Techniques to solve linear systems with zillions of variables but relatively few nonzero coefficients. Using complex functions to morph one plane region into another. You can't put a round Earth on a flat piece of paper without distortion, but some ways are better than others for various purposes. A new way to solve linear programming problems that promises to be faster than the simplex method for large problems. An ordinal number is just a well-ordered set.
Finite ordinals are the familiar whole numbers.
But all kinds of infinities are easy to define and do all the standard arithmetic with. A generalization of complex numbers. Multiplication is noncommutative. Quaternions can be used to represent rotations in 3D. Lebesgue integral. This is the definition of integral mathemeticians really use.