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), the quantity of information required to distinguish two mutually exclusive states 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, NOT Tautology: 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  pq p  q  p  q(p  q)  (  p  q) The circuit diagram for a binary half adder, which adds two bits together, producing sum and carry bits. p  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  0×2 0 = 0 1  1×2 0 = 1 10  1× ×2 0 = 2 11  1× ×2 0 =  1× × ×2 0 =  1× × × × × × ×2 0 = 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 = ) = =  ) = 23 10

← ) ← ← – 0 = 0; 0 – 1 = 1 (with borrow); 1 – 0 = 1; 1 – 1 = ←  ) ← ← 87 10

PC Keyboard Commands: 8-bit binary ASCII code (American Standard Code for Information Interchange) A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a b c d e f g h i j k l m

A Brief History: Indian mathematician: Pingala Chinese: BaGua traditional African divination systems: Ifá Francis Bacon Gottfried Leibniz George Boole Claude Shannon George Stibitz Alan Turing

The ancient Indian mathematican Pingala 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 Gbe-MejiI I II Yeku-MejiII II IIII Woli-MejiII I III Di-MejiI II III Abla-MejiI II II II Akla-MejiII II III Loso-MejiI I IIII Wele-MejiII II II Guda-MejiI I I II Sa-MejiII I I I Lete-MejiI I II I Tula-MejiI II I I Turukpe-Meji II II I II ka-Maji II I II II Ce-Meji I III II Fu-Meji II III 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 1561 Death: 9 April 1626 School/Tradition: British Empiricism Field: English philosopher Achievement: knighted in 1603 Best Known As: a philosophical advocate and defender of the scientific revolution Sir Francis Bacon a AAAAA g AABBA n ABBAA t BAABA b AAAAB h AABBB o ABBAB u-v BAABB c AAABA i-j ABAAA p ABBBA w BABAA d AAABB k ABAAB q ABBBB x BABAB e AABAA l ABABA r BAAAA y BABBA f 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 Leipzig, Saxony Died: November 14, 1716 Hannover, Hanover Residence: Germany Nationality: German Field: mathematician and philosopher Institution: University of Leipzig Alma Mater: University of Altdorf Academic Advisor: Erhard Weigel Notable Students: Jacob Bernoulli Known 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 science Main Interests: mathematics, logic, philosophy of mathematics Notable Ideas: Boolean algebra Influences: Aristotle, Spinoza, Newton Influenced: 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, 2001 School: University of Michigan 1932–1936 MIT 1936–1940 Field: mathematics, electrical engineering, Notable Ideas and Influences: analog computer, digital circuitry, data and signal processing Award: 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, 1995 School: Bachelor's degree: Denison University in Granville, Ohio, Master's degree: Union College in 1927 Ph.D.in mathematical physics: Cornell University : in 1930 Field: mathematics, computer science, Notable Ideas and Influences: in 1940 used 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, 1954 School: King's College, Cambridge 1931–1934 Princeton University 1937–1938 Field: mathematics, cryptographer Notable 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

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