Binary:

1
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.

2

• 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.

3

• 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)

4
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).

5

• 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.

6
The circuit diagram for a binary half adder,
which adds two bits together, producing sum and
carry bits.
p ? p q p?q ?p?q (p?q)?(
?p?q) 1 0 1 1 1 1 0 1 0 0 1
0 1 0 0 0 0 0 0 1 1 0 1 0
7
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

8

• 0 ? 020 0
• 1 ? 120 1
• 10 ? 121 020 2
• 11 ? 121 120 3
• 101 ? 122 021 120 5
• 1001101 ? 126 025 024 123
• 122 021
  120
• 77

9
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

10

• 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

11

• PC Keyboard Commands 8-bit binary
• ASCII code (American Standard Code
• for Information Interchange).

78 01001110 N 79 01001111 O 80
01010000 P 81 01010001 Q 82
01010010 R 83 01010011 S 84
01010100 T 85 01010101 U 86
01010110 V 87 01010111 W 88
01011000 X 89 01011001 Y 90 01011010
Z
97 01100001 a 98 01100010 b 99
01100011 c 100 01100100 d 101
01100101 e 102 01100110 f 103
01100111 g 104 01101000 h 105
01101001 i 106 01101010 j 107
01101011 k 108 01101100 l 109
01101101 m

• 65 01000001 A
• 66 01000010 B
• 67 01000011 C
• 68 01000100 D
• 69 01000101 E
• 70 01000110 F
• 71 01000111 G
• 72 01001000 H
• 73 01001001 I
• 74 01001010 J
• 75 01001011 K
• 76 01001100 L
• 77 01001101 M

12
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

13

• 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.

14

• 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 4 Gbe-Meji I I
I I Yeku-Meji II II II II Woli-Meji II
I I II Di-Meji I II II I
Abla-Meji I II II II Akla-Meji II
II II I Loso-Meji I I
II II Wele-Meji II II I I
Guda-Meji I I I II Sa-Meji II
I I I Lete-Meji I I II
I Tula-Meji I II I I
Turukpe-Meji II II I
II ka-Maji II I II
II Ce-Meji I II I II Fu-Meji
II I II I
15

• 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.

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• 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.

18
Born July 1 1646 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
19

• 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.

20
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
21

• 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.

22
Birth April 30, 1916 (Petoskey,
Michigan) Death February 24, 2001 School
University of Michigan 19321936 MIT
19361940 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
23

• 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.

24
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.
George Robert Stibitz the father of modern
digital computer
25

• 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.

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• 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.

27
Birth June 23, 1912 (London) Death June
7, 1954 School King's College, Cambridge
19311934 Princeton University
19371938 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-desig
ned 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
28
replica of a bombe machine
German military Enigma machine
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• Decoding
• 01001101 01000001
• 01010100 01001000

30

• Decoding
• 01001101 01000001
• 77 65
• 01010100 01001000
• 84 72

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• Decoding
• 01001101 01000001
• 77 65
• M A
• 01010100 01001000
• 84 72
• T H