Page 13 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Digital Circuits and Logic Design

Notes Unit and Number
The terms that you learned in the decimal and binary sections are also used with the octal system.
The unit remains a single object, and the number is still a symbol used to represent one or more
units.
Base (Radix)

As with the other systems, the radix, or base, is the number of symbols used in the system. The
octal system uses eight symbols—0 through 7. The base, or radix, is indicated by the subscript 8.

Positional Notation
The octal number system is a positional notation number system. Just as the decimal system
uses powers of 10 and the binary system uses powers of 2, the octal system uses power of 8 to
determine the value of a number’s position. The following bar graph shows the positions and
the power of the base:
Remember that the power, or exponent, indicates the number of times the base is multiplied by
itself. The value of this multiplication is expressed in base 10 as shown below:
8 = 8 × 8 × 8, or 512
3
10
8 = 8 × 8, or 64
2
10
8 = 8
1
10
8 = 1 10
0
1
8 = , or .125
–1
8 10
1 1
8 = , or , or .015625
–2
88 64 10
´
1 1
8 = , or , or .0019531
–3
888 512 10
´´
All numbers to the left of the radix point are whole numbers, and those to the right are fractional
numbers.
About 1672, Gottfried Wilhelm von Leibniz (German) perfected a machine
that could perform all the basic operations (add, subtract, multiply, divide),
as well as extract the square root. Modern electronic digital computers still
use von Leibniz’s principles.
1.5 Conversion from One Number System to Another
Numbers expressed in decimal number system are much more meaningful to us than are numbers
expressed in any other number system. This is because we have been using decimal numbers in
our day-to-day life, right from childhood; however, we can represent any number in one number
system in any other number system. Because the input and final output values are to be in decimal,
computer professionals are often required to convert number in other systems to decimal and
vice versa. Many methods can be used to convert numbers from one base to another. A method
of converting from another base to decimal, and a method of converting from decimal to another
base are described here:

1.5.1 Converting from Another Base to Decimal

The following steps are used to convert a number in any other base to a base 10 (decimal) number:
Step 1: Determine the column (positional) value of each digit (this depends on the position of the
digit and the base of the number system).

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