|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Question 4 |
|
|
|
|
|
|
|
a) Largest positive |
|
|
|
|
|
|
|
|
|
|
|
0 |
1110 |
1111
1111 111 |
|
|
|
|
|
|
|
|
|
exponent: |
14 - 7 = 7 |
|
|
|
|
|
mantissa: |
1.1111 1111 111 |
|
|
|
|
|
= 2 - 2-11 |
|
|
|
|
|
number: |
27 * (2 - 2-11) = 28 - 2-4 |
|
|
|
|
|
|
|
|
|
b) smallest positive
normalized |
|
|
|
|
|
|
|
|
|
|
|
0 |
0001 |
0000
0000 000 |
|
|
|
|
|
|
|
|
|
|
exponent: |
1 - 7 = -6 |
|
|
|
|
|
|
mantissa: |
1.0 |
|
|
|
|
|
|
number: |
2-6 * 1 = 2-6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
c) smallest denormalized |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0 |
0000 |
0000
0000 001 |
|
|
|
|
|
|
|
|
|
|
|
|
exponent: |
-6 |
|
|
|
|
|
mantissa: |
1.0 * 2-11 |
|
|
|
|
number: |
2-6 * 2-11 = 2-17 |
|
|
|
|
|
|
|
|
|
|
|
d) Smallest gap between
normalized: |
|
|
|
|
|
|
|
|
|
|
0 |
0001 |
0000
0000 001 |
|
|
|
|
|
0 |
0001 |
0000
0000 000 |
|
|
|
|
|
|
|
|
|
|
|
|
exponent: |
-6 |
|
|
|
|
|
|
|
mantissa: |
1.0000 0000 001 - 1 |
|
|
|
|
difference: |
2-6 * 2-11 = 2-17 |
|
|
|
|
|
|
|
|
|
|
|
e) Largest gap: |
|
|
|
|
|
|
|
|
|
|
0 |
1110 |
1111
1111 111 |
|
|
|
|
0 |
1110 |
1111
1111 110 |
|
|
|
|
|
|
|
|
|
exponent: |
14 - 7 = 7 |
|
|
|
|
|
|
mantissa: |
1.1111 1111 111 - 1.1111
1111 110 |
|
|
|
|
difference: |
27 * 2-11 = 2-4 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|