Solutions8
Computer Architecture
Tutorial 4 – Floating Point Numbers – Answers
1) Binary fractions are:
a) 5.5 is 101.1
b) 8.25 is 1000.01
c) 9 is 1001
0.3 0.6, 1.2, 0.4, 0.8, 1.6, 1.2 01001 1001 1001 etc.
9.3 is 1001. 01001 1001 1001 repeating etc.
d) 11.46875 is 1011.01111
2) Convert the binary number 1001.1010101 to decimal.
1001 binary is 9 decimal
. 1 0 1 0 1 0 1
128 64 32 16 8 4 2 1 Sum=85
Fraction = 85 /128 = 0.6640625
Number = 9.6640625
3) a) 101.1 = 1.011 x 2
2
b) 1000.01 = 1.00001 x 2
3
c) 0.00010101 = 1.0101 x 2
–4
4) Convert –31.3 to IEEE Single Precision format.
First convert to a binary number -31.3 = -11111.01001 1001 1001
Next Normalise
1.11110 1001 1001 1001 1001 1001 x 2
4
Significand field is 1111 0100 1100 1100 1100 110 (23 bits with 1. omitted)
Exponent field is 4+127 = 131 = 1000 0011
Number is -ve therefore Sign field is 1
Sign Exponent Significand
1 1000 0011 1111 0100 1100 1100 1100 110
5) Convert the IEEE Single Precision format hex value C154 0000 to decimal.
C154 0000 = 1100 0001 0101 0100 0000 0000 0000 00000
Sign Exponent Significand
1 1000 0010 1010 1000 0000 0000 0000 000
Exponent field = 1000 0010 = 130 => Exponent = 130 – 127 = 3
Significand field = 10101 Adding Hidden Bit => 1.10101
Therefore number is 1.10101 x 2
3
= 1101.01 = Decimal 13.25
Sign is 1 therefore number is -13.25
6) Carry out the operation 31.3 + 13.25 in IEEE single precision arithmetic
Number Sign Exponent Significand
31.3 0 1000 0011 1111 0100 1100 1100 1100 110
13.25 0 1000 0010 1010 1000 0000 0000 0000 000
Significand of Larger Number = 1.1111 0100 1100 1100 1100 110
Significand of Smaller Number= 1.1010 1000 0000 0000 0000 000
Exponents differ by 1. Therefore shift binary point of Smaller Number 1 place.
Significand of Larger Number = 1.1111 0100 1100 1100 1100 1100
Significand of Smaller Number= 0.1101 0100 0000 0000 0000 0000
Significand of Sum = 10.1100 1000 1100 1100 1100 1100
Sum = 10.1100 1000 1100 1100 1100 1100 x 2
4
Normalise 1.01100 1000 1100 1100 1100 1100 x 2
5
Sign Exponent Significand
0 1000 0100 0110 0100 0110 0110 0110 011
7)
Fraction Binary Decimal
1/4 0.01 0.25
3/8 0.011 0.375
23/16 1.0111 1.4375
45/16 10.1101 2.8125
11/8 1.011 1.375
45/8 101.101 5.625
49/16 11.0001 3.0625