Understanding floating point representation errors; what's wrong with my thinking?

Question

I\'m having some trouble understanding why some figures can\'t be represented with floating point number.

As we know, a normal float would have sign bit, exponent,

Answer 1

Each floating-point number in the IEEE 754 standard is, in effect, some integer multiplied by some integer power of two. E.g., 3 is represented by 3 * 2⁰, 96 is represented by 3 * 2³, and 3/16 is represented by 3 * 2^-4.

There are no integers x and y such that .1 = x * 2^y, therefore .1 cannot be exactly represented by a floating-point number. Proof: If .1 = x * 2^y, then 10x = 2^-y. 2^-y is clearly positive, so x is positive. It is also an integer, so 10x is divisible by 10, so it is divisible by 5. Therefore 2^-y is a power of two that is divisible by 5, which is clearly impossible.