You can totally define division by zero.

"There are mathematical structures in which [dividing by zero] is defined. [...] However, such structures do not satisfy every ordinary rule of arithmetic (the field axioms)." https://wikipedia.org/wiki/Division_by_zero

You're familiar with the ordinary rules of arithmetic (ORA!) applied to the real numbers. However, there are many number systems that follow different rules.

The rules of arithmetic for floats, wraparound ints, elliptic curves, and asymptotic analysis are all different from ORA! (and each other) but they are useful in their own ways.

IIRC division by zero throws an exception.

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.

>both of those be NaN

Totally NOT. NaN is defined as 0.0/0.0. That's it.

IEEE754 has a separate +Infinity and -Infinity to go along with SNaN and QNaN.

For instance:

- as parent said 1/0 is Inf.

- as parent said 1/(-0) and log(0) is -Inf.

- sqrt(-1) and 0/0 is NaN.

[1] https://www.gnu.org/software/libc/manual/html_node/Infinity-...

this is wrong. ieee defines them

not in ieee-754, only in reality