Appendix A: Math review

A.1: Math is a way of writing language

Just like we use a spoken and written language such as English, Spanish, or Chinese to communicate verbally, we use mathematics to communicate ideas in science. Mathematical notation has been developed to accurately and efficiently communicate these ideas. Let’s start with a simple example. You could state:

Twelve plus four divided by eight is equal to two.

This could be interpreted multiple ways: are you adding twelve and four, then dividing that result by eight? Do you first divide four by eight and add that result to twelve? You need some punctuation for clearer communication, such as:

Twelve plus four, divided by eight, is two.

Just two commas change your sentence from something ambiguous into a statement of fact. As the problems you are working on become more complex, the risk of miscommunication increases.

Mathematical notation exists to eliminate (or at least minimize) the potential for miscommunication due to imprecision in verbal language. Our previous statement becomes

\[ \frac{12 + 4}{8} = 2 \]

Each symbol, and how they are grouped and organized has precise, unambiguous meaning.

Now let’s try some algebra:

The total of twelve and some unknown quantity, divided by eight, is equal to two.

Determine the unknown quantity.

In mathematical notation, we have

\[ \frac{12 + x}{8} = 2 \]

Working through the steps to determine the unknown quantity is a matter of re-phrasing the statement:*

The total of twelve plus some unknown quantity is equal to the product of two and eight (two groups of eight)

\[ 12 + x = (2)(8) \]

In other words, the total of twelve and some unknown quantity is equal to sixteen

\[ 12 + x = 16 \]

So, some unknown quantity is equal to sixteen minus twelve (twelve less than sixteen)

\[ x = 16 – 12 \]

Therefore, the unknown quantity is four!

\[ x = 4 \]

Once you start thinking about math the same way you think about language, working with and using math can become much less daunting—it’s just a matter of translation! An addition sign means “take this and add it to that.” An exponent translates to “take this and multiply it by itself that many times.” The sine of an angle is nothing more than two particular sides of a right triangle, divided by each other.**

To take this analogy one step farther, you can think of operations (addition, subtraction, etc.) as verbs, and order of operations as grammar and punctuation. And while languages can have very complicated grammatical structures and rules, which nearly always have exceptions, mathematical “grammar rules” are relatively simple and rarely—if ever—have exceptions.

In the rest of this appendix, we will look at some specific examples of the math used in this book.

*This explanation was inspired by the post Simplify! on the Math with Bad Drawings blog by Ben Orlin

**Even into calculus, this mentality can help: the “derivative” of a function is just the slope of a line at some specific spot; an “integral” is the area under a line between two particular points.