Digital logic design lecture 2

Digital Logic Design
PROF. SHEHZAD ALI

Overview
Number system
Binary Numbers
Boolean Algebra
Switching Algebra
Logic Gates
Boolean Function

Number System
A decimal number such as 7392 represents a quantity equal to 7 thousands plus 3 hundreds,
plus 9 tens, plus 2 units. The thousands, hundreds, etc. are powers of 10 implied
by the position of the coefficients. To be more exact, 7392 should be written as
7 x 103 + 3 x 102 + 9 x 10' + 2 x 10°
In general, a number expressed in base-r system has coefficients multiplied by powers
of r:

Binary Systems (Binary Arithmetic)

NUMBER BASE CONVERSIONS
A binary number can be converted to decimal by forming the sum of the powers of 2
of those coefficients whose value is 1. For example
Similarly, a number expressed in base r can be converted to its decimal equivalent by multiplying
each coefficient with the corresponding power of r and adding. The following is an example of
octal-to-decimal conversion:

NUMBER BASE CONVERSIONS
Decimal to other Number System
To convert a decimal number(base 10) into any base r , divide that number with base r and write
the remainder value from bottom to top.

Conversions
Fractional decimal number to binary
4065171011

OCTAL AND HEXADECIMAL NUMBERS
The conversion from and to binary, octal, and hexadecimal plays an important part in digital
computers. Since 2^3 = 8 and 2^4 = 16, each octal digit corresponds to three binary digits and
each hexadecimal digit corresponds to four binary digits. The conversion from binary to octal is
easily accomplished by partitioning the binary number into groups of three digits each, starting
from the binary point and proceeding to the left and to the right. The corresponding octal digit is
then assigned to each group. The following example illustrates the procedure:

OCTAL AND HEXADECIMAL NUMBERS
Conversion from octal or hexadecimal to binary is done by a procedure reverse to the above.
Each octal digit is converted to its three-digit binary equivalent. Similarly, each hexadecimal digit
is converted to its four-digit binary equivalent. This is illustrated in the following examples:

Boolean Algebra OR Switching Algebra
In 1854 George Boole introduced a systematic treatment of logic and developed for this
purpose an algebraic system now called Boolean algebra.
The Boolean algebra we present is an algebra dealing with binary variables and logic
operations.
In 1938 C. E. Shannon introduced a two-valued Boolean algebra called switching algebra, in
which he demonstrated that the properties of bi-stable electrical switching circuits can be
represented by this algebra.
For the formal definition of Boolean algebra, we shall employ the postulates formulated by E. V.
Huntington in 1904.

Boolean Algebra OR Switching Algebra
A Boolean expression is an algebraic expression formed by using binary variables, the constants
0 and 1, the logic operation symbols, and parentheses.
A Boolean function can be described by a Boolean equation consisting of a binary variable
identifying the function followed by an equals sign and a Boolean expression.

Logic Gates
Logic gates are electronic circuits that operate on one or more input signals to produce an
output signal.
The graphics symbols used to designate the three types of gates—AND, OR, and NOT—are
shown in Figure 1(a).
The gates are electronic circuits that produce the equivalents of logic-1 and logic-0 output
signals in accordance with their respective truth tables if the equivalents of logic-1 and logic-0
input signals are applied.

Basic Identities of Boolean Algebra

Digital logic design lecture 2

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