Binary: Binary basically means “twos”, two parts or two pieces. The binary system is also known as the base-2 system. The binary representation of any.

Binary:

Binary basically means “twos”, two parts or two pieces.The binary system is also known as the base-2 system. The binary representation of any number has only two digits, 0 and 1.

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.

A bit of storage is like a light switch; it can be either on (1) or off (0), thequantity of information required to distinguish two mutually exclusivestates from each other.Bits can be represented in many forms. For example:circuitry (electrical levels)tapes, cassettes (magnetically)CD-ROMs, CDs (pits, grounds)

Other Applications:

Computer: bit operation using Boolean logic operators: AND, OR, XOR, NOTTautology: statement of propositional logic two value principle: true (1)

false(0).

e.g. p : Let p be the proposition “The book is in the library” p : Then its negation p means “The book is not in the library”

p ( p) : “The book is or is not in the library” is a true proposition.

p ( p) : “The book is and is not in the library” is a false proposition.

[ p ( p) ] : “It is false that the book both is and is not in the library” is true.

p p q pq pq (pq)( pq)1 0 1 1 1 10 1 0 0 1 01 0 0 0 0 00 1 1 0 1 0

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

p

p

q

q

Conversion between decimal and binary:

Counting in binary starts with the first digit, using 0 or 1, then move to the next higher digit to the left. e.g. 0 1 10 11 101 1001101

0 0×20 = 0

1 1×20 = 1

10 1×21 + 0×20 = 2

11 1×21 + 1×20 = 3

101 1×22 + 0×21 + 1×20 = 5

1001101 1×26 + 0×25 + 0×24 + 1×23

+ 1×22 + 0×21 + 1×20

= 77

Binary Arithmetic: After a digit reaches 1 in binary, an

increment resets it to 0 and at same

time carries an increment of the next

digit to it left.

e.g. 011012 = 1310

+) 101112 = 2310

11011102 = 11010

) 101112 = 2310

0 1 1 0 1 ← 1310

+) 1 0 1 1 1 ← 2310

1 0 0 1 0 0 ← 3610

0 – 0 = 0; 0 – 1 = 1 (with borrow);

1 – 0 = 1; 1 – 1 = 0

1 1 0 1 1 1 0 ← 11010

) 1 0 1 1 1 ← 2310

1 0 1 0 1 1 1 ← 8710

PC Keyboard Commands: 8-bit binary

ASCII code (American Standard Code

for Information Interchange).

65 01000001 A66 01000010 B67 01000011 C68 01000100 D69 01000101 E70 01000110 F71 01000111 G72 01001000 H73 01001001 I74 01001010 J75 01001011 K76 01001100 L77 01001101 M

78 01001110 N79 01001111 O80 01010000 P81 01010001 Q82 01010010 R83 01010011 S84 01010100 T85 01010101 U86 01010110 V87 01010111 W88 01011000 X89 01011001 Y90 01011010 Z

97 01100001 a98 01100010 b99 01100011 c100 01100100 d101 01100101 e102 01100110 f103 01100111 g104 01101000 h105 01101001 i106 01101010 j107 01101011 k108 01101100 l109 01101101 m

A Brief History:

Indian mathematician: PingalaChinese: BaGuatraditional African divination systems: IfáFrancis BaconGottfried LeibnizGeorge BooleClaude ShannonGeorge StibitzAlan Turing

The ancient Indian mathematicanPingala presented the first known description of a binary numeral system around 800 BC.

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.

Sixteen Principal Afa-du(Yeveh Vodoun) Name 1 2 3 4Gbe-Meji I I I IYeku-Meji II II II IIWoli-Meji II I I IIDi-Meji I II II I

Abla-Meji I II II IIAkla-Meji II II II ILoso-Meji I I II IIWele-Meji II II I I

Guda-Meji I I I IISa-Meji II I I ILete-Meji I I II ITula-Meji I II I I

Turukpe-Meji II II I IIka-Maji II I II IICe-Meji I II I IIFu-Meji II I II I

In 1605 Francis Bacon used a system by which letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as random text.

Birth: 22 January 1561Death: 9 April 1626School/Tradition: British EmpiricismField: English philosopherAchievement: knighted in 1603Best Known As: a philosophical advocate and defender of the scientific revolution

Sir Francis Bacon

a AAAAA g AABBA n ABBAA t BAABAb AAAAB h AABBB o ABBAB u-v BAABBc AAABA i-j ABAAA p ABBBA w BABAAd AAABB k ABAAB q ABBBB x BABABe AABAA l ABABA r BAAAA y BABBAf AABAB m ABABB s BAAAB z BABBB

Bacon's cipher:

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.

Born: July 1 1646 Leipzig, SaxonyDied: November 14, 1716 Hannover, HanoverResidence: GermanyNationality: GermanField: mathematician and philosopherInstitution: University of LeipzigAlma Mater: University of AltdorfAcademic Advisor: Erhard WeigelNotable Students: Jacob BernoulliKnown For: infinitesimal calculus, calculus, monad, theodicy, optimism

Gottfried Wilhelm von Leibniz

In 1854, British mathematician George Boole published a landmark paper detailing a system of logic that would become known as Boolean algebra. His logical system proved instrumental in the development of the binary system, particularly in its implementation in electronic circuitry.

Birth: November 2, 1815 (Lincoln, Lincolnshire, England)Death: December 8, 1864 (Ballintemple, County Cork, Ireland)School/Tradition: mathematical foundations of computer scienceMain Interests: mathematics, logic, philosophy of mathematicsNotable Ideas: Boolean algebraInfluences: Aristotle, Spinoza, NewtonInfluenced: modern computer scientists: Jevons, De Morgan, Peirce, Johnson, Shannon

George Boole

the founders of the field of computer science

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.

Birth: April 30, 1916 (Petoskey, Michigan)Death: February 24, 2001School: University of Michigan 1932–1936

MIT 1936–1940

Field: mathematics, electrical engineering,Notable Ideas and Influences: analog computer, digital circuitry, data and signal processingAward: Alfred Noble Prize, 1940 Morris Liebmann Memorial Award, 1949 Research Corporation Award, 1956 Golden Plate Award, 1967 Joseph Jacquard Award, 1978 Harold Pender Award, 1978 Audio Engineering Society Gold Medal, 1985 Eduard Rhein Prize, 1991 National Inventors Hall of Fame inducted, 2004

Claude Elwood Shannon

the father of information theory

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. 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.

George Robert Stibitz

the father of modern digital computer

Birth: April 20, 1904 (York, Pennsylvania)Death: January 31, 1995School: Bachelor's degree: Denison University in Granville, Ohio, Master's degree: Union College in 1927 Ph.D.in mathematical physics: Cornell University : in 1930Field: mathematics, computer science,Notable Ideas and Influences: in 1940used a teletype to send commands remotely to the Complex Number Calculator (a computing machine) in New York over telephone lines.

In 1931, Alan Turing reformulated Kurt Gödel's results on the limits of proof and computation, substituting Gödel's universal arithmetic-based formal language by what are now called Turing machines.

The Turing Machine that he envisioned is essentially the same as today's multi-purpose computers. He described a machine that would read a series of ones and zeros from a tape. These ones and zeros described the steps that needed to be done to solve a particular problem or perform a certain task.

Birth: June 23, 1912 (London)Death: June 7, 1954School: King's College, Cambridge 1931–1934 Princeton University 1937–1938 Field: mathematics, cryptographerNotable Ideas and Influences: *1936: Submitted momentous paper: “On Computable Numbers with an Application to the Entscheidungsproblem” creating Turing machines for study in theory of computation.*Worked on Government Code and Cypher School. *Devised an electromechanical machine “the bombe” which helped to break cipher machine “Enigma”.*Co-designed a portable machine to allow secure voice communications.*1950 Wrote paper: describing “the Turing Test“.*1952: Published paper: “The Chemical Basis of Morphogenesis” putting forth the Turing hypothesis of pattern formation.*1945: awarded the OBE

Alan Mathison Turing

the father of modern computer science

replica of a bombe machine

German military Enigma machine

Decoding:

01001101 01000001

01010100 01001000

Decoding:

01001101 01000001

77 65

01010100 01001000

84 72

Decoding:

01001101 01000001

77 65

M A

01010100 01001000

84 72

T H