Example Questions and Solutions
Selected solutions for various example questions.
$ \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\RP}{\operatorname{Re}} \newcommand{\IP}{\operatorname{Im}} \newcommand{\abs}[1]{\left| #1 \right|} \newcommand{\p}[1]{\left( #1 \right)} \newcommand{\qed}{\blacksquare} $
Chapter 1: Example Chapter
Exercise 1.1 Prove that $\sqrt{2}$ is irrational.
Proof
Suppose that $\sqrt 2$ is rational. Then it can be written as a fraction $$ \sqrt 2 = \frac{a}{b} $$ where $\gcd(a,b)$ = 1. By squaring both sides we see $$ a^2 = 2 b^2. $$ Since $a^2$ is a multiple of $2$, and $2$ is prime, we can conclude $a$ is even. Let $a = 2k$ for some $k \in \Z$. We now have: $$ a^2 = (2k)^2 = 4k^2 = 2b^2 \implies b^2 = 2k^2. $$ For the same reasons, $b$ is also even. Thus $\gcd(a, b) \geq 2$, contradicting our initial assumption.
$\qed$
Exercise 1.2 Question for exercise 1.2 goes here.
Proof
Proof content for exercise 1.2 goes here...
$\qed$