Page 55 - Computer Science V2.0 Class 11

(0.3) = (0.010011001)
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Recurring digits
So, we can write, (0.3) as (0.0 1001 1001) if 9-bit accuracy of the fraction part is required, (0.3) as (0.0 1001
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2
1001 100) if 12-bit accuracy of the fraction part is required, and (0.3) as (0.0 1001 1001 1001 1001 1001) if 21-bit
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2
2
accuracy of the fraction part is required.
Similarly, the binary equivalent of (0.8) is (0.110011001100110011) to 18 decimal places as the digits '1100' are
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2
being repeated.
The same method will apply when the fraction is represented as (p/q) . Let us illustrate it on 3/7 .
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Example 16: Convert 3/7 to an equivalent binary number.
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Fraction part Integer part
(3/7) × 2 = 6/7 6/7 0
(6/7) × 2 = 12/7 5/7 1
(5/7) × 2 = 10/7 3/7 1
(3/7) × 2 = 6/7 6/7 0
(6/7) × 2 = 12/7 5/7 1
(5/7) × 2 = 10/7 3/7 1

3/7 = (0.011011) Recurring digits
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2
To obtain the binary representation of a decimal number having an integer and the fractional part, then apply the
following steps:
1. Convert the integer part to an equivalent binary number (say, a)
2. Convert the fraction part between 0 and 1 to an equivalent binary number between 0 and 1 (say, b).
3. Write the binary representation of the integer part, followed by the binary representation of the fraction part.
Example 17: Convert (101.25) to its equivalent binary number.
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Whole number part - 101

2 101 Remainder
2 50 1
2 25 0

2 12 1
= (1100101)
2 6 0 2

2 3 0
2 1 1
0 1

Fractional Part Integer part
0.25 × 2 = 0.50 0.50 0
(.01)
0.50 × 2 = 1.00 0.0 1 2
(101.25) = (1100101.01) 2
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Find binary representation of (0.25) , (0.75) and (0.125) .
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Number Systems and Encoding Schemes 41