3 Fixedpoint numbers a 10 marks A fixedpoint binary number s

3. Fixed-point numbers

a) [10 marks] A fixed-point binary number system has \'m\' whole bits and \'n\' fractional bits. What is the range of numbers represented, assuming an unsigned format? Does every binary rational in this range have a finite decimal expansion? ___ (yes/no) Justify your answer.

b) [10 marks] A fixed-point binary number system has 4 whole bits and 12 fractional bits. Represent the decimal fractions 0.75, 0.3, and 1/20 in this format. Recipe: Deal with the integer part separately. Now, take the fractional part, multiply by 2, record the integer part as one digit of your answer, and repeat until the fractional part is 0, or you run out of binary digits. Practice on 0.6875 = 11/16 = (0000.101100000000)---or just (0.1011). Do not round. Show answers in hexadecimal.

Solution

a)

For assuming an unsigned format integer 3 bits allocated for integer part and 12 bits for fractional part.
Yes, for every binary rational number we have a finite decimal expansion. It will depend on how many bits we are representing. As system takes only definite bits , there will be definitely finite decimal expansion.

b)

0.75 -> 0.11000000 --> 0.C0000000

0.3 --> 0.01001101 -> 0.4CCCCCCD

1/20 --> 0.00001101 --> 0.0CCCCCCD