Introduction to Digital Systems
Digital systems have such a prominent role in everyday life that we refer to the present
technological period as the digital age. Digital systems are used in communication,
business transactions, traffic control, spacecraft guidance, medical treatment, weather
monitoring, the Internet, and many other commercial, industrial, and scientific
enterprises. We have digital telephones, digital televisions, digital versatile discs, digital
cameras, handheld devices, and, of course, digital computers.
One characteristic of digital systems is their ability to represent and manipulate discrete
elements of information. Any set that is restricted to a finite number of elements
contains discrete information. Examples of discrete sets are the 10 decimal digits, the 26
letters of the alphabet, the 52 playing cards, and the 64 squares of a chessboard.
Discrete elements of information are represented in a digital system by physical
quantities called signals. Electrical signals such as voltages and currents are the most
common. The signals in most present-day electronic digital systems use just two
discrete values and are therefore said to be binary. A binary digit, called a bit, has two
values: 0 and 1. Discrete elements of information are represented with groups of bits
called binary codes.
A decimal number such as 8,473 represents a quantity equal to 8 thousands, plus 4
hundreds, plus 7 tens, plus 3 units. The thousands, hundreds, etc., are powers of 10
implied by the position of the coefficients (symbols) in the number. To be more exact,
8,473 is a shorthand notation for what should be written as
8 X 103 + 4 X 102 + 7 X 101 + 3 X 100
However, the convention is to write only the numeric coefficients and, from their
position, deduce the necessary powers of 10 with powers increasing from right to left.
In general, a number with a decimal point is represented by a series of coefficients:
a5a4a3a2a1a0 . a-1a-2a-3
The coefficients aj are any of the 10 digits (0, 1, 2, …,9), and the subscript value j gives
the place value and, hence, the power of 10 by which the coefficient must be multiplied.
Thus, the preceding decimal number can be expressed as
105a5 + 104a4 + 103a3 + 102a2 + 101a1 + 100a0 + 10-1a-1 + 10-2a-2 + 10-3a-3
With a3 = 8, a2 = 4, a3=7, a4=3
The decimal number system is said to be of base, or radix, 10 because it uses 10 digits
and the coefficients are multiplied by powers of 10.
The binary system is a different number system. The coefficients of the binary number
system have only two possible values: 0 and 1. Each coefficient aj is multiplied by a
power of the radix, e.g., 2j, and the results are added to obtain the decimal equivalent of
CSE & IT Dept., PEC