Logic and Proof
Most people think that mathematics is all about manipulating numbers and formulas to compute something. While numbers play a starring role (like Brad Pitt or Angelina Jolie) in math, it is also important to understand why things work the way they do. That means at its core, math is more about logic than it is about numbers (or Hollywood movie stars, for that matter).
For example, the area of a rectangle can be found by multiplying its length times its width. Big whoop. But why does area work like that? Once we know the answer to that question, we can approach related questions. Quietly, though, because we don't want to scare them away.
To explain this "why" (em cee ay?), mathematicians have developed the idea of "proof." Proof is a touching drama starring Gwyneth Paltrow, Jake Gyllenhaal, and Anthony Hopkins. Or, you know, an explanation of why something is true.
First, you list your assumptions, and then you argue your way to a desired conclusion, explaining each step along the way. By the end, you should have convinced yourself and anybody who may be reading your proof, like Anthony Hopkins (it could happen!) or your math teacher (far more likely) that you're right. It feels wonderful to bask in the glow of mathematical truth, doesn't it?
But don't break out your suntan lotion just yet. Before we can prove anything, we need to clarify the language we'll use. After all, it's hard to argue with somebody who speaks a different language, right? Don't worry if it seems abstract and mysterious at first. Just like any language, you have to use it in order to really learn it, so it will become much more natural as we start working out some proofs later.