[lolcatuser]

The reason division by zero can be a number is that floating-point numbers aren't individual numbers but ranges of numbers. So 0.0 isn't really the integer 0, it is the range of numbers between 0 and the smallest representable non-zero number. So division by 0.0 may be division by zero, or it may be division by a very small number, in which case it would approach infinity.

The same goes for the floating-point infinity: it doesn't represent the concept of infinity, it is a placeholder for every number between the largest representable number and infinity. That's how dividing a very large number by a very small number can result in infinity, a number which is really not a number.

This is the philosophy by which IEEE floating-point numbers were designed, and it's the explanation behind which negative zeroes and infinities make sense in floating point.

The way I find it easiest to reason about is by taking a graph of an asymptote and rounding it to the nearest units. You somehow need a way to say "there is an asymptote here!" even though you might not have all the units, and so you introduce infinities and negative zeroes to maintain as much precision as possible.

[adgjlsfhk1]

this isn't true. floats don't have semantics of internals. they have semantics of exact points that do math with rounding (otherwise the result of subtracting 2 floats would sometimes be more than 1 float)

[throwawaymaths]

Moreover the spec defines rounding modes (rtnz, rtg, rtfz rtl) that are designed to exactly do interval arithmetic