A friend proposes the following scheme for representing rational numbers (numbers that can be represented as p/q, where p and q are both integers and q = 0) using 13 bits. The leftmost 6 bits will be the signed binary integer p; the rightmost 7 bits will be the signed binary integer q.
a. What is the largest value representable using this scheme?
b. What is the smallest positive (non-zero) value representable using this scheme?
c. What's the smallest value (i.e., most negative) representable using this scheme?
d. Give a pair of distinct bit patterns that both represent the same non-zero value under this scheme.
e. Give an unusable bit pattern under this scheme (a pattern that does not represent a rational number).
f. Propose a change to the scheme that keeps it simple but increases the number of distinct values representable.