Why do 1000.5, 1/16 and 1.5/32 have an exact representation in an arbitrary (finite) normalized binary floating point number system but 123.4, 0.025 and 1/10 don't? How can this easily been seen without trying to create the complete floint point number?
[Math] Exact representation of floating point numbers
binaryfloating point
Best Answer
Written as fractions in lowest terms, the denominator is a power of $2$ for those having a finite binary representation
So
while
all having non-powers of $2$ in the denominator.
By comparison, for decimal fractions to have a finite representation, the denominator of the lowest terms fraction should be a a power of $2$ times a power of $5$ since the the prime factorisation of $10$ is $2 \times 5$