Could "$\infty$" be understood by taking the reciprocals of the Hyperreal numbers? When learning mathematics we are told that infinity is undefined. (*)
Recently I read about the infinitesimal version of Calculus and how we can in fact treat $dy/dx$ as a fraction under this approach (something we can't do with limits). This is achieved by constructing the Hyperreal numbers $^*\mathbb{R}$, which contain the real numbers and the infinitesimal numbers (all positive numbers greater than zero, yet less than any real number).
Thinking back to (*) I wonder, could sense of infinity be made somehow by including the reciprocal infinitesimal numbers? Presumably these numbers would be defined since the infinitesimal numbers are never zero.

In fact, infinite numbers are a part of the Hyperreal Numbers. Just check this out on Wikipedia.
