Around 250B.C., the Greek mathematician Archimedes proved that . Had he had access to a...

Around 250B.C., the Greek mathematician Archimedes proved that . Had he had access to a computer and the standard library , he would have been able to determine that the single-precision floating-point approximation of π has the hexadecimal representation 0x40490FDB. Of course, all of these are just approximations, since π is not rational.

A. What is the fractional binary number denoted by this floating-point value?

B. What is the fractional binary representation of ? Hint: See Problem 2.82.

C. At what bit position (relative to the binary point) do these two approximations to π diverge?

Ref prb:

Consider numbers having a binary representation consisting of an infinite string of the form 0.y y y y y y . . ., where y is a k-bit sequence. For example, the binary representation of

is 0.001100110011 . . . (y = 0011).

A. Let Y = B2U_k(y), that is, the number having binary representation y. Give a formula in terms of Y and k for the value represented by the infinite string.

Hint: Consider the effect of shifting the binary point k positions to the right.

B. What is the numeric value of the string for the following values of y?

(a) 101

(b) 0110

(c) 010011