Why is it mathematically consistent to allow imaginary numbers but prohibit division by zero?

Question

Couldn’t the result of division by zero be “defined”, just like the square root of -1?

Edit: Wow, thanks for all the great answers! This thread was really interesting and I learned a lot from you all. While there were many excellent answers, the ones that mentioned Riemann Sphere were exactly what I was looking for:

https://en.wikipedia.org/wiki/Riemann\_sphere

TIL: There are many excellent mathematicians on Reddit!

In: 1691

Answer 1

Division by zero is undefined in a very different way from the square root of -1.

For the square root of -1 there is no real value it could be, so if you invent a value for it to be that’s the issue solved.

For dividing by zero there are too many values it could be. 0/0 could be literally any value from +infinity to -infinity (or even an imaginary or complex value). Creating a new value for it to be just makes the problem worse.