Stupid math question, but why is the solution undefined when you divide finite numbers by zero?
Why can't you just define the answer as "what you get when you divide a finite number by zero"
It seems like a useful thing to have
Good news: you can! It's called the "projective real line", which is why I was making a joke with the word "projecting".
There are a few quirks. One is that you only get one infinity, not two -- infinity and -infinity are the same number.
Why is it called "projective"? Imagine a circle sitting on top of the real line at 0. Now put a lamp at the tip-top of the circle. Any point* on the circle will "cast a shadow" from that lamp down to the real line. In this way we can "project" each point from the line to a point on the circle...
*with the exception of the top point where the lamp is. That's our infinity.
Why visualize things in this way? Well, now 1/x is just flipping the circle upside down. 0 and infinity are just opposite ends of the circle, with the positive reals down the right and negative reals down the left.
This idea extends to projective planes (that have a line at infinity instead of a single point) and the projective complex plane (the lamp is on top of a sphere sitting on the complex plane, and rotations of the sphere have special names called Mobius transforms) and the whole field of projective geometry.
