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0.235 = (2 x 10"') + (3 x 10"2) + (5 x 10"3) and
68.53 = (6 x 10') + (8x 10°) +(5 + (3 x 10"2)
Similarly, in the binary number system,
0.101 =(1 x2"') + (0x2"2) 10.01 =(1 x21) + <0x2°) + x2"3)and
10.01 = (1x21) + (0x20) + 0x11) + (1x22)
Thus, the binary point serves the same purpose as the decimal point. Some of the
positional values in the binary number system are given below.
Position
4
4
Position Value 24
24
Quantity
16
1/16
Represented 3 2 1 0 Binary Point 1 2 3 23 22 21 20 Binary Point 21 22 23 8 4 2 1 Binary Point ½ ¼ 1/8 In general, a number in a number system with base b would be written as:
an an.i... ao. a.i a_2... a.m
and would be interpreted to mean
an x bn + an., x b"'1 + ... + ao x b° + a4 x b"1 + a.2 x b"2 + .;. + a.m x b"m
The symbols an, an.h ..., a.m used in the above representation should be one of the b
symbols allowed in the number system.
Thus, as per the above mentioned general rule,
46.328 = (4 x 81) + (6 x 8°) + (3 x 8'1) + (2 x 8"2) and
5A.3C16 = (5 x 161) + (A x 16°) + (3 x 16"1) + (C x 16"2)
Example 3.26.
Find the decimal equivalent of the binary number 110.101
Solution:
110.1012 = 1 x22+ 1 x2' + 0x2° + 1 x2" 0 x 2"1 +
0x22 + 1 x 23
= 4 + 2 + 0 + 0.5 + 0 + 0.125
= 6 + 0.5 + 0.125
= 6.62510
Points to Remember
1. Number systems are basically of two types: nonpositional and positional. 2. In a nonpositional number system, 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. It is very difficult to perform arithmetic with such a number system.
3. 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.
• The digit itself,
• The position of the digit in t...
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This document was uploaded on 04/07/2014.
 Spring '14

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