pre-math-camp.Rmd

---
title: "Introduction to Math for Political Scientists"
subtitle: "AKA Math Camp"
author: "J. Alexander Branham"
date: "Fall 2017"
mainfont: Linux Libertine
linkcolor: "blue"
urlcolor: "blue"
output:
    pdf_document:
        latex_engine: xelatex
        toc: false
---

Welcome to the Department of Government at the University of Texas!
I'm looking forward to meeting everyone. I'll be your instructor for
Introduction to Math for Political Scientists, otherwise known as Math
Camp. In the past our incoming students have told us their math skills
are rusty and they would like to be better prepared for the Stats I
class that is required of all Government Department graduate students.
Math Camp will give you an opportunity to brush up on your
quantitative skills, provide an introduction to some of the
statistical software you'll be using next year, and is great chance to
get to know your fellow first-year students.

This year, we're meeting August 21st through the 29th. Classes
at UT start August 30th. There is a full (tentative) schedule for the
class [on the class's website](https://jabranham.com/math-camp). 

Here is a checklist of items that you should review before your
arrival. Please feel free to email me with any problems you run into.

* Review what math knowledge is assumed 
    + Complete the exercises in that section to refresh your memory if needed 
* Install R and Rstudio on your computer using the [instructions](https://jabranham.com/math-camp) on the website
    + Complete the exercises below to get acquainted with R

Other than that, welcome to Texas and the the Government Department!
I'm looking forward to meeting everyone!

# Assumed math knowledge 

Students come to the department with a wide range of comfort with and
exposure to math. I assume a very minimal exposure to math, certainly
nothing beyond what was on the GRE. The following material is
knowledge that I assume you are familiar with. Odds are that you've
seen all of this before; I suggest you read through and do the
exercises anyway to refamiliarize yourself with math, especially if
it's been a while since you've dealt with it.

## Arithmetic

I assume you have a working knowledge of arithmetic (addition,
subtraction, multiplication, division). I also assume that you know
the standard order of operations (PEMDAS - parentheses, exponents,
multiplication/division, addition/subtraction). I assume you are
comfortable with some of the various properties of arithmetic - that
addition, subtraction, multiplication, and division are communicative
and associative, and that multiplication and division are distributive
as well. Finally, I assume that you understand that equalities are
symmetric and transitive. For a review of any of this, I have
[several slides that should bring you up to speed](/math-camp/slides/0-arithmetic.pdf)

### Arithmetic exercises

1. Simplify the following expression to a single real number:

$$(1 + 3)^2 - 4 + \left(\frac{3}{4} * \frac{16}{3} \right)$$

2. Is the following statement true or false: 
   We know that $y>x$ and we know that $x <z$ therefore we know that
   $y>z$. 
   
## Basic algebra

A basic understanding of high school level algebra is assumed. That
is, you understand the basics of using symbols in math to denote
variables or unknowns (in fact, you used this in one of the above
questions already!). I also assume that you know and understand how to
solve for a variable in a given equality (e.g. $4x=3$, find $x$). No
need to memorize the quadratic formula or anything complicated. 

### Algebra exercises

1. Find $x$:

$$(1+3)^2 + 4x - 16 = 173$$

2. Find $y$:

$$y(2+7) - 7(y-3) = 21 $$

3. Find $z$:

$$z + (16-7)^{\frac{1}{2}}= 201$$

4. Find $x$ and $y$:

$$2x + y = 4$$

$$x - y = -1$$

5. Solve for $z$:

$$z^{2} + 3(1 - 2) = 2$$

## Basic geometry

Geometry is used much less than the above math skills in our
coursework. I assume a very basic familiarity with geometry, like how
to calculate the are of a rectangle. 

### Geometry exercises 

1. You are given a rectangle and told that its area is 1 square kilometers
   and that its width is 3 kilometers. What is its length? 

# Programs 

If you have a personal laptop, please install R and an editor (RStudio
is probably the most popular in the department). R is free and
open-source software. Installation of both programs is
straightforward; instructions are on the class website. Please email
me if you run into any problems.

## R Exercises 

No familiarity with R is assumed before math camp starts, so you are
not obligated to do these exercices, though they might help you get
familiar with the basics of how R works. If you run into any issues,
feel free to email me. 

R can be used as a calculator. Open up R and try adding 2 and 2. I've
included what R will print out after two `##`:

```{r}
2 + 2
```

R does that and prints the output, 4 in this case. Don't worry for now
about the [1].

R also understands the standard order of operations. As an example,
you can see what this statement evaluates to:

```{r}
(3 / 4) + 2 * (1 + 3)
```

Once R prints out `8.75`, it "forgets" the result. If you wanted to
save the result to access later, you can assign the result of that
statement to an object. `<-` is called the "assignment operator" and
it assigns the right side to the left side.[^3] So this tells R to
make `x` hold the value of the mathematical expression on the right:

```{r}
x <- (3 / 4) + 2 * (1 + 3)
```

If all goes well, `x` now holds the result of that expression (which
we know is `8.75`). Notice that R doesn't print anything out. If you
want to see what `x` is, you can just type it into R:

```{r}
x
```

Of course, we're oftentimes working with data where each row is an
observation and each column is a variable. R refers to this type of
data as a `data.frame`. R includes several `data.frame`s as toy
examples. One famous one is the `mtcars` dataset, which we can look at
with the following command (`head` will list the first 6 observations.
If you want to look at the whole thing, just type `mtcars` and R will
print out the whole dataset). 

```{r}
head(mtcars)
```

We might be interested in the average fuel efficiency of these cars.
To get that information, we use the `mean` function and give it the
mpg data that we're interest in. This is how to do that in R:

```{r}
mean(mtcars$mpg)
```

The `$` symbol tells R to look inside the object on the left for the
object on the right. In our case, this means that R will look inside
`mtcars` for `mpg`. Since it finds it, it then calls the `mean`
function and prints the mean mpg (about 20mpg in this case). 

R can also produce publication-quality figures. We'll go over the
basics of this, and here's an example.[^2]

```{r}
library(ggplot2)

ggplot(mtcars, aes(x = mpg, y = wt)) +
  geom_point() +
  geom_smooth() +
  labs(x = "Miles per Gallon",
       y = "Weight")
```


# Answers

## Math

### Arithmetic 

1. 16
2. False 

### Algebra

1. $x=\frac{173}{4}$
2. $y=0$
3. $z=198$
4. $x=1\qquad y=2$
5. $z = \sqrt{5}$

### Geometry

1. $\frac{1}{3}$

[^2]: If you try to do this on your computer, you'll have to install
    the `ggplot2` package with `install.packages("ggplot2")`
    beforehand. Don't worry about what this means --- we'll cover it
    in math camp. 

[^3]: If you have experience with programming in other languages (like
    python), this symbol may look a little strange to use for
    assignment. Many other languages use the equals sign (`=`). You
    can do that in R, but it's not considered best practices