10 8 unit ii introduction to digital electronics 21

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UNIT II
INTRODUCTION TO DIGITAL ELECTRONICS
2.1 Number systems and conversions
Number systems provide the basis for all operations in information processing
systems. In a number system the information is divided into a group of symbols; for
example, 26 English letters, 10 decimal digits etc. In conventional arithmetic, a
number system based upon ten units (0 to 9) is used. However, arithmetic and logic
circuits used in computers and other digital systems operate with only 0's and 1's
because it is very difficult to design circuits that require ten distinct states. The
number system with the basic symbols 0 and 1 is called binary. ie. A binary system
uses just two discrete values. The binary digit (either 0 or 1) is called a bit.
A group of bits which is used to represent the discrete elements of information is a
symbol. The mapping of symbols to a binary value is known a binary code. This
mapping must be unique. For example, the decimal digits 0 through 9 are represented
in a digital system with a code of four bits. Thus a digital system is a system that
manipulates discrete elements of information that is represented internally in binary
form. Computers process data in binary form, or base 2.
Number systems are classified on the basis of radix/base, as shown in Table 2.1.
Table 2.1 Classification of number systems
Binary 0, 1
Ternary 0, 1, 2
Quaternary 0, 1, 2, 3
Octal 0, 1, 2, 3, 4, 5, 6, 7
Decimal 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Hexadecimal 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B,
C, D, E, F
Decimal Numbers
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The invention of decimal number system has been the most important factor in the
development of science and technology. The decimal number system uses positional
number representation, which means that the value of each digit is determined by its
position in a number.
The base, also called the radix of a number system is the number of symbols that the
system contains. The decimal system has ten symbols: 0,1,2,3,4,5,6,7,8,9. In other
words, it has a base of 10. Each position in the decimal system is 10 times more
significant than the previous position. The numeric value of a decimal number is
determined by multiplying each digit of the number by the value of the position in
which the digit appears and then adding the products. Thus the number 2734 is
interpreted as
Here 4 is the least significant digit (LSD) and 2 is the most significant digit (MSD).
In general in a number system with a base or radix r, the digits used are from 0 to r-1
and the number can be represented as
Equation (1) is for all integers and for the fractions (numbers between 0 and 1), the
following equation holds.
Thus for decimal fraction 0.7123
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Binary Numbers
The binary number has a radix of 2. As r = 2, only two digits are needed, and these
are 0 and 1. Like the decimal system, binary is a positional system, except that each
bit position corresponds to a power of 2 instead of a power of 10. In digital systems,
the binary number system and other number systems closely related to it are used
almost exclusively. Hence, digital systems often provide conversion between decimal
and binary numbers. The decimal value of a binary number can be formed by
multiplying each power of 2 by either 1 or 0 followed by adding the values together.