There are some contexts in which division by zero is defined in a meaningful way. The most common such situation is the real projective line or its complex analogue, the Riemann sphere [0]. We’ll focus on the latter, since it has a richer structure. The Riemann sphere is essentially just the complex plane “closed up” (compactified) by adding a point at infinity. The metric is defined such that points of very large norm are all close to infinity, and hence to one another.

Any nonzero number divided by zero is infinity. Zero divided by zero is undefined. The operation of adding 1 has a parabolic fixed point at infinity.

The Riemann sphere is not a group, but it still has many very nice properties. For instance, the set of analytic maps from the Riemann sphere to itself is precisely the set of rational functions (together with the constant map infinity).

A much more profound statement is Thurston’s characterization of rational functions [1], which states that every orientation preserving branched covering map F from the sphere to itself is equivalent (topologically conjugate) to a rational map on the Riemann sphere, unless F satisfies a certain computable property.

In other words, maps on one of the most ubiquitous topological spaces, namely the sphere, are often best understood by introducing a structure where division by zero is allowed.

[0] https://en.m.wikipedia.org/wiki/Riemann_sphere

[1] https://pi.math.cornell.edu/~hubbard/ThurstonRatMaps.pdf