Complex Numbers

Complex numbers were invented so that we can take square roots of negative numbers. The idea is to assume that we have a square root of $-1$, denoted by $i$, that obeys the rules of arithmetic. We have:

$$i^2=-1$$

Definition: Complex Numbers

• A complex number is an ordered pair $(a,b)$ where $a,b\in \mathbb{R}$, but we write it as $a+bi$.
• The set of all complex numbers is denoted by $\mathbb{C}$:

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

• Addition and multiplication on $\mathbb{C}$ are defined by:

$$\begin{array}{r}(a+bi)+(c+di)=(a+c)+(b+d)i\\ (a+bi)(c+di)=(ac-bd)+(ad+bc)i\end{array}$$

If $a\in \mathbb{R}$, we identify $a=a+0i$. Thus, $\mathbb{R}$ is a subset of $\mathbb{C}$.

Properties

Properties of complex arithmetic:

• Commutativity: $\alpha +\beta =\beta +\alpha$ and $\alpha \beta =\beta \alpha$ for all $\alpha ,\beta \in \mathbb{C}$
• Identities: $\lambda +0=\lambda$ and $\lambda 1=\lambda$ for all $\lambda \in \mathbb{C}$
• Additive inverse: For every $\alpha \in \mathbb{C}$, there exists a unique $\beta \in \mathbb{C}$ such that $\alpha +\beta =0$
• Multiplicative inverse: For every $\alpha \in \mathbb{C}$ with $\alpha \ne 0$, there exists a unique $\beta \in \mathbb{C}$ such that $\alpha \beta =1$
• Distributive property: $\lambda (\alpha +\beta )=\lambda \alpha +\lambda \beta$ for all $\lambda ,\alpha ,\beta \in \mathbb{C}$

Subtraction and Divsion

We can use the additive and multiplicative inverses to define subtraction and division operations with complex numbers.

Suppose $\alpha ,\beta \in \mathbb{C}$.

• Let $-\alpha$ denote the additive inverse of $\alpha$. Thus, $-\alpha$ is the unique complex number such that $\alpha +(-\alpha )=0$.
• Subtraction on $\mathbb{C}$ is defined by

$$\beta -\alpha =\beta +(-\alpha )$$

• For $\alpha \ne 0$, let $1/\alpha$ and $\frac{1}{\alpha }$ define the multiplicative inverse of $\alpha$. Thus, $1/\alpha$ is the unique complex number such that $\alpha (1/\alpha )=1$.
• For $\alpha \ne 0$, division by $\alpha$ is defined by

$$\frac{\beta }{\alpha }=\beta \left(\frac{1}{\alpha }\right)$$

$\mathbb{F}$

So that we can conveniently make definitions and prove theorems that apply to both real and complex numbers, we adopt the notation of $\mathbb{F}$, which stands for either $\mathbb{R}$ or $\mathbb{C}$. The letter $\mathbb{F}$ is used because $\mathbb{R}$ and $\mathbb{C}$ are examples of fields.

Thus, if we prove a theorem involving $\mathbb{F}$, we will know that it holds when $\mathbb{F}$ is replaced by $\mathbb{R}$, and when $\mathbb{F}$ is replaced with $\mathbb{C}$.

Elements of $\mathbb{F}$ are called scalars. Scalar objects are numerical values, as opposed to vectors.

For $\alpha \in \mathbb{F}$ and a positive integer $m$, we define ${\alpha }^m$ to denote the product of 𝛼 with itself $m$ times:

$${\alpha }^m=\alpha \underset{m\text{ times}}{\underbrace{\cdot \cdot \cdot }}\alpha$$

This implies that

$$({\alpha }^m{)}^n={\alpha }^{mn}\text{and}(\alpha \beta {)}^m={\alpha }^m{\beta }^m$$

for all $\alpha ,\beta \in \mathbb{F}$ and all positive integers $m$, $n$.