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

Nothing, and they have. The most common definitions are the Real Projective Line, and Riemann Sphere, which defines z/0 = ∞ for all non-zero real or complex numbers z.

There is also a less well known structure called a Wheel that fully defines division by adding 2 elements: ∞ = z/0 and ⊥ = 0/0.

As others have alluded to, these structures are not nuce to work with in general, which is why we don’t use them as much. In contrast, the complex numbers turn out to be suprisingly nice to work with in general, with few downsides, so we pretty much always assume they are available.