what is the most negative fraction that can be represented by a n bit binary number in 2's complement?

Question

I am reading computer architecture by Carl Hamacher. The following lines are confusing me

If we use a 32-bit binary number to represent a signed integer in 2's complement then the range of values that can be represented is -2^31 to 2^31 - 1. //fine

The same 32-bit patterns can be interpreted as fractions in the range -1 to +1 - 2^-31 if we assume that the implied binary point is just to the right of the sign bit.

I don't understand how the range has been calculated for the fraction. I myself calculated the range for positive fractions with the highest fraction being 2^31-1. But I am unable to calculate the lowest value (mentioned as -1).

Answer 1

Shifting the binary point is tantamount to dividing by a power of two. So if we shift the binary point 31 positions, we are effectively representing fractions whose denominators are 2³¹ and whise numerators are the unshifted integers. Thus the range becomes -2³¹/2³¹ to (2³¹-1)/2³¹, which is -1 to 1-2^-31.

Another way to look at it: In 2's complement integer representation, the bit positions have weights (left to right) of -2³¹, 2³⁰, 2²⁹, …, 2⁰ (where only the first weight is negative). Shifting the binary point to just after the high-order (and negated) bit -- "the sign bit" -- makes rhe weights -2⁰, 2^-1, 2^-2, …, 2^-31 (again, only the first weight is negative). Again, it's clear that the "most negative" is 100…0, which has the value -1, the weight for the only one-bit