binary number

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A binary number is a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
expressed in the base-2
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing number A number is a mathematical object used to count, measure, and label. The original example ...
or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (
zero 0 (zero) is a number, and the numerical digit used to represent that number in numeral system, numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. A ...
) and "1" ( one). 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 radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral syste ...
with a
radix In a positional numeral system, the radix or base is the number of unique numerical digit, digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (b ...
of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using
logic gate A logic gate is an idealized or physical device implementing a Boolean function, a logical operation performed on one or more Binary number, binary inputs that produces a single binary output. Depending on the context, the term may refer to an id ...
s, the binary system is used by almost all modern 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.

# 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 astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of ...
, Juan Caramuel y Lobkowitz, and
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
. 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 ''Book of Changes'' or ''Classic of Changes'', is an ancient Chinese divination text that is among the oldest of the Chinese classics. Originally a divination manual in the Western Z ...
''.

## Egypt

The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions (not related to the binary number system) and 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 Horus or Heru, Hor, Har in Ancient Egyptian, is one of the most significant ancient Egyptian deities who served many functions, most notably as god of kingship and the sky. He was worshipped from at least the late prehistoric Egypt until the Pto ...
, 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 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 The Fifth Dynasty of ancient Egypt (notated Dynasty V) is often combined with Dynasties Third Dynasty of Egypt, III, Fourth Dynasty of Egypt, IV and Sixth Dynasty of Egypt, VI under the group title the Old Kingdom of Egypt, Old Kingdom. The Fifth ...
, 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 Egypt Ancient Egypt was a civilization in Northeast Africa situated in the Nile Valley. Ancie ...
, approximately 1200 BC. 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.

## China

The ''
I Ching The ''I Ching'' or ''Yi Jing'' (, ), usually translated ''Book of Changes'' or ''Classic of Changes'', is an ancient Chinese divination text that is among the oldest of the Chinese classics. Originally a divination manual in the Western Z ...
'' 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 period (geology), periods of the Cenozoic era (geology), Era in the geologic time scale of the International Commission on Stratigraphy (ICS). It follows the Neogene Period and spa ...
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 occultic, standardized process or ritual. Used in various forms throughout history ...
technique. It is based on taoistic duality of
yin and yang Yin and yang ( and ) is a Chinese philosophical concept that describes opposite but interconnected forces. In Chinese cosmology, the universe creates itself out of a primary chaos of material energy, organized into the cycles of yin and ya ...
. Eight trigrams (Bagua) 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 Dynasties in Chinese history, imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song following his usurpation of the throne of the Later Zhou. ...
scholar
Shao Yong Shao Yong (; 1011–1077), courtesy name Yaofu (堯夫), named Shào Kāngjié (邵康節) was a Chinese cosmologist, historian, philosopher, and poet who greatly influenced the development of Neo-Confucianism across China during the Song dynast ...
(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, bit numbering is the convention used to identify the bit positions in a binary numeral system, binary number. Bit significance and indexing In computing, the least significant bit (LSB) is the bit position in a Binary numeral sy ...
on top of single hexagrams in 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. 3rd2nd century Common Era, BCE) was an ancient Indian poet and Indian mathematics, mathematician, and the author of the ' (also called the ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody. The ' is a wo ...
(c. 2nd century BC) developed a binary system for describing prosody. 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 to Character encoding, encode Written language, text characters as standardized sequences of two different signal durations, called ''dots'' and ''dashes'', or ''dits'' and ''dahs''. Morse cod ...
.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 radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral syste ...
. 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 values.

## Other cultures

The residents of the island of
Mangareva Mangareva is the central and largest island of the Gambier Islands in French Polynesia. It is surrounded by smaller islands: Taravai in the southwest, Aukena and Akamaru (Island), Akamaru in the southeast, and islands in the north. Mangareva ha ...
in
French Polynesia )Territorial motto: ( en, "Great Tahiti of the Golden Haze") , anthem = , song_type = Regional anthem , song = "Ia Ora 'O Tahiti Nui" , image_map = French Polynesia on the globe (French Polynesia centered).svg , map_alt = Location of French ...
were using a hybrid binary-
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
system before 1450.
Slit drum A slit drum or slit gong is a hollow percussion instrument. In spite of the name, it is not a true drum but an idiophone, usually carved or constructed from bamboo or wood into a box with one or more slits in the top. Most slit drums have one slit ...
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 as well as in
medieval In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire ...
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, rock (geology), rocks, or sand. The most prevalent ...
.

## Western predecessors to Leibniz

In the late 13th century
Ramon Llull Ramon Llull (; c. 1232 – c. 1315/16) was a philosopher A philosopher is a person who practices or investigates philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about e ...
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 and Lord Chancellor of England England is a Countries of ...
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 representing information within another message or physical object, in such a manner that the presence of the information is not evident to ...
.)
John Napier John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinisation of names, L ...
in 1617 described a system he called
location arithmetic Location arithmetic (Latin ''arithmeticae localis'') is the additive (non-positional) binary arithmetic, binary numeral systems, which John Napier explored as a computation technique in his treatise ''Rabdology'' (1617), both symbolically and on ...
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 astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of ...
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, 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 in Chinese legend and Chinese mythology, mythology, credited along with his sister and wife Nüwa with List of protoplasts, creating humanity and the invention of music, hunting, fishin ...
"''.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 China, Chinese culture, Chinese language, History of China, Chinese history, and/or Chinese people. Those with professional training and practice in the study of China are refer ...
, 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 is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
he admired. The relation was a central idea to his universal concept of a language or
characteristica universalis The Latin term ''characteristica universalis'', commonly interpreted as ''universal characteristic'', or ''universal character'' in English, is a universal language, universal and formal language imagined by Gottfried Leibniz able to express ma ...
, 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, Mathematical logic, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the fath ...
and
George Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely Autodidacticism, self-taught English people, English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics ...
in forming modern symbolic logic. Leibniz was first introduced to the ''
I Ching The ''I Ching'' or ''Yi Jing'' (, ), usually translated ''Book of Changes'' or ''Classic of Changes'', is an ancient Chinese divination text that is among the oldest of the Chinese classics. Originally a divination manual in the Western Z ...
'' through his contact with the French Jesuit
Joachim Bouvet Joachim Bouvet (, courtesy name: 明远) (July 18, 1656, in Le Mans – June 28, 1730, in Peking) was a French Jesuit China missions, Jesuit who worked in China, and the leading member of the Figurist movement. China Bouvet came to China in 1687, ...
, 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 a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome ...
'' or creation out of nothing.

## Later developments

In 1854, British mathematician
George Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely Autodidacticism, self-taught English people, English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics ...
published a landmark paper detailing an
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
ic system of
logic Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...
that would become known as
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denote ...
. 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 people, American mathematician, electrical engineering, electrical engineer, and cryptography, cryptographer known as a "father of information theory". As a 21-year-o ...
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 "A Symbolic Analysis of Relay and Switching Circuits" is the title of a master's thesis written by computer science pioneer Claude Elwood Shannon, Claude E. Shannon while attending the Massachusetts Institute of Technology (MIT) in 1937. In his th ...
'', Shannon's thesis essentially founded practical
digital circuit In theoretical computer science, a circuit is a model of computation in which input values proceed through a sequence of gates, each of which computes a function (computer science), function. Circuits of this kind provide a generalization of Boolean ...
design. In November 1937,
George Stibitz George Robert Stibitz (April 30, 1904 – January 31, 1995) was a Bell Labs researcher internationally recognized as one of the fathers of the modern digital computer. He was known for his work in the 1930s and 1940s on the realization of Boolea ...
, then working at
Bell Labs Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984), then AT&T Bell Laboratories (1984–1996) and Bell Labs Innovations (1996–2007), is an American industrial research and scientific development company A com ...
, 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, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
. 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 university, private research university in Hanover, New Hampshire. Established in 1769 by Eleazar Wheelock, it is one of the nine colonial colleges chartered before the American Revolution. Although founded 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- ...
. 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 mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cover ...
,
John Mauchly John William Mauchly (August 30, 1907 – January 8, 1980) was an American physicist who, along with J. Presper Eckert, designed ENIAC, the first general-purpose electronic digital computer, as well as EDVAC, BINAC and UNIVAC I, the first com ...
and
Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...
, who wrote about it in his memoirs. The Z1 computer, which was designed and built by
Konrad Zuse Konrad Ernst Otto Zuse (; 22 June 1910 – 18 December 1995) was a German civil engineer, List of pioneers in computer science, pioneering computer scientist, inventor and businessman. His greatest achievement was the world's first programmab ...
between 1935 and 1938, used
Boolean logic In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
and binary floating point numbers. (12 pages)

# Representation

Any number can be represented by a sequence of bits (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, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), w ...
s; on a
magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
disk, magnetic polarities may be used. A "positive", " yes", 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 digits: , , , , , , , , and . They are the most commonly used symbols to write Decimal, decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers ...
, 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 In a positional numeral system, the radix or base is the number of unique numerical digit, digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (b ...
. 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 (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
counting system as a frame of reference.

## Decimal counting

Decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
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 exact same procedure, and again the incremental substitution begins with the least significant 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, :0001, (rightmost bit starts over, and next digit is incremented) :0010, 0011, (rightmost two bits start over, and next bit is incremented) :0100, 0101, 0110, 0111, (rightmost three bits start over, and the next bit is incremented) :1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ... In the binary system, each bit represents an increasing power of 2, with the rightmost bit 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" bit. 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 only if 2 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 la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
. 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, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an Integer (computer science), integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. ...
. 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 () is an elementary part of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their chang ...
in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.

### 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 Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science ...
operation $\lor$. The difference is that $1 \lor 1 = 1$, while $1+1=10$.

## Subtraction

Subtraction Subtraction is an Arithmetic#Arithmetic operations, 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 ...
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, a negative number represents an opposite. In the real number system, a negative number is a number that is inequality (mathematics), less than 0 (number), zero. Negative numbers are often used to represent the magnitude of a loss ...
of equal
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign (mathematics), sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative number, negative (in which cas ...
. Computers use
signed number representations In computing, signed number representations are required to encode negative numbers in binary number systems. In mathematics, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in RAM or CPU Process ...
to handle negative numbers—most commonly the
two's complement Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the Most Significant Bit, binary digit with the greatest place value (the leftmos ...
notation. Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation subtraction can be summarized by the following formula: :

## Multiplication

Multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
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 point: 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 the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science ...
operation $\land$.

## Division

Long division In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu–Arabic numeral system, Hindu-Arabic numerals (Positional notation) that is simple enough to perform by hand. It breaks down a division (mathem ...
in binary is again similar to its decimal counterpart. In the example below, the
divisor In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
is 1012, or 5 in decimal, while the
dividend A dividend is a distribution of Profit (accounting), profits by a corporation to its shareholders. When a corporation earns a profit or surplus, it is able to pay a portion of the profit as a dividend to shareholders. Any amount not distributed ...
is 110112, or 27 in decimal. The procedure is the same as that of decimal
long division In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu–Arabic numeral system, Hindu-Arabic numerals (Positional notation) that is simple enough to perform by hand. It breaks down a division (mathem ...
; 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 ----- 0 1 0 Thus, the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division (mathematics), division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to a ...
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. 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-l ...
; the logical operators AND, OR, and XOR may be performed on corresponding bits in two binary numerals provided as input. The logical 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 In computer programming, an arithmetic shift is a Bitwise_operation#Bit_shifts, shift operator, sometimes termed a signed shift (though it is not restricted to signed operands). The two basic types are the arithmetic left shift and the arith ...
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

To convert from a base-10
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
to its base-2 (binary) equivalent, the number is divided by two. The remainder is the
least-significant bit In computing, bit numbering is the convention used to identify the bit positions in a binary numeral system, binary number. Bit significance and indexing In computing, the least significant bit (LSB) is the bit position in a Binary numeral sy ...
. 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.

## 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 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. 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 $\frac$, the second $(\frac)^2 = \frac$, etc. So if there is a 1 in the first place after the decimal, then the number is at least $\frac$, 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, $(\frac)_$, 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 computer, computing machinery. It includes the study and experimentation of algorithmic processes, and development of both computer hardware , hard ...
. 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: $\begin x & = & 1100&.1\overline\ldots \\ x\times 2^6 & = & 1100101110&.\overline\ldots \\ x\times 2 & = & 11001&.\overline\ldots \\ x\times(2^6-2) & = & 1100010101 \\ x & = & 1100010101/111110 \\ x & = & (789/62)_ \end$ 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 simply 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 ...
, 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, the radix or base is the number of unique numerical digit, digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (b ...
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 The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, uses a base-10 numbe ...
numeral system, since octal uses a radix of 8, which is a
power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number 2, two as the Base (exponentiation), base and integer  as the exponent. In a context where only integers are considered, is re ...
(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 simply 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 ...
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 simply 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 ...
: :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 A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
(called a
decimal point A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
in the decimal system). For example, the binary number 11.012 means: For a total of 3.25 decimal. All 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 In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
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 (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 0.111111... is the sum of the
geometric series In mathematics, a geometric series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \, ...
2−1 + 2−2 + 2−3 + ... which is 1. Binary numerals which neither terminate nor recur represent
irrational number In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
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 $\sqrt$, the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
, another irrational. It has no discernible pattern.

* Balanced ternary *
Binary code A binary code represents plain text, text, instruction set, computer processor instructions, or any other data using a two-symbol system. The two-symbol system used is often "0" and "1" from the binary number, binary number system. The binary cod ...
*
Binary-coded decimal In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computer, computing machinery. It includes the study and experimentation of algorithmic processes, and development of both computer hardware , ha ...
* Finger binary *
Gray code The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray (researcher), Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For ...
*
IEEE 754 The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard Floating-point arithmetic#IEEE 754 ...
*
Linear-feedback shift register In computing, a linear-feedback shift register (LFSR) is a shift register whose input bit is a Linear#Boolean functions, linear function of its previous state. The most commonly used linear function of single bits is exclusive-or (XOR). Thus, ...
* Offset binary * Quibinary * Reduction of summands * Redundant binary representation *
Repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose Numerical digit, digits are periodic function, periodic (repeating its values at regular intervals) and the infinity, infinitely repeated portion is not zero. It ...
*
Two's complement Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the Most Significant Bit, binary digit with the greatest place value (the leftmos ...