# More Math Tips

## Part 2: The Language of Math

### Know the definitions

Math has its own language with its own vocabulary. You wouldn’t expect to do well on a cell biology test if you don’t know what a ribosome or an endoplasmic reticulum is. And chances are, you won’t get the best grade you can in your geometry class if you never learn the difference between corresponding angles and alternate interior angles. Most math textbooks make it very clear what their definitions are. They are usually inside a big colored rectangle and have the word Definition at the top. If you’re not understanding what a problem is asking, ask yourself if you know what all the words mean. If you don’t, and it seems like it might be a math word, go back through your book chapter and look through the definitions to see if it appears there. Or look up the word in the index at the back of the book, maybe it was in an earlier chapter and you just forgot about it. Make sure you periodically review the definitions you’ve learned so far to make sure you have them memorized, or at least can remember them with a little bit of thought.

### Be lazy

Besides the definitions, you should memorize nothing else if at all possible. There are a few exceptions to this, but for the most part you want to reduce the number of formulas you need to memorize by simply knowing how to do the procedure that the formula is a “shortcut” for. Sometimes you will use a formula so much that you will just know it, and that’s fine. What if you forget the formula, though? If you know the procedure that produced the formula, you can re-derive it from first principles. This is the difference between a definition (or an axiom) and a theorem. A theorem is something that’s proved. A theorem follows from our basic assumptions. We can always get the theorem back if we understand the basics and use our brain. For instance I may forget the identity $\large \cos^2x = 1-\sin^2x.$  But if I remember the unit circle and the pythagorean theorem, I can get $\large \sin^2x + \cos^2x = 1$ and do some algebra to get back $\large \cos^2x = 1-\sin^2x$. Or I might have forgotten the quadratic formula. That’s okay, if I know how to complete the square and I use the square root property, I don’t need the quadratic formula.

### Understand the notation

“Notation” is just a fancy word meaning “the way we write math.” Understanding the standard way of writing things in math is obviously important if you’re going to get anywhere. Imagine trying to read a book without knowing the alphabet!

Misunderstanding notation actually can get you the wrong answer sometimes. Imagine if I was asked to find $\sin^{-1}(1)$. I remember that my teacher said $\sin^2(x)$ is just funny mathematical notation that means the same thing as $(\sin(x))^2$. That must mean that $\sin^{-1}(1)$ means the same thing as $(\sin(1))^{-1}$ which means the same thing as 1/sin(1), right? I plug that into my calculator and get 57, which is the wrong answer. It’s the wrong answer because in this context, the power of -1 is not actually an exponent at all, but really just means “the inverse sine function,” and so $\sin^{-1}(x)$ gives me the angle between -90 degrees and +90 degrees that I would need to put into sin(x) to get 1. (in this case, the answer would be 90 degrees).

“But that’s dumb! Why does a power of negative 1 mean one thing sometimes, and another thing a different time?” Well, it’s all about context. Mathematicians have adopted a convention (an agreed upon way of doing things) to interpret -1 in this way because it’s suggestive. $\frac{a}{b}$ and $\frac{b}{a}$ (which can also be written $(\frac{a}{b})^{-1}$) are inverses because they “undo” each other. If I multiply 5 by $\frac{2}{3}$ and then multiply that by $\frac{3}{2}$ I will get back 5, the thing I started with. Similarly, $\sin(x)$ and $\sin^{-1}(x)$ are inverses because $\sin^{-1}(\sin(x)) = x$ and $\sin(\sin^{-1}(x)) = x$. In other words plugging x into the sin function and then plugging that into the $\sin^{-1}$ function, we get back x, the thing we started with.

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