Found this on the door of an awesome math prof at my university

Found this on the door of an awesome math prof at my university

 

"hyperreals"

The hyperreal number are a so-called "extension" of the real numbers which means that they contain all the real numbers and some neat extra numbers. You have probably encountered hyperreal numbers in calculus - for example infinity and minus infinity.

"infinitesimals"

The second important type of hyperreal numbers are the infinitesimal numbers - or numbers that are closer to zero than any real number and yet not exactly zero. If youve done some integrals then youllved (English grammar, mathematician style) the pleasure to meat "dx", an infinitely small change in some variable x. That is an infinitesimal.

"standard part"

As Ive said, the hyperreals are "essentially" just the reals with some extra numbers. The standard part of any hyperreal is simply the real number that is closest to it - or you could say that it extracts its real component. For example: The standard part of any infinitesimal is zero because they are extremely close to the real number zero. Similarly, the standard part of something like "5 + dx" would be 5 since that is the real number that its closest to.

Why is this useful?

Essentially it is useful because it allows you to formulate calculus and many things that build upon it without any limits and fuzzy mathematics only by using the standard part function. This is especially handy when you are directly manipulating the infinitesimals which is bogus mathematics in the "ordinary" formulation of calculus but perfectly valid in the hyperreal formulation.