binary numeral system

8/3/2019 Binary Numeral System

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Binary numeral system

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Numeral systemsby culture

Hindu-Arabic numerals

Western Arabic

Eastern Arabic

Indian family

Burmese

Khmer

Mongolian

Thai

East Asian numerals

Chinese

JapaneseSuzhou

Korean

VietnameseCounting rods

Alphabetic numerals

AbjadArmenian

ryabhaaCyrillic

Ge'ezGreek(Ionian)

Hebrew

Other systems

Aegean

AtticBabylonian

Brahmi

Egyptian

Etruscan

Inuit

MayanQuipu

Roman

Sumerian

Urnfield

List of numeral system topics

Positional systemsbybase

Decimal(10)
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1,2,3,4,5,8,12,16,20,60more

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The binary numeral system, or base-2 number system, represents numeric values using twosymbols,0and1. More specifically, the usualbase-2system is apositional notationwith aradix

of 2. Owing to its straightforward implementation indigital electronic circuitryusinglogic gates,the binary system is used internally by all moderncomputers.

Contents

[hide]

1 History 2 Representation 3 Counting in binary 4 Fractions in binary 5 Binary arithmetic

o 5.1 Addition 5.1.1 Addition table

o 5.2 Subtractiono 5.3 Multiplication

5.3.1 Multiplication tableo 5.4 Division

6 Bitwise operations 7 Conversion to and from other numeral systemso 7.1 Decimal

o 7.2 Hexadecimalo 7.3 Octal

8 Representing real numbers 9 On-line live converters 10 See also 11 Notes 12 References 13 External links

[edit] History

The Indian scholarPingala(circa5th2nd centuries BC) developed advanced mathematical

concepts for describingprosody, and in so doing presented the first known description of abinary numeral system.[1][2]He used binary numbers in the form of short and long syllables (the

latter equal in length to two short syllables), making it similar toMorse code.[3][4]
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iki/Binary_numeral_system#Hexadecimalhttp://en.wikipedia.org/wiki/Binary_numeral_system#Octalhttp://en.wikipedia.org/wiki/Binary_numeral_system#Octalhttp://en.wikipedia.org/wiki/Binary_numeral_system#Representing_real_numbershttp://en.wikipedia.org/wiki/Binary_numeral_system#Representing_real_numbershttp://en.wikipedia.org/wiki/Binary_numeral_system#On-line_live_convertershttp://en.wikipedia.org/wiki/Binary_numeral_system#On-line_live_convertershttp://en.wikipedia.org/wiki/Binary_numeral_system#See_alsohttp://en.wikipedia.org/wiki/Binary_numeral_system#See_alsohttp://en.wikipedia.org/wiki/Binary_numeral_system#Noteshttp://en.wikipedia.org/wiki/Binary_numeral_system#Noteshttp://en.wikipedia.org/wiki/Binary_numeral_system#Referenceshttp://en.wikipedia.org/wiki/Binary_numeral_system#Referenceshttp://en.wikipedia.org/wiki/Binary_numeral_system#External_linkshttp://en.wikipedia.org/wiki/Binary_numeral_system#External_linkshttp://en.wikipedia.org/w/index.php?title=Binary_numeral_system&action=edit&section=1http://en.wikipedia.org/w/index.php?title=Binary_numeral_system&action=edit&section=1http://en.wikipedia.org/wiki/Pingalahttp://en.wikipedia.org/wiki/Pingalahttp://en.wikipedia.org/wiki/Pingalahttp://en.wikipedia.org/wiki/Circahttp://en.wikipedia.org/wiki/Circahttp://en.wikipedia.org/wiki/Circahttp://en.wikipedia.org/wiki/Prosody_(poetry)http://en.wikipedia.org/wiki/Prosody_(poetry)http://en.wikipedia.org/wiki/Prosody_(poetry)http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-0http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-0http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-0http://en.wikipedia.org/wiki/Morse_codehttp://en.wikipedia.org/wiki/Morse_codehttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-2http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-2http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-2http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-2http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-2http://en.wikipedia.org/wiki/Morse_codehttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-0http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-0http://en.wikipedia.org/wiki/Prosody_(poetry)http://en.wikipedia.org/wiki/Circahttp://en.wikipedia.org/wiki/Pingalahttp://en.wikipedia.org/w/index.php?title=Binary_numeral_system&action=edit&section=1http://en.wikipedia.org/wiki/Binary_numeral_system#External_linkshttp://en.wikipedia.org/wiki/Binary_numeral_system#Referenceshttp://en.wikipedia.org/wiki/Binary_numeral_system#Noteshttp://en.wikipedia.org/wiki/Binary_numeral_system#See_alsohttp://en.wikipedia.org/wiki/Binary_numeral_system#On-line_live_convertershttp://en.wikipedia.org/wiki/Binary_numeral_system#Representing_real_numbershttp://en.wikipedia.org/wiki/Binary_numeral_system#Octalhttp://en.wikipedia.org/wiki/Binary_numeral_system#Hexadecimalhttp://en.wikipedia.org/wiki/Binary_numeral_system#Decimalhttp://en.wikipedia.org/wiki/Binary_numeral_system#Conversion_to_and_from_other_numeral_systemshttp://en.wikipedia.org/wiki/Binary_numeral_system#Bitwise_operationshttp://en.wikipedia.org/wiki/Binary_numeral_system#Divisionhttp://en.wikipedia.org/wiki/Binary_numeral_system#Multiplication_tablehttp://en.wikipedia.org/wiki/Binary_numeral_system#Multiplicationhttp://en.wikipedia.org/wiki/Binary_numeral_system#Subtractionhttp://en.wikipedia.org/wiki/Binary_numeral_system#Addition_tablehttp://en.wikipedia.org/wiki/Binary_numeral_system#Additionhttp://en.wikipedia.org/wiki/Binary_numeral_system#Binary_arithmetichttp://en.wikipedia.org/wiki/Binary_numeral_system#Fractions_in_binaryhttp://en.wikipedia.org/wiki/Binary_numeral_system#Counting_in_binaryhttp://en.wikipedia.org/wiki/Binary_numeral_system#Representationhttp://en.wikipedia.org/wiki/Binary_numeral_system#Historyhttp://en.wikipedia.org/wiki/Binary_numeral_systemhttp://en.wikipedia.org/wiki/Computerhttp://en.wikipedia.org/wiki/Logic_gatehttp://en.wikipedia.org/wiki/Digital_electronic_circuithttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Positional_notationhttp://en.wikipedia.org/wiki/2_(number)http://en.wikipedia.org/wiki/Base_(mathematics)http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/0_(number)http://en.wikipedia.org/w/index.php?title=Template:Numeral_systems&action=edithttp://en.wikipedia.org/wiki/Template_talk:Numeral_systemshttp://en.wikipedia.org/wiki/Template:Numeral_systemshttp://en.wikipedia.org/wiki/Category:Positional_numeral_systemshttp://en.wikipedia.org/wiki/Sexagesimalhttp://en.wikipedia.org/wiki/Vigesimalhttp://en.wikipedia.org/wiki/Hexadecimalhttp://en.wikipedia.org/wiki/Duodecimalhttp://en.wikipedia.org/wiki/Octalhttp://en.wikipedia.org/wiki/Quinaryhttp://en.wikipedia.org/wiki/Quaternary_numeral_systemhttp://en.wikipedia.org/wiki/Ternary_numeral_systemhttp://en.wikipedia.org/wiki/Unary_numeral_system


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A set of eighttrigramsand a set of 64hexagrams, analogous to the three-bit and six-bit binary

numerals, were known inancient Chinathrough theclassic textI Ching. In the 11th century,scholar and philosopherShao Yongdeveloped a method for arranging the hexagrams which

corresponds to the sequence 0 to 63, as represented in binary, with yin as 0, yang as 1 and the

least significant biton top. There is, however, no evidence that Shao understood binary

computation. The ordering is also thelexicographical orderonsextuplesof elements chosenfrom a two-element set.[5]

Similar sets of binary combinations have also been used in traditionalAfricandivination systems

such asIfas well as inmedievalWesterngeomancy. The base-2 system utilized in geomancy

had long been widely applied in sub-Saharan Africa.

In 1605Francis Bacondiscussed 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.[6]

Importantly for the general theory of binary encoding, he added that thismethod 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, andany instruments of like nature".[6]

(SeeBacon's cipher.)

The modern binary number system was fully documented byGottfried Leibnizin the17th

centuryin his articleExplication de l'Arithmtique Binaire. Leibniz's system uses 0 and 1, likethe modern binary numeral system. As aSinophile, Leibniz was aware of theI Ching and 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 ofphilosophicalmathematicshe admired.[7]

In 1854,BritishmathematicianGeorge Boolepublished a landmark paper detailing analgebraic

system oflogicthat would become known asBoolean algebra. His logical calculus was tobecome instrumental in the design of digital electronic circuitry.[8]

In 1937,Claude Shannonproduced his master's thesis atMITthat implemented Boolean algebra

and binary arithmetic using electronic relays and switches for the first time in history. EntitledA

Symbolic Analysis of Relay and Switching Circuits, Shannon's thesis essentially foundedpracticaldigital circuitdesign.[9]

In November 1937,George Stibitz, then working atBell Labs, completed a relay-based computerhe dubbed the "Model K" (for "Kitchen", where he had assembled it), which calculated using

binary addition.[10]Bell Labs thus authorized a full research programme 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 theAmerican Mathematical Societyconference atDartmouth Collegeon September 11, 1940, Stibitz was able to send the Complex Number

Calculator remote commands over telephone lines by ateletype. It was the first computing

machine ever used remotely over a phone line. Some participants of the conference whowitnessed the demonstration wereJohn Von Neumann,John MauchlyandNorbert Wiener, who

wrote about it in his memoirs.[11][12][13]
http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-2http://en.wikipedia.org/wiki/Ba_guahttp://en.wikipedia.org/wiki/Ba_guahttp://en.wikipedia.org/wiki/Ba_guahttp://en.wikipedia.org/wiki/Hexagram_(I_Ching)http://en.wikipedia.org/wiki/Hexagram_(I_Ching)http://en.wikipedia.org/wiki/Hexagram_(I_Ching)http://en.wikipedia.org/wiki/Zhou_Dynastyhttp://en.wikipedia.org/wiki/Zhou_Dynastyhttp://en.wikipedia.org/wiki/Zhou_Dynastyhttp://en.wikipedia.org/wiki/Chinese_classic_textshttp://en.wikipedia.org/wiki/Chinese_classic_textshttp://en.wikipedia.org/wiki/I_Chinghttp://en.wikipedia.org/wiki/I_Chinghttp://en.wikipedia.org/wiki/I_Chinghttp://en.wikipedia.org/wiki/Shao_Yonghttp://en.wikipedia.org/wiki/Shao_Yonghttp://en.wikipedia.org/wiki/Shao_Yonghttp://en.wikipedia.org/wiki/Least_significant_bithttp://en.wikipedia.org/wiki/Least_significant_bithttp://en.wikipedia.org/wiki/Lexicographical_orderhttp://en.wikipedia.org/wiki/Lexicographical_orderhttp://en.wikipedia.org/wiki/Lexicographical_orderhttp://en.wikipedia.org/wiki/Sextuplehttp://en.wikipedia.org/wiki/Sextuplehttp://en.wikipedia.org/wiki/Sextuplehttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-4http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-4http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-4http://en.wikipedia.org/wiki/Africanhttp://en.wikipedia.org/wiki/Africanhttp://en.wikipedia.org/wiki/Africanhttp://en.wikipedia.org/wiki/If%C3%A1http://en.wikipedia.org/wiki/If%C3%A1http://en.wikipedia.org/wiki/If%C3%A1http://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/Geomancy#Divination.2C_Geomancy.2C_fractals_and_modern_computershttp://en.wikipedia.org/wiki/Geomancy#Divination.2C_Geomancy.2C_fractals_and_modern_computershttp://en.wikipedia.org/wiki/Geomancy#Divination.2C_Geomancy.2C_fractals_and_modern_computershttp://en.wikipedia.org/wiki/Francis_Baconhttp://en.wikipedia.org/wiki/Francis_Baconhttp://en.wikipedia.org/wiki/Francis_Baconhttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-Bacon1605-5http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-Bacon1605-5http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-Bacon1605-5http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-Bacon1605-5http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-Bacon1605-5http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-Bacon1605-5http://en.wikipedia.org/wiki/Bacon%27s_cipherhttp://en.wikipedia.org/wiki/Bacon%27s_cipherhttp://en.wikipedia.org/wiki/Bacon%27s_cipherhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/17th_centuryhttp://en.wikipedia.org/wiki/17th_centuryhttp://en.wikipedia.org/wiki/17th_centuryhttp://en.wikipedia.org/wiki/17th_centuryhttp://en.wikipedia.org/wiki/Explication_de_l%27Arithm%C3%A9tique_Binairehttp://en.wikipedia.org/wiki/Explication_de_l%27Arithm%C3%A9tique_Binairehttp://en.wikipedia.org/wiki/Explication_de_l%27Arithm%C3%A9tique_Binairehttp://en.wikipedia.org/wiki/Sinophilehttp://en.wikipedia.org/wiki/Sinophilehttp://en.wikipedia.org/wiki/Sinophilehttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-6http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-6http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-6http://en.wikipedia.org/wiki/United_Kingdomhttp://en.wikipedia.org/wiki/United_Kingdomhttp://en.wikipedia.org/wiki/United_Kingdomhttp://en.wikipedia.org/wiki/George_Boolehttp://en.wikipedia.org/wiki/George_Boolehttp://en.wikipedia.org/wiki/George_Boolehttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Logichttp://en.wikipedia.org/wiki/Logichttp://en.wikipedia.org/wiki/Logichttp://en.wikipedia.org/wiki/Boolean_algebra_(logic)http://en.wikipedia.org/wiki/Boolean_algebra_(logic)http://en.wikipedia.org/wiki/Boolean_algebra_(logic)http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-7http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-7http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-7http://en.wikipedia.org/wiki/Claude_Shannonhttp://en.wikipedia.org/wiki/Claude_Shannonhttp://en.wikipedia.org/wiki/Claude_Shannonhttp://en.wikipedia.org/wiki/MIThttp://en.wikipedia.org/wiki/MIThttp://en.wikipedia.org/wiki/MIThttp://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_and_Switching_Circuitshttp://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_and_Switching_Circuitshttp://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_and_Switching_Circuitshttp://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_and_Switching_Circuitshttp://en.wikipedia.org/wiki/Digital_circuithttp://en.wikipedia.org/wiki/Digital_circuithttp://en.wikipedia.org/wiki/Digital_circuithttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-8http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-8http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-8http://en.wikipedia.org/wiki/George_Stibitzhttp://en.wikipedia.org/wiki/George_Stibitzhttp://en.wikipedia.org/wiki/George_Stibitzhttp://en.wikipedia.org/wiki/Bell_Labshttp://en.wikipedia.org/wiki/Bell_Labshttp://en.wikipedia.org/wiki/Bell_Labshttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-9http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-9http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-9http://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/American_Mathematical_Societyhttp://en.wikipedia.org/wiki/American_Mathematical_Societyhttp://en.wikipedia.org/wiki/American_Mathematical_Societyhttp://en.wikipedia.org/wiki/Dartmouth_Collegehttp://en.wikipedia.org/wiki/Dartmouth_Collegehttp://en.wikipedia.org/wiki/Teletypehttp://en.wikipedia.org/wiki/Teletypehttp://en.wikipedia.org/wiki/Teletypehttp://en.wikipedia.org/wiki/John_Von_Neumannhttp://en.wikipedia.org/wiki/John_Von_Neumannhttp://en.wikipedia.org/wiki/John_Von_Neumannhttp://en.wikipedia.org/wiki/John_Mauchlyhttp://en.wikipedia.org/wiki/John_Mauchlyhttp://en.wikipedia.org/wiki/John_Mauchlyhttp://en.wikipedia.org/wiki/Norbert_Wienerhttp://en.wikipedia.org/wiki/Norbert_Wienerhttp://en.wikipedia.org/wiki/Norbert_Wienerhttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-10http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-10http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-12http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-12http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-12http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-10http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-10http://en.wikipedia.org/wiki/Norbert_Wienerhttp://en.wikipedia.org/wiki/John_Mauchlyhttp://en.wikipedia.org/wiki/John_Von_Neumannhttp://en.wikipedia.org/wiki/Teletypehttp://en.wikipedia.org/wiki/Dartmouth_Collegehttp://en.wikipedia.org/wiki/American_Mathematical_Societyhttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-9http://en.wikipedia.org/wiki/Bell_Labshttp://en.wikipedia.org/wiki/George_Stibitzhttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-8http://en.wikipedia.org/wiki/Digital_circuithttp://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_and_Switching_Circuitshttp://en.wikipedia.org/wiki/A_Symbolic_Analysis_of_Relay_and_Switching_Circuitshttp://en.wikipedia.org/wiki/MIThttp://en.wikipedia.org/wiki/Claude_Shannonhttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-7http://en.wikipedia.org/wiki/Boolean_algebra_(logic)http://en.wikipedia.org/wiki/Logichttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/George_Boolehttp://en.wikipedia.org/wiki/United_Kingdomhttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-6http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Sinophilehttp://en.wikipedia.org/wiki/Explication_de_l%27Arithm%C3%A9tique_Binairehttp://en.wikipedia.org/wiki/17th_centuryhttp://en.wikipedia.org/wiki/17th_centuryhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Bacon%27s_cipherhttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-Bacon1605-5http://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-Bacon1605-5http://en.wikipedia.org/wiki/Francis_Baconhttp://en.wikipedia.org/wiki/Geomancy#Divination.2C_Geomancy.2C_fractals_and_modern_computershttp://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/If%C3%A1http://en.wikipedia.org/wiki/Africanhttp://en.wikipedia.org/wiki/Binary_numeral_system#cite_note-4http://en.wikipedia.org/wiki/Sextuplehttp://en.wikipedia.org/wiki/Lexicographical_orderhttp://en.wikipedia.org/wiki/Least_significant_bithttp://en.wikipedia.org/wiki/Shao_Yonghttp://en.wikipedia.org/wiki/I_Chinghttp://en.wikipedia.org/wiki/Chinese_classic_textshttp://en.wikipedia.org/wiki/Zhou_Dynastyhttp://en.wikipedia.org/wiki/Hexagram_(I_Ching)http://en.wikipedia.org/wiki/Ba_gua


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[edit] Representation

A binary number can be represented by any sequence ofbits(binary digits), which in turn maybe represented by any mechanism capable of being in two mutually exclusive states. The

following sequences of symbols could all be interpreted as the binary numeric value of667:

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

Abinary clockmight useLEDsto express binary values. In this clock, each column of LEDs

shows abinary-coded decimalnumeral of the traditionalsexagesimaltime.

The numeric value represented in each case is dependent upon the value assigned to eachsymbol. In a computer, the numeric values may be represented by two differentvoltages; on a

magneticdisk, magneticpolaritiesmay 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 usingArabic numerals, binary numbersare commonly written using the symbols 0 and 1. When written, binary numerals are often

subscripted, prefixed or suffixed in order to indicate their base, or radix. The following notations

are equivalent:

100101 binary (explicit statement of format)

100101b (a suffix indicating binary format)

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 fromdecimal 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 to refer to the numeral asone hundred.

[edit] Counting in binary

Decimal Binary

0 0

1 1

2 10

3 11
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Counting in binary is similar to counting in any other number system.

Beginning with a single digit, counting proceeds through each symbol, inincreasing 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 theleft) is incremented, and counting starts over at 0. Indecimal, counting

proceeds like so:

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, ...

Since binary is a base-2 system, each digit represents an increasing power of 2, with therightmost digit representing 20, the next representing 21, then 22, and so on. To determine the

decimal representation of a binary number simply take the sum of the products of the binary

digits and the powers of 2 which they represent. For example, the binary number:

100101

is converted to decimal form by:

[(1) 2

5

] + [(0) 2

4

] + [(0) 2

3

] + [(1) 2

2

] + [(0) 2

1

] + [(1) 2

0

] =

[1 32] + [0 16] + [0 8] + [1 4] + [0 2] + [1 1] = 37

To create higher numbers, additional digits are simply added to the left side of the binary

representation.

[edit] Fractions in binary

4 100

5 101

6 110

7 111

8 1000

9 1001

10 1010

11 1011

12 1100

13 1101

14 1110

15 1111

16 10000
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Fractions in binary only terminate if the denominator has2as the onlyprime factor. As a result,

does not have a finite binary representation, and this causes 10(0.1) to be not precisely equal to 1infloating point arithmetic. As an example, to interpret the binary expression for 1/3 =

.010101..., this means: 1/3 = 0 21

+ 1 22

+ 0 23

+ 1 24

+ ... = 0.3125 + ... An exact

value cannot be found with a sum of a finite number of inverse powers of two, and zeros and

ones alternate forever.

Fraction Decimal Binary Fractional Approx.

1/1 1 or 0.9999... 1 or 0.1111... 1/1

1/2 0.5 0.1 1/2

1/3 0.333... 0.010101... 1/4+1/16+1/64...

1/4 0.25 0.01 1/4

1/5 0.2 0.00110011... 1/8+1/16+1/128...

1/6 0.1666... 0.0010101... 1/8+1/32+1/128...

1/7 0.142857142857... 0.001001... 1/8+1/64+1/512...

1/8 0.125 0.001 1/8

1/9 0.111... 0.000111000111... 1/16+1/32+1/64...

1/10 0.1 0.000110011... 1/16+1/32+1/256...

1/11 0.090909... 0.0001011101000101101... 1/16+1/64+1/128...

1/12 0.08333... 0.00010101... 1/16+1/64+1/256...

1/13 0.07692376923... 0.000100111011000100111011... 1/16+1/128+1/256...

1/14 0.0714285714285... 0.0001001001... 1/16+1/128+1/1024...

1/15 0.0666... 0.00010001... 1/16+1/256...

1/16 0.0625 0.0001 1/16

[edit] Binary arithmetic

Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction,

multiplication, and division can be performed on binary numerals.

[edit] Addition

Thecircuit diagramfor a binaryhalf adder, which adds two bits together, producing sum and

carry bits.

The simplest arithmetic operation in binary isaddition. Adding two single-digit binary numbers

is relatively simple, using a form of carrying:

0 + 0 0

0 + 1 11 + 0 1

1 + 1 10, carry 1 (since 1 + 1 = 0 + 1 binary 10)
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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, theprocedure 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

binary numeral system

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