Page 7 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Digital Circuits and Logic Design

Notes Non-Positional Number Systems
In early days, human beings counted on fingers. When ten fingers were not adequate, stones,
pebbles or sticks were used to indicate values. This method of counting uses an additive approach
or the non-positional number system. In this system, we have symbols such as I for 1, II for 2, III
for 3, IIII for 4, IIIII for 5, etc. Each symbol represents the same value regardless of its position
in the number and the symbols are simply added to find out the value of a particular number.
Since it is very difficult to perform arithmetic with such a number system, positional systems
were developed.
Positional Number Systems

In a positional number system, there are only a few symbols called digits, and these symbols
represent different values depending on the position they occupy in the number. The value of
each digit in such a number is determined by three considerations:

1. The digit itself,
2. The position of the digit in the number, and

3. The base of the number system.

Decimal, Binary, Octal, Hexadecimal number systems.

1.1 Decimal Number System

The decimal number system is composed of 10 numerals or symbols. These 10 symbols are 0, 1,
2, 3, 4, 5, 6, 7, 8, and 9; using these symbols as digitals of a number, we can express any quantity.
The decimal system, also called the base-10 system because it has 10 digits, has evolved naturally
as a result of the fact that man has 10 fingers.

The decimal number system is a positional-value system in which the value of a digit depends
on its position. For example, consider the decimal number 453. We know that the digit 4 actually
represents 4 hundreds, the 5 represents 5 tens and the 3 represents 3 units. In essence, the 4 carries
the most weight of the three digital; it is referred to as the most significant digit (MSD). The 3
carries the least weight and is called the least significant digit (LSD).
Consider another example, 27.35. This number is actually equal to 2 tens plus 7 units plus 3 tenths
plus 5 hundredths, or 2 * 10 + 7 * 1 + 3 * 0.1 + 5 * 0.01. The decimal point is used to separate the
integer and fractional parts of the number.
More rigorously, the various positions relative to the decimal point carry weights that can
be express as powers of 10. This is illustrated in Figure 1.1, where the number 2745.214 is
represented. The decimal point separates the positive powers of 10 from the negative powers.
The number 2745.214 is thus equal to (2*10 ) + (7*10 ) + (4*10 ) + (5*10 ) + (2*10 ) + (1*10 )
+1
+2
+3
0
–2
–1
+ (4 * 10 ). In general, any number W simply the sum of the products of each digit value and
–3
its positional value.
Decimal Counting: When counting, in the decimal system, we start with 0 in the unit’s position
and take each symbol (digit) in progression until we reach 9. Then we add a 1 to the next higher
position and start over with zero in the first position (see Figure 1.1). This process continues until
2 LOVELY PROFESSIONAL UNIVERSITY