What’s stopping mathematicians from defining a number for 1 ÷ 0, like what they did with √-1?

Question

What’s stopping mathematicians from defining a number for 1 ÷ 0, like what they did with √-1?

In: Mathematics

Answer 1

It *is* defined when working in geometry with complex numbers and it does have useful properties. There is something called the Riemann sphere, in which you have all complex numbers (which you already know about based on the end of your question and one more number ∞, where we have rules such as

> z/0 = ∞ and z/∞ = 0 when z is any nonzero complex number

> z ± ∞ = ∞ and ∞ ± ∞ = ∞ when z is any complex number.

A related video: https://www.youtube.com/watch?v=hhI8fVxvmaw