Page 8 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Unit 1: Number Systems

the count of 99 is reached. Then we add first to the third position and start over with zeros in Notes
the first two positions. The same pattern is followed continuously as high as we wish to count.
It is important to note that in decimal counting the units position (LSD) changes upward with
each step in the count, the tens position changes upward every 10 steps in the count, the hundreds
position changes upward every 100 steps in the count, and so on.
Another characteristic of the decimal system is that using only two decimal places we can count
through 10 = 100 different numbers (0 to 99). With three places we can count through 1000
2
numbers (0 to 999); and so on.
Figure 1.1: Decimal Counting

In general, with N places or digits we can count through 10 different numbers, starting with and
N
including zero. The largest number will always be 10 N–1 .
Blaise Pascal (French) invented the first adding machine in 1642. Twenty years
later, an Englishman, Sir Samuel Moreland, developed a more compact device
that could multiply, add, and subtract.

1.2 Binary Number System

Unfortunately, the decimal number system does not lend itself to convenient implementation in
digital systems. For example, it is very difficult to design electronic equipment so that it can work
with 10 different voltage levels (each one representing one decimal character, 0 through 9). On
the other hand, it is very easy to design simple, accurate electronic circuits that operate with only
two voltage levels. For this reason, almost every digital system uses the binary number system
(base 2) as the basic number system of its operations, although other systems are often used in
conjunction with binary.

In the binary system there are only two symbols or possible digit values, 0 and 1. Even so, this
base-2 system can be used to represent any quantity that can be represented in decimal or other
number systems. In general though, it will take a greater number of binary digits to express a
given quantity.
All the statements made earlier concerning the decimal system are equally applicable to the binary
system. The binary system is also a positional-value system, wherein each binary digit has its

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