Integer vs floating point numbers in IEEE single format

Question

Can any integer number between the smallest and largest positive normal floating-point numbers in IEEE single format be stored exactly in this floating-point system?
If yes, please explain, possibly providing an example.

Answer 1

No.

In IEEE-754 binary32, a finite number is represented in the format ± x.xxx…xxx₂ • 2^e, where each x is a bit, there are 23 bits after the “.”, and −126 ≤ e ≤ 127. It is in normal form if the first x is 1. So the largest positive normal number representable is + 1.111…111₂ • 2¹²⁷ = (2 − 2⁻²³) • 2¹²⁷ = 2¹²⁸ − 2¹⁰⁴.

The smallest positive normal number is + 1.000…000₂ • 2⁻¹²⁶ = 2⁻¹²⁶. Between these lies the integer 2²⁴+1 = 16,777,217. This integer is not representable because, in binary, it is 1000…0001₂, where there are 23 zero bits between the ones, making 25 bits in total, so it cannot be represented in the form ± x.xxx…xxx₂ • 2^e, which has just 24 bits in the significand.