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I'm currently trying to get better at math. I've decided to do research on every facet of mathematics, starting at the source: what is a number? And going from there, getting progressively more advanced.

Instead of just trying to memorize formulas or learn how to do problems from a math book, like in school, I want to focus on the WHY. Why do the formulas work? Why does this theorem work? etc.

I think if I understand the underlying logic, I will start to understand all aspects of math better.

I've broken math into the following main branches:

Algebra

Geometry

Statistics

Calculus

Number theory

Combinatorics

Topology

How would you define each of those branches?