In this section we review of the basics of working with complex numbers.

## Arithmetic with complex numbers

A complex number is a linear combination of $1$ and $i=\sqrt{-1}$, typically written in the form $a+bi$. Complex numbers can be added, subtracted, multiplied and divided, just like we are used to doing with real numbers, including the restriction on division by zero. We will not define these operations carefully, but instead illustrate with examples.

Example ACN: Arithmetic of complex numbers.

In this example, we used $6+4i$ to convert the denominator in the fraction to a real number. This number is known as the conjugate, which we define in the next section. We will often exploit the basic properties of complex number addition, subtraction, multiplication and division, so we will carefully define the two basic operations, together with a definition of equality, and then collect nine basic properties in a theorem.

Definition CNE (Complex Number Equality) The complex numbers $\alpha=a+bi$ and $\beta=c+di$ are equal, denoted $\alpha=\beta$, if $a=c$ and $b=d$.
(This definition contains Notation CNE.)

Definition CNA (Complex Number Addition) The sum of the complex numbers $\alpha=a+bi$ and $\beta=c+di$ , denoted $\alpha+\beta$, is $(a+c)+(b+d)i$.
(This definition contains Notation CNA.)

Definition CNM (Complex Number Multiplication) The product of the complex numbers $\alpha=a+bi$ and $\beta=c+di$ , denoted $\alpha\beta$, is $(ac-bd)+(ad+bc)i$.
(This definition contains Notation CNM.)

Theorem PCNA (Properties of Complex Number Arithmetic) The operations of addition and multiplication of complex numbers have the following properties.

• If $\alpha,\beta\in\complexes$, then $\alpha+\beta\in\complexes$.
• If $\alpha,\beta\in\complexes$, then $\alpha\beta\in\complexes$.
• For any $\alpha,\,\beta\in\complexes$, $\alpha+\beta=\beta+\alpha$.
• For any $\alpha,\,\beta\in\complexes$, $\alpha\beta=\beta\alpha$.
• For any $\alpha,\,\beta,\,\gamma\in\complexes$, $\alpha+\left(\beta+\gamma\right)=\left(\alpha+\beta\right)+\gamma$.
• For any $\alpha,\,\beta,\,\gamma\in\complexes$, $\alpha\left(\beta\gamma\right)=\left(\alpha\beta\right)\gamma$.
• For any $\alpha,\,\beta,\,\gamma\in\complexes$, $\alpha(\beta+\gamma)=\alpha\beta+\alpha\gamma$.
• There is a complex number $0=0+0i$ so that for any $\alpha\in\complexes$, $0+\alpha=\alpha$.
• There is a complex number $1=1+0i$ so that for any $\alpha\in\complexes$, $1\alpha=\alpha$.
• For every $\alpha\in\complexes$ there exists $-\alpha\in\complexes$ so that $\alpha+\left(-\alpha\right)=0$.
• For every $\alpha\in\complexes$, $\alpha\neq 0$ there exists $\frac{1}{\alpha}\in\complexes$ so that $\alpha\left(\frac{1}{\alpha}\right)=1$.

## Conjugates of Complex Numbers

Definition CCN (Conjugate of a Complex Number) The conjugate of the complex number $\alpha=a+bi\in\complex{\null}$ is the complex number $\conjugate{\alpha}=a-bi$.

Example CSCN: Conjugate of some complex numbers.

Notice how the conjugate of a real number leaves the number unchanged. The conjugate enjoys some basic properties that are useful when we work with linear expressions involving addition and multiplication.

Theorem CCRA (Complex Conjugation Respects Addition) Suppose that $\alpha$ and $\beta$ are complex numbers. Then $\conjugate{\alpha+\beta}=\conjugate{\alpha}+\conjugate{\beta}$.

Theorem CCRM (Complex Conjugation Respects Multiplication) Suppose that $\alpha$ and $\beta$ are complex numbers. Then $\conjugate{\alpha\beta}=\conjugate{\alpha}\conjugate{\beta}$.

Theorem CCT (Complex Conjugation Twice) Suppose that $\alpha$ is a complex number. Then $\conjugate{\conjugate{\alpha}}=\alpha$.

## Modulus of a Complex Number

We define one more operation with complex numbers that may be new to you.

Definition MCN (Modulus of a Complex Number) The modulus of the complex number $\alpha=a+bi\in\complex{\null}$, is the nonnegative real number \begin{equation*} \modulus{\alpha}=\sqrt{\alpha\conjugate{\alpha}}=\sqrt{a^2+b^2}. \end{equation*}

Example MSCN: Modulus of some complex numbers.

The modulus can be interpreted as a version of the absolute value for complex numbers, as is suggested by the notation employed. You can see this in how $\modulus{-3}=\modulus{-3+0i}=3$. Notice too how the modulus of the complex zero, $0+0i$, has value $0$.