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

Imaginary numbers are very real its just a bad name. They show up in physics constantly and not just because they are convenient to use. Quantum mechanics is falsifiable without the use of imaginary numbers for instance.

https://arxiv.org/abs/2101.10873