a.
1. 25.875
--convert the integer part 25 to binary by dividing it with 2 until the quotient becomes 0.
Division=quotient remainder
25÷2= 12 1
12÷2= 6 0
6÷2=3 0
3÷2=1 1
1÷2=0 1
-- take the remainder values from down to up.
11001
---convert the fraction part .875 to binary by multiplying with 2 and take integer part of result until the fraction part becomes 0.
Multiplication = integer part fraction part
0.875 × 2 = 1 0.75
0.75 × 2 =1 0.5
0.5 × 2 =1 0
Take integer part value = 111
Total integer+ fraction value = 11001.111
Move decimal point to left for 4 places= 1.1001111×2^4 here mantissa is 1001111.
E=127+4 = 131(10000011)
Floating point representation:
Sign E Mantissa
0 10000011 1001 111 1111 1111 1111 1111
2. -7.0625
Follow the same steps as above.
Integer part 7
Division = quotient remainder
7÷2 = 3 1
3÷2 = 1 1
1÷2 = 0 1
7 value is 111
-- now fraction part .0625
Multiplication = integerpart fraction part
0.625×2 = 0 0.125
0.125×2 = 0 0.25
0.25×2 = 0 0.5
0.5×2 = 1 0
0.625 value is : 0001
Total value : 111.0001
Decimal point move to left for 2 places: 1.110001×2^2
E=127+2=129 (10000001)
Floating point representation:
S E Mantissa
1 10000001 1100 0100 0000 0000 0000 000
b.
1.given : C 1 0 B 0 0 0 0
- convert to binary
1100 0001 0000 1011 0000 0000 0000 0000
S E Mantissa
1st bit ---- S
Next 8 bits -----E
Next 23 bits -----mantissa
E= 10000010 =130
Exponential bias = 130-127 = 3
Mantissa part 000 1011 0000 ...........
2^-4 + 2^-6 + 2^-7 = 0.0859375
Decimal value will be obtained by the following formula :
(-1)^S × (1+Mantissa ) × 2^E
(-1)^1 × (1+0.0859375)×2^3
=-8.6875
2. Given : 4 6 A B 0 0 0 0
Binary value :
0100 0110 1010 1011 0000 0000 0000 0000
S E Mantissa
1st bit ---S
Next 8 bits ---E
Next 23 bits -- Mantissa
E= 10001101 = 141
Exponential bias : 141-127=14
Mantissa part : 010 1011 00000 .......
2^-2 +2^-4 + 2^-6 +2^-7 = 0.3359375
Binary value will be obtained from the formula :
(-1)^S × (1+Mantissa )×2^E
(-1)^0 × (1+0.3359375) × 2^ 14
=21888
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