Nonstandard analysis

Nonstandard analysis is a special type of math that helps us understand things in a different way. Imagine you have some really small numbers, like 1/3 or even smaller. And imagine you have some really big numbers, like a million or even bigger. Standard math only lets us work with these small and big numbers in certain ways, and we have to use a lot of rules to make them work. But with nonstandard analysis, we can make up new rules that let us work with these numbers in different ways that are easier to understand.

To do this, we use something called "infinitesimals". These are really, really small numbers that are so small we can't even see them or measure them like we do with normal numbers. But with nonstandard analysis, we can imagine that these infinitesimals are part of our number system, like 1, 2, 3 and so on. This means we can work with small numbers that are so tiny we couldn't do it before.

We can also use "infinite numbers". These are numbers that are so big we can't even imagine them or count that high. But with nonstandard analysis, we can also work with these infinite numbers like they are just regular numbers.

Nonstandard analysis lets us see things in a new way that can help us solve math problems that we couldn't solve before. It's kind of like having a new set of tools to use that can make math easier and more fun!