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The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2. Owing to its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by all modern computers. Image File history File links Merge-arrows. ... The term binary code can mean several different things: There are a variety of different methods of coding numbers or symbols into strings of, including fixed-length binary numbers, prefix codes such as Huffman code, and other arithmetic coding. ... This article is about different methods of expressing numbers with symbols. ... I like cream cheese, it tastes good on toast. ... The Eastern Arabic numerals (also called Eastern Arabic numerals, Arabic-Indic numerals, Arabic Eastern Numerals) are the symbols (glyphs) used to represent the Hindu-Arabic numeral system in conjunction with the Arabic alphabet in Egypt, Iran, Pakistan and parts of India, and also in the no longer used Ottoman Turkish... Khmer numerals are the numerals used in the Khmer language of Cambodia. ... India has produced many numeral systems. ... The Brahmi numerals are an indigenous Indian numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). ... Today, speakers of Chinese use three numeral systems: the ubiquitous system of Arabic numerals, along with two ancient Chinese numeral systems. ... The counting rods (Traditional Chinese: , Simplified Chinese: , pinyin: chou2) were used by ancient Chinese before the invention of the abacus. ... The Abjad numerals are a decimal numeral system which was used in the Arabic-speaking world prior to the use of the Hindu-Arabic numerals from the 8th century, and in parallel with the latter until Modern times. ... Cyrillic numerals was a numbering system derived from the Cyrillic alphabet, used by South and East Slavic peoples. ... Note: This article contains special characters. ... The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. ... Attic numerals were used by ancient Greeks, possibly from the 7th century BC. They were also known as Herodianic numerals because they were first described in a 2nd century manuscript by Herodian. ... Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. ... The Etruscan numerals were used by the ancient Etruscans. ... Mayan numerals. ... Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ... During the beginning of the Urnfield culture, around 1200 BC, a series of votive sickles of bronze with marks that have been interpreted as a numeral system, appeared in Central Europe. ... This is a list of numeral system topics, by Wikipedia page. ... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... For other uses, see Decimal (disambiguation). ... Quaternary is the base four numeral system. ... The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ... Base32 is a derivation of Base64 with the following additional properties: The resulting character set is all uppercase, which can often be beneficial when using a case-sensitive filesystem. ... In computing, base64 is a data encoding scheme whereby binary-encoded data is converted to printable ASCII characters. ... The unary numeral system is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen symbol is repeated N times. ... Ternary or trinary is the base-3 numeral system. ... Nonary is a base 9 numeral system, typically using the digits 0-8, but not the digit 9. ... The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. ... The vigesimal or base-20 numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten). ... As there are 24 hours in a day a numbering system based upon 24, and as the base 12 is convenient here some examples of the base 24 (quadrovigesimal) system. ... Base 30 or trigesimal is a positional numeral system using 30 as the radix. ... Base 36 is a positional numeral system using 36 as the radix. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... This article is about different methods of expressing numbers with symbols. ... Zero redirects here. ... This article is about the number one. ... In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ... This article does not cite any references or sources. ... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... A logic gate performs a logical operation on one or more logic inputs and produces a single logic output. ... This article is about the machine. ...

History

The ancient Indian writer Pingala developed advanced mathematical concepts for describing prosody, and in doing so presented the first known description of a binary numeral system, possibly as early as the 8th century BC.^[1] Others place him much later; R. Hall, Mathematics of Poetry, has "c. 200 BC". The numeration system was based on the Eye of Horus Old Kingdom numeration system.^{[citation needed]} Pingala (पिङ्गल ) is the supposed author of the Chandas shastra (, also Chandas sutra ), a Sanskrit treatise on prosody considered one of the Vedanga. ... Prosody may mean several things: Prosody consists of distinctive variations of stress, tone, and timing in spoken language. ... The Wadjat - later called The Eye of Horus The Eye of Horus (previously wadjet and the Eye of the Moon; and afterward as The Eye of Ra)[1] is an ancient Egyptian symbol of protection and royal power from deities, in this case from Horus or Ra. ...

A full set of 8 trigrams and 64 hexagrams, analogous to the 3-bit and 6-bit binary numerals, were known to the ancient Chinese in the classic text I Ching. Similar sets of binary combinations have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. For other uses, such as the Peruvian province or town, see Bagua (disambiguation). ... A hexagram is any of the sixty-four sets of solid and broken lines used in the Chinese classic text I Ching. ... Chinese classic texts or Chinese canonical texts are the classical literature in Chinese culture that are considered to be the best or the most valuable. ... Alternative meaning: I Ching (monk) The I Ching (Traditional Chinese: 易經, pinyin y jīng; Cantonese IPA: jɪk6gɪŋ1; Cantonese Jyutping: jik6ging1; alternative romanizations include I Jing, Yi Ching, Yi King) is the oldest of the Chinese classic texts. ... ‹ The template below (Citations missing) is being considered for deletion. ... Geomancer redirects here. ...

An arrangement of the hexagrams of the I Ching, ordered according to the values of the corresponding binary numbers (from 0 to 63), and a method for generating the same, was developed by the Chinese scholar and philosopher Shao Yong in the 11th century. However, there is no evidence that Shao understood binary computation; the ordering is also the lexicographical order on sextuples of elements chosen from a two-element set. Shao Yung (邵雍) was one of the most remarkable men who has ever probed the hidden, metaphysical secrets of life. ... In mathematics, the lexicographic or lexicographical order, (also known as dictionary order, alphabetic order or lexicographic(al) product), is a natural order structure of the Cartesian product of two ordered sets. ...

In 1605 Francis Bacon discussed a system by which 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".^[2] (See Bacon's cipher.) For other persons named Francis Bacon, see Francis Bacon (disambiguation). ... Bacons cipher or the Baconian cipher is a method of steganography (a method of hiding a secret message as opposed to a true cipher) devised by Francis Bacon. ...

The modern binary number system was fully documented by Gottfried Leibniz in the 17th century in his article Explication de l'Arithmétique Binaire. Leibniz's system used 0 and 1, like the modern binary numeral system. Leibniz redirects here. ... External Links Baron Gottfried Wilhelm von Leibniz (1646-1716) [1] ...

In 1854, British mathematician George Boole published a landmark paper detailing an algebraic system of logic that would become known as Boolean algebra. His logical calculus was to become instrumental in the design of digital electronic circuitry. Not to be confused with George Boolos. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... Boolean algebra is the finitary algebra of two values. ...

In 1937, Claude Shannon produced his master's thesis at MIT that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled A Symbolic Analysis of Relay and Switching Circuits, Shannon's thesis essentially founded practical digital circuit design. Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001), an American electrical engineer and mathematician, has been called the father of information theory,[1] and was the founder of practical digital circuit design theory. ... Mapúa Institute of Technology (MIT, MapúaTech or simply Mapúa) is a private, non-sectarian, Filipino tertiary institute located in Intramuros, Manila. ... Boolean algebra is the finitary algebra of two values. ... In his 1937 MIT masters thesis, A Symbolic Analysis of Relay and Switching Circuits, Claude Elwood Shannon proved that Boolean algebra and binary arithmetic could be used to simplify the arrangement of the electromechanical relays then used in telephone routing switches, then turned the concept upside down and also... Digital circuits are electric circuits based on a number of discrete voltage levels. ...

In November of 1937, George Stibitz, then working at Bell Labs, completed a relay-based computer he dubbed the "Model K" (for "Kitchen", where he had assembled it), which calculated using binary addition. Bell Labs thus authorized a full research program in late 1938 with Stibitz at the helm. Their Complex Number Computer, completed January 8, 1940, was able to calculate complex numbers. In a demonstration to the American Mathematical Society conference at Dartmouth College on September 11, 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a teletype. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were John Von Neumann, John Mauchly, and Norbert Wiener, who wrote about it in his memoirs. George Stibitz George Robert Stibitz (April 20, 1904 – January 31, 1995) is internationally recognized as a father of the modern digital computer. ... Bell Laboratories (also known as Bell Labs and formerly known as AT&T Bell Laboratories and Bell Telephone Laboratories) was the main research and development arm of the United States Bell System. ... is the 8th day of the year in the Gregorian calendar. ... Year 1940 (MCMXL) was a leap year starting on Monday (link will display the full 1940 calendar) of the Gregorian calendar. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... The American Mathematical Society (AMS) is dedicated to the interests of mathematical research and education, which it does with various publications and conferences as well as annual monetary awards to mathematicians. ... Dartmouth College is a private, coeducational university located in Hanover, New Hampshire, USA. Incorporated as Trustees of Dartmouth College,[6][7] it is a member of the Ivy League and one of the nine colonial colleges founded before the American Revolution. ... is the 254th day of the year (255th in leap years) in the Gregorian calendar. ... Year 1940 (MCMXL) was a leap year starting on Monday (link will display the full 1940 calendar) of the Gregorian calendar. ... A teleprinter (teletypewriter, teletype or TTY) is a now largely obsolete electro-mechanical typewriter which can be used to communicate typed messages from point to point through a simple electrical communications channel, often just a pair of wires. ... For other persons named John Neumann, see John Neumann (disambiguation). ... Eckert and Mauchly examine a printout of ENIAC results in a newsreel from February 1946. ... Norbert Wiener Norbert Wiener (November 26, 1894, Columbia, Missouri – March 18, 1964, Stockholm Sweden) was an American theoretical and applied mathematician. ...

Representation

A binary number can be represented by any sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. The following sequences of symbols could all be interpreted as the same binary numeric value of 667: This article is about the unit of information. ... 600 (six hundred) is the natural number following five hundred and ninety-nine and preceding six hundred and one. ...

 1 0 1 0 0 1 1 0 1 1 | - | - - | | - | | x o x o o x x o x x y n y n n y y n y y 

A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.

The numeric value represented in each case is dependent upon the value assigned to each symbol. In a computer, the numeric values may be represented by two different voltages; on a magnetic 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. Image File history File links Binary_clock. ... Image File history File links Binary_clock. ... A binary clock is a clock which displays traditional sexagesimal time in a binary format. ... LED redirects here. ... In computing and electronic systems, binary-coded decimal (BCD) is an encoding for decimal numbers in which each digit is represented by its own binary sequence. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ... For the indie-pop band, see The Magnetic Fields. ... Disk storage is a general category of a computer storage mechanisms, in which data is recorded on planar, round and rotating surfaces (disks, discs, or platters). ... The polarity of an object is, in general, its physical alignment of atoms. ...

In keeping with customary representation of numerals using Arabic numerals, binary numbers are commonly written using the symbols 0 and 1. When written, binary numerals are often subscripted, prefixed or suffixed in order to indicate their base, or radix. The following notations are equivalent: For other uses, see Arabic numerals (disambiguation). ...

100101 binary (explicit statement of format)

100101b (a suffix indicating binary format)

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)

0b100101 (a prefix indicating binary format, common in programming languages)

When spoken, binary numerals are usually read digit-by-digit, in order to distinguish them from decimal numbers. 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 is equal to the decimal value four, it would be confusing, and numerically incorrect, to refer to the numeral as one hundred.

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. Decimal counting uses the symbols 0 through 9, while binary only uses the symbols 0 and 1.

When the symbols for the first digit are exhausted, the next-higher digit (to the left) is incremented, and counting starts over at 0. In decimal, counting proceeds like so: For other uses, see Decimal (disambiguation). ...

000, 001, 002, ... 007, 008, 009, (rightmost digit starts over, and next digit is incremented)

010, 011, 012, ...

   ...

090, 091, 092, ... 097, 098, 099, (rightmost two digits start over, and next digit is incremented)

100, 101, 102, ...

After a digit reaches 9, an increment resets it to 0 but also causes an increment of the next digit to the left. In binary, counting is the same except that only the two symbols 0 and 1 are used. 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, ...

Binary arithmetic

Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.

Addition

The circuit diagram for a binary half adder, which adds two bits together, producing sum and carry bits.

The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple: Image File history File links Half-adder. ... Image File history File links Half-adder. ... The circuit diagram for a 4 bit TTL counter, a type of state machine A circuit diagram (also known as an electrical diagram, elementary diagram, or electronic schematic) is a simplified conventional pictorial representation of an electrical circuit. ... In electronics, an adder or summer is a digital circuit that performs addition of numbers. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ...

0 + 0 → 0

0 + 1 → 1

1 + 0 → 1

1 + 1 → 0, carry 1 (since 1 + 1 = 0 + 1 × 10 in binary)

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

5 + 5 → 0, carry 1 (since 5 + 5 = 0 + 1 × 10)

7 + 9 → 6, carry 1 (since 7 + 9 = 6 + 1 × 10)

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

 1 1 1 1 1 (carried digits) 0 1 1 0 1 + 1 0 1 1 1 ------------- = 1 0 0 1 0 0 

In this example, two numerals are being added together: 01101₂ (13 decimal) and 10111₂ (23 decimal). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 10₂. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 10₂ again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 11₂. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 100100₂ (36 decimal).

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

Subtraction

Subtraction works in much the same way: 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ...

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 

Subtracting a positive number is equivalent to adding a negative number of equal absolute value; computers typically use two's complement notation to represent negative values. This notation eliminates the need for a separate "subtract" operation. The subtraction can be summarized with this formula: A negative number is a number that is less than zero, such as −2. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... The twos complement of a binary number is defined as the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ...

A - B = A + not B + 1

For further details, see two's complement. The twos complement of a binary number is defined as the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ...

Multiplication

Multiplication in binary is similar to its decimal counterpart. Two numbers A and B can be multiplied by partial products: for each digit in B, the product of that digit in A is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in B that was used. The sum of all these partial products gives the final result. In mathematics, multiplication is an elementary arithmetic operation. ...

Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:

• If the digit in B is 0, the partial product is also 0
• If the digit in B is 1, the partial product is equal to A

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

 1 0 1 1 (A) × 1 0 1 0 (B) --------- 0 0 0 0 ← Corresponds to a zero in B + 1 0 1 1 ← Corresponds to a one in B + 0 0 0 0 + 1 0 1 1 --------------- = 1 1 0 1 1 1 0 

Binary numbers can also be multiplied with bits after a binary point: The binary point is simply the binary equivalent of the decimal point. ...

 1 0 1.1 0 1 (A) (5.625 in decimal) × 1 1 0.0 1 (B) (6.25 in decimal) ------------- 1 0 1 1 0 1 ← Corresponds to a one in B + 0 0 0 0 0 0 ← Corresponds to a zero in B + 0 0 0 0 0 0 + 1 0 1 1 0 1 + 1 0 1 1 0 1 ----------------------- = 1 0 0 0 1 1.0 0 1 0 1 (35.15625 in decimal) 

See also Booth's multiplication algorithm. Booths multiplication algorithm is a multiplication algorithm that multiplies two signed binary numbers in twos complement notation. ...

Division

Binary division is again similar to its decimal counterpart: In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...

 __________ 1 0 1 | 1 1 0 1 1 

Here, the divisor is 101₂, or 5 decimal, while the dividend is 11011₂, or 27 decimal. The procedure is the same as that of decimal long division; 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: In arithmetic, long division is a procedure for calculating the division of one integer, called the dividend, by another integer called the divisor, to produce a result called the quotient. ...

 1 __________ 1 0 1 | 1 1 0 1 1 − 1 0 1 ----- 0 1 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 ----- 0 1 1 − 0 0 0 ----- 1 1 1 − 1 0 1 ----- 1 0 

Thus, the quotient 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, 27 divided by 5 is 5, with a remainder of 2.

Bitwise operations

Main article: bitwise operation

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; 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 left of a binary number is the equivalent of multiplication by a (positive, integral) power of 2. In computer programming, a bitwise operation operates on one or two bit patterns or binary numerals at the level of their individual bits. ... In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ... In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ... In computer programming, a bitwise operation operates on one or two bit patterns or binary numerals at the level of their individual bits. ... AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ... OR logic gate. ... Exclusive disjunction, also known as exclusive or and symbolized by XOR or EOR, is a logical operation on two operands that results in a logical value of true if and only if one of the operands, but not both, has a value of true. ... Negation (i. ... In telecommunication, an arithmetic shift is a shift, applied to the representation of a number in a fixed radix numeration system and in a fixed-point representation system, and in which only the characters representing the fixed-point part of the number are moved. ...

Conversion to and from other numeral systems

Decimal

To convert from a base-10 integer numeral to its base-2 (binary) equivalent, the number is divided by two, and the remainder is the least-significant bit. The (integer) result is again divided by two, its remainder is the next most significant bit. This process repeats until the result of further division becomes zero. The binary representation of decimal 149, with the LSB highlighted. ...

For example, 118₁₀, in binary, is:

Operation	Remainder
118 ÷ 2 = 59	0
59 ÷ 2 = 29	1
29 ÷ 2 = 14	1
14 ÷ 2 = 7	0
7 ÷ 2 = 3	1
3 ÷ 2 = 1	1
1 ÷ 2 = 0	1

Reading the sequence of remainders from the bottom up gives the binary numeral 1110110₂.

This method works for conversion from any base, but there are better methods for bases which are powers of two, such as octal and hexadecimal given below. The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ...

To convert from base-2 to base-10 is the reverse algorithm. The bits of the binary number are used one by one, starting with the most significant bit. Beginning with the value 0, repeatedly double the prior value and add the next bit to produce the next value. This can be organized in a multi-column table. For example to convert 110010101101₂ to decimal:

Prior value	× 2 +	Next Bit	Next value
			= 0
0	× 2 +	1	= 1
1	× 2 +	1	= 3
3	× 2 +	0	= 6
6	× 2 +	0	= 12
12	× 2 +	1	= 25
25	× 2 +	0	= 50
50	× 2 +	1	= 101
101	× 2 +	0	= 202
202	× 2 +	1	= 405
405	× 2 +	1	= 811
811	× 2 +	0	= 1622
1622	× 2 +	1	= 3245

The result is 3245₁₀. This method is an application of the Horner scheme. In the mathematical subfield of numerical analysis, the Horner scheme or Horner algorithm, named after William George Horner, is an algorithm for the efficient evaluation of polynomials in monomial form. ...

 Binary: 1 1 0 0 1 0 1 0 1 1 0 1 Decimal: 1×2^11 + 1×2^10 + 0×2^9 + 0×2^8 + 1×2^7 + 0×2^6 + 1×2^5 + 0×2^4 + 1×2^3 + 1×2^2 + 0×2^1 + 1×2^0 = 3245 

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 .11010110101₂, the first digit is , the second , etc. So if there is a 1 in the first place after the decimal, then the number is at least , 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, ₁₀, in binary, is:

Converting	Result
	0.
	0.0
	0.01
	0.010
	0.0101

Thus the repeating decimal fraction 0.3... is equivalent to the repeating binary fraction 0.01... .

Or for example, 0.1₁₀, in binary, is:

Converting	Result
0.1	0.
0.1 × 2 = 0.2 < 1	0.0
0.2 × 2 = 0.4 < 1	0.00
0.4 × 2 = 0.8 < 1	0.000
0.8 × 2 = 1.6 ≥ 1	0.0001
0.6 × 2 = 1.2 ≥ 1	0.00011
0.2 × 2 = 0.4 < 1	0.000110
0.4 × 2 = 0.8 < 1	0.0001100
0.8 × 2 = 1.6 ≥ 1	0.00011001
0.6 × 2 = 1.2 ≥ 1	0.000110011
0.2 × 2 = 0.4 < 1	0.0001100110

This is also a repeating binary fraction 0.000110011... . 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 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. A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ...

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:

x	=	1100	.101110011100...
	=	1100101110	.0111001110...
	=	11001	.0111001110...
	=	1100010101
x	=	(789/62)10

Another way of converting from binary to decimal, often quicker for a person familiar with hexadecimal, is to do so indirectly—first converting (x in binary) into (x in hexadecimal) and then converting (x in hexadecimal) into (x in decimal). In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ...

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, we divide 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, we split it into two pieces of about the same size, convert each to binary, then multiply the first piece by 10^k and add them, where k is the number of digits in the least-significant (rightmost) piece. This article is about the string operation of computer programming. ...

Hexadecimal

Binary may be converted to and from hexadecimal somewhat more easily. This is because the radix of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 2⁴, so it takes four digits of binary to represent one digit of hexadecimal. In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ...

The following table shows each hexadecimal digit along with the equivalent decimal value and four-digit binary sequence:

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). For example: This article is about padding in fashion. ...

01010010₂ = 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 the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is equivalent to octal 7, and so forth. The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ...

Octal	Binary
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111

Converting from octal to binary proceeds in the same fashion as it does for hexadecimal: In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ...

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 thus means: In mathematics, radix point refers to the symbol used in numerical representations to separate the integral part of the number (to the left of the radix) from its fractional part (to the right of the radix). ... The decimal separator is used to mark the boundary between the integer and the fractional parts of a decimal numeral. ...

1 × 21	(1 × 2 = 2)	plus
1 × 20	(1 × 1 = 1)	plus
0 × 2-1	(0 × ½ = 0)	plus
1 × 2-2	(1 × ¼ = 0.25)

For a total of 3.25 decimal.

All dyadic rational numbers 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 In mathematics, a dyadic fraction or dyadic rational is a rational number that when written as a fraction has denominator a power of two, i. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...

= = 0.0101010101...₂

= = 0.10110100 10110100 10110100...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. 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 2^-1 + 2^-2 + 2^-3 + ... which is 1. For other uses, see Decimal (disambiguation). ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

Binary numerals which neither terminate nor recur represent irrational numbers. For instance, In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ...

• 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 , the square root of 2, another irrational. It has no discernible pattern, although a proof that is irrational requires more than this. See irrational number.

In mathematics, a square root (√) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ...

References

See also

• Two's complement
• Finger binary
• Binary-coded decimal
• Gray code
• linear feedback shift register

The twos complement of a binary number is defined as the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ... Finger binary is a system for counting and displaying binary numbers on the fingers of one or more hands. ... In computing and electronic systems, binary-coded decimal (BCD) is an encoding for decimal numbers in which each digit is represented by its own binary sequence. ... The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one digit. ... A linear feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state. ...

External links

• A brief overview of Leibniz and the connection to binary numbers
• Binary System at cut-the-knot
• Conversion of Fractions at cut-the-knot
• Binary Digits at Math Is Fun
• How to Convert from Decimal to Binary at wikiHow

cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ... cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ... Math Is Fun (or Maths Is Fun in British English) is an educational website maintained by Rod Pierce devoted to the concept that mathematics is, indeed, fun. ... wikiHow is a wiki-based community with a database of how-to guides. ...