The hyperreal number system is a way of treating infinities and infinitesimals in a rigorous way that is consistent with the way that we treat ordinary numbers.

I recently posted a short video introducing the concept in a simple way. I thought some of you might be interested.

I find the hyperreals interesting for a number of reasons. First of all, I think that infinities and infinitesimals are somewhat of the equivalent of Intelligent Design for mathematics. Infinitesimals were essentially banned from mathematics in the 1800s because it was said that they were inconsistent and non-rigorous (this is why calculus switched from infinitesimals to limits). This move was largely philosophically motivated, with Hilbert and others trying to naturalize mathematics.

However, in the 1960s, infinitesimals were proved to be equally mathematically rigorous as other standard mathematical entities.

In any case, from a practical side, infinitesimals make calculus, limits, and other sorts of mathematical ideas a LOT simpler to work with. In fact, using these types of numbers, Calculus becomes pretty much identical with Algebra.