a charting success for United Kingdom, British singer Dusty Springfield, after an absence of nearly t ...
on 11 September 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a
teletype A teleprinter (teletypewriter, teletype or TTY) is an electromechanical device that can be used to send and receive typed messages through various communications channels, in both point-to-point (telecommunications), point-to-point and point-t ...
. 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 von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American Hungarian Americans (Hungarian language, Hungarian: ''amerikai magyarok'') are United States, Americans of Hungarian p ...

,
John Mauchly John William Mauchly (August 30, 1907 – January 8, 1980) was an American physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, ...
and
Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( an ...

, who wrote about it in his memoirs. The Z1 computer, which was designed and built by
Konrad Zuse Konrad Zuse (; 22 June 1910 – 18 December 1995) was a German civil engineer A civil engineer is a person who practices civil engineering Civil engineering is a Regulation and licensure in engineering, professional engineering discipli ...

between 1935 and 1938, used
Boolean logic In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, respectively. Instead of elementary ...
and binary floating point numbers.

# Representation

Any number can be represented by a sequence of
bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algo ...
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: The numeric value represented in each case is dependent upon 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. In a modern computer, the numeric values may be represented by two different
voltage Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ...

s; on a
magnetic Magnetism is a class of physical attributes that are mediated by magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For in ...

disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
, magnetic polarities may be used. A "positive", "
yes Yes or YES may refer to: * An affirmative particle in the English language; see yes and no ''Yes'' and ''no'', or word pairs with similar words, are expressions of the affirmative and the negative, respectively, in several languages, includi ...
", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use. In keeping with customary representation of numerals using
Arabic numerals Arabic numerals are the ten numerical digit A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a Positional notation, positional numeral sy ...

, binary numbers are commonly written using the symbols 0 and 1. When written, binary numerals are often subscripted, prefixed or suffixed in order 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) * 100101B (a suffix indicating binary format) * bin 100101 (a prefix indicating binary format) * 1001012 (a subscript indicating base-2 (binary) notation) * %100101 (a prefix indicating binary format; also known as Motorola convention) * 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, in order 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 The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
counting system as a frame of reference.

## Decimal counting

Decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
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) :010, 011, 012, ... :   ... :090, 091, 092, ... 097, 098, 099, (rightmost two digits are reset to zeroes, and next digit is incremented) :100, 101, 102, ...

## Binary counting

Binary counting follows the same procedure, except that only the two symbols ''0'' and ''1'' are available. Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left: :0000, :0001, (rightmost digit starts over, and next digit is incremented) :0010, 0011, (rightmost two digits start over, and next digit is incremented) :0100, 0101, 0110, 0111, (rightmost three digits start over, and the next digit is incremented) :1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ... In the binary system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. The value of a binary number is the sum of the powers of 2 represented by each "1" digit. For example, the binary number 100101 is converted to decimal form as follows: :1001012 = ( 1 ) × 25 + ( 0 ) × 24 + ( 0 ) × 23 + ( 1 ) × 22 + ( 0 ) × 21 + ( 1 ) × 20 :1001012 = 1 × 32 + 0 × 16 + 0 × 8 + 1 × 4 + 0 × 2 + 1 × 1 :1001012 = 3710

# Fractions

Fractions in binary arithmetic
terminate Terminate may refer to: *Electrical termination, ending a wire or cable properly to prevent interference *Termination of employment, the end of an employee's duration with an employer *Terminate with extreme prejudice, a euphemism for assassinatio ...
only if is the only
prime factor A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
in the
denominator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
. As a result, 1/10 does not have a finite binary representation (10 has prime factors 2 and 5). This causes 10 × 0.1 not to precisely equal 1 in
floating-point arithmetic In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
. As an example, to interpret the binary expression for 1/3 = .010101..., this means: 1/3 = 0 × 2−1 + 1 × 2−2 + 0 × 2−3 + 1 × 2−4 + ... = 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.

# Binary arithmetic

Arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.

xor Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, ...
y = (x + y) mod 2 for any two bits x and y allows for very fast calculation, as well.

### Long carry method

The binary addition table is similar, but not the same, as the
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra (logic), Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expression (mathematics) ...

of the
logical disjunction In logic, disjunction is a logical connective typically notated \lor whose meaning either refines or corresponds to that of natural language expressions such as "or". In classical logic, it is given a truth functional semantics of logic, seman ...
operation $\lor$. The difference is that $1$$\lor$$1=1$, while $1+1=10$.

## Subtraction

Subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of equal
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

. Computers use
signed number representations In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and softw ...
to handle negative numbers—most commonly the
two's complement Two's complement is a mathematical operation In mathematics, an operation is a Function (mathematics), function which takes zero or more input values (called ''operands'') to a well-defined output value. The number of operands is the arity of the ...
notation. Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation subtraction can be summarized by the following formula: A − B = A + not B + 1

## Multiplication

Multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

in binary is similar to its decimal counterpart. Two numbers and can be multiplied by partial products: for each digit in , the product of that digit in is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in 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 is 0, the partial product is also 0 * If the digit in is 1, the partial product is equal to For example, the binary numbers 1011 and 1010 are multiplied as follows: 1 0 1 1 () × 1 0 1 0 () --------- 0 0 0 0 ← Corresponds to the rightmost 'zero' in + 1 0 1 1 ← Corresponds to the next 'one' in + 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 pointIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
: 1 0 1 . 1 0 1 (5.625 in decimal) × 1 1 0 . 0 1 (6.25 in decimal) ------------------- 1 . 0 1 1 0 1 ← Corresponds to a 'one' in + 0 0 . 0 0 0 0 ← Corresponds to a 'zero' in + 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

The binary multiplication table is the same as the
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra (logic), Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expression (mathematics) ...

of the
logical conjunction In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
operation $\land$.

## Division

Long division In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'ar ...

in binary is again similar to its decimal counterpart. In the example below, the
divisor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

is 1012, or 5 in decimal, while the
dividend A dividend is a distribution of profit Profit may refer to: Business and law * Profit (accounting), the difference between the purchase price and the costs of bringing to market * Profit (economics), normal profit and economic profit * Profit ...
is 110112, or 27 in decimal. The procedure is the same as that of decimal
long division In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'ar ...

; here, the divisor 1012 goes into the first three digits 1102 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 ----- 1 0 Thus, the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
of 110112 divided by 1012 is 1012, as shown on the top line, while the remainder, shown on the bottom line, is 102. 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 taking a binary square root digit by digit is the same as for a decimal square root and is explained
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies Here Technologies (trading as A trade name, trading name, or business name is a pseudonym A pseudonym () or alias () (originally: ...
. An example is: 1 0 0 1 --------- √ 1010001 1 --------- 101 01 0 -------- 1001 100 0 -------- 10001 10001 10001 ------- 0

# Bitwise operations

Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using Boolean logical operators. When a string of binary symbols is manipulated in this way, it is called a
bitwise operation In computer programming, a bitwise operation operates on a bit string, a bit array or a Binary numeral system, binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher le ...
; the logical operators
AND And or AND may refer to: Logic, grammar, and computing * Conjunction (grammar) In grammar In linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study o ...
, OR, and
XOR Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, ...
may be performed on corresponding bits in two binary numerals provided as input. The logical 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 300px, A left arithmetic shift of a binary number by 1. The empty position in the least significant bit is filled with a zero. In computer programming Computer programming is the process of designing and building an executable computer p ...
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 convert from a base-10
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
to its base-2 (binary) equivalent, the number is divided by two. The remainder is the
least-significant bit In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
. 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)10 is expressed as (101100101)2. 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 100101011012 to decimal: The result is 119710. The first Prior Value of 0 is simply an initial decimal value. This method is an application of the
Horner scheme Horner is a surname * An English surname of Anglo-Saxon origin that derives from the Middle English Middle English (abbreviated to ME) was a form of the English language spoken after the Norman conquest of England, Norman conquest (1066) until ...
. 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 0.110101101012, the first digit is $\begin \frac \end$, the second $\begin \left(\frac\right)^2 = \frac \end$, etc. So if there is a 1 in the first place after the decimal, then the number is at least $\begin \frac \end$, 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, $\begin \left(\frac\right) \end$10, in binary, is: Thus the repeating decimal fraction 0.... is equivalent to the repeating binary fraction 0.... . Or for example, 0.110, in binary, is: This is also a repeating binary fraction 0.0... . 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 0.1 + ... + 0.1, (10 additions) differs from 1 in
floating point arithmetic In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
. 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: : $\begin x & = & 1100&.1\overline\ldots \\ x\times 2^6 & = & 1100101110&.\overline\ldots \\ x\times 2 & = & 11001&.\overline\ldots \\ x\times\left(2^6-2\right) & = & 1100010101 \\ x & = & 1100010101/111110 \\ x & = & \left(789/62\right)_ \end$ Another way of converting from binary to decimal, often quicker for a person familiar with
hexadecimal In mathematics and computing, the hexadecimal (also base 16 or hex) numeral system is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system repres ...
, is to do so indirectly—first converting ($x$ in binary) into ($x$ in hexadecimal) and then converting ($x$ in hexadecimal) into ($x$ 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''k'', 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 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''k'' and added to the second converted piece, where ''k'' is the number of decimal digits in the second, least-significant piece before conversion.

Binary may be converted to and from hexadecimal more easily. This is because the
radix In a positional numeral system Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any of the (or ). More generally, a positional system is a numeral system in which the contribution ...

of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 24, 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: :3A16 = 0011 10102 :E716 = 1110 01112 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 Padding is thin cushioned material sometimes added to clothes. Padding may also be referred to as batting when used as a layer in lining quilts or as a packaging or stuffing material. When padding is used in clothes, it is often done in an attempt ...
). For example: :10100102 = 0101 0010 grouped with padding = 5216 :110111012 = 1101 1101 grouped = DD16 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: :C0E716 = (12 × 163) + (0 × 162) + (14 × 161) + (7 × 160) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,38310

## 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, 23, 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 mathematics and computing, the hexadecimal (also base 16 or hex) numeral system is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system repres ...
in the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is equivalent to octal 7, and so forth. Converting from octal to binary proceeds in the same fashion as it does for
hexadecimal In mathematics and computing, the hexadecimal (also base 16 or hex) numeral system is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system repres ...
: :658 = 110 1012 :178 = 001 1112 And from binary to octal: :1011002 = 101 1002 grouped = 548 :100112 = 010 0112 grouped with padding = 238 And from octal to decimal: :658 = (6 × 81) + (5 × 80) = (6 × 8) + (5 × 1) = 5310 :1278 = (1 × 82) + (2 × 81) + (7 × 80) = (1 × 64) + (2 × 8) + (7 × 1) = 8710

# Representing real numbers

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 11.012 means: For a total of 3.25 decimal. All dyadic fraction, dyadic rational numbers $\frac$ 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 :$\frac = \frac = 0.01010101\overline\ldots\,_2$ :$\frac = \frac = 0.10110100 10110100\overline\ldots\,_2$ 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 The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
. 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 2−1 + 2−2 + 2−3 + ... which is 1. Binary numerals which neither terminate nor recur represent irrational numbers. 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 $\sqrt$, the square root of 2, another irrational. It has no discernible pattern.

* Balanced ternary * 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