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Jawaharlal nehru technological university anantapur college of engineering
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LECTURE NOTES
ON
SWITCHING THEORY AND LOGIC DESIGN
2016 - 2017
II B.Tech II semester (JNTUH-R15)
ELECTRONICS AND COMMUNICATION ENGINEERING

UNIT - 1
NUMBER SYSTEMS & CODES
NUMBER SYSTEMS & CODES
Philosophy of number systems
Complement representation of negative numbers
Binary arithmetic
Binary codes
Error detecting & error correcting codes
Hamming codes
HISTORY OF THE NUMERAL SYSTEMS:
A numeral system (or system of numeration) is a linguistic system and mathematical notation for
representing numbers of a given set by symbols in a consistent manner. For example, It allows the
numeral "11" to be interpreted as the binary numeral for three, the decimal numeral for eleven, or other
numbers in different bases.
Ideally, a numeral system will:
Represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers)
Give every number represented a unique representation (or at least a standard representation)
Reflect the algebraic and arithmetic structure of the numbers.
For example, the usual decimal representation of whole numbers gives every whole number a unique
representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction,
multiplication and division) being present as the standard algorithms of arithmetic. However, when
decimal representation is used for the rational or real numbers, the representation is no longer unique:
many rational numbers have two numerals, a standard one that terminates, such as 2.31, and another that
recurs, such as 2.309999999... . Numerals which terminate have no non-zero digits after a given position.
For example,numerals like 2.31 and 2.310 are taken to be the same, except in the experimental sciences,
where greater precision is denoted by the trailing zero.

The most commonly used system of numerals is known as Hindu-Arabic numerals.Great Indian
mathematicians Aryabhatta of Kusumapura (5th Century) developed the place value notation.
Brahmagupta (6th Century) introduced the symbol zero.
BINARY
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.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. 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 thesame, was developed by
the Chinese scholar and philosopher Shao Yong in the 11th century.
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. 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.
Binary codes
Binary codes are codes which are represented in binary system with modification from the original ones.
Weighted Binary codes
Non Weighted Codes
Weighted binary codes are those which obey the positional weighting principles, each position of the
number represents a specific weight. The binary counting sequence is an example.

Reflective Code
A code is said to be reflective when code for 9 is complement for the code for 0, and so is for 8 and 1
codes, 7 and 2, 6 and 3, 5 and 4. Codes 2421, 5211, and excess-3 are reflective, whereas the 8421 code is
not.
Sequential Codes
A code is said to be sequential when two subsequent codes, seen as numbers in binary
representation, differ by one. This greatly aids mathematical manipulation of data. The 8421 and Excess-3
codes are sequential, whereas the 2421 and 5211 codes are not.
Non weighted codes
Non weighted codes are codes that are not positionally weighted. That is, each position within the binary
number is not assigned a fixed value. Ex: Excess-3 code
Excess-3 Code
Excess-3 is a non weighted code used to express decimal numbers. The code derives its name from the
fact that each binary code is the corresponding 8421 code plus 0011(3).
Gray Code
The gray code belongs to a class of codes called minimum change codes, in which only one bit in the
code changes when moving from one code to the next. The Gray code is non-weighted code, as the
position of bit does not contain any weight. The gray code is a reflective digital code which has the
special property that any two subsequent numbers codes differ by only one bit. This is also called a unitdistance code. In digital Gray code has got a special place.

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## Sandeep Kumar

10 months ago00