Chapter 1 - Vector Spaces

Definition 1.1 Complex Numbers $\mathbb{C}$

A complex number is an ordered pair $(a,b)$, where $a,b\in \mathbb{R}$, but usually written as $a+bi$.

• The set of all complex numbers is denoted as $\mathbb{C}$:

$$\mathbb{C}=\{a+bi:a,b\in \mathbb{R}\}$$

• Addition and multiplication on $\mathbb{C}$ are defined by: $$\begin{array}{rl}(a+bi)+(c+di)& =(a+c)+(b+d)i\\ (a+bi)(c+di)& =(ac-bd)+(ad+bc)i\end{array}$$ Where $a,b,c,d\in \mathbb{R}$.

Note that $\mathbb{R}\subset \mathbb{C}$, since $\mathbb{R}=\{a+bi:b=0,a\in \mathbb{R}\}$.

Theorem 1.3 Properties of Complex Arithmetic

We may derive these properties from the definitions above.

1. Commutativity. For all $\alpha ,\beta \in \mathbb{C}$,

$$\begin{gathered} \alpha +\beta =\beta +\alpha \\ \alpha \beta =\beta \alpha \end{gathered}$$

2. Associativity. For all $\alpha ,\beta ,\lambda \in \mathbb{C}$,

$$\begin{gathered} (\alpha +\beta )+\lambda =\alpha +(\beta +\lambda ) \\ (\alpha \beta )\lambda =\alpha (\beta \lambda ) \end{gathered}$$

3. Identities. For all $\lambda \in \mathbb{C}$,

$$\begin{gathered} \lambda +0=\lambda \\ \lambda 1=\lambda \end{gathered}$$

4. Additive Inverse is unique. For every $\alpha \in \mathbb{C}$, there exists a unique $\beta \in \mathbb{C}$ such that $\alpha +\beta =0$.

5. Multiplicative Inverse is unique. For every $\alpha \in \mathbb{C}$ with $\alpha \ne 0$, there exists a unique $\beta \in \mathbb{C}$ such that $\alpha \beta =1$.

6. Distributive Property. $\lambda (\alpha +\beta )=\lambda \alpha +\lambda \beta$ for all $\lambda ,\alpha ,\beta \in \mathbb{C}$.