Section CNO  Complex Number Operations

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

$\begin{array}{llll}\hfill \left(2+5i\right)+\left(6-4i\right)& =\left(2+6\right)+\left(5+\left(-4\right)\right)i=8+i& \hfill & \\ \hfill \left(2+5i\right)-\left(6-4i\right)& =\left(2-6\right)+\left(5-\left(-4\right)\right)i=-4+9i& \hfill & \\ \hfill \left(2+5i\right)\left(6-4i\right)& =\left(2\right)\left(6\right)+\left(5i\right)\left(6\right)+\left(2\right)\left(-4i\right)+\left(5i\right)\left(-4i\right)=12+30i-8i-20i^2& \hfill & \\ \hfill & =12+22i-20\left(-1\right)=32+22i& \hfill & \\ \multicolumn{4}{c}{\text{ Division takes just a bit more care. We multiply the denominator by a complex number chosen to produce a real number and then we can produce a complex number as a result.}}\\ \\ \hfill \frac{2+5i}{6-4i}& =\frac{2+5i}{6-4i}\frac{6+4i}{6+4i}=\frac{-8+38i}{52}=-\frac{8}{52}+\frac{38}{52}i=-\frac{2}{13}+\frac{19}{26}i& \hfill & \end{array}$

$⊠$

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 $\left(a+c\right)+\left(b+d\right)i$ .

(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 $\left(ac-bd\right)+\left(ad+bc\right)i$ .

(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.

$\square$

Proof   We could derive each of these properties of complex numbers with a proof that builds on the identical properties of the real numbers. The only proof that might be at all interesting would be to show Property MICN since we would need to trot out a conjugate. For this property, and especially for the others, we might be tempted to construct proofs of the identical properties for the reals. This would take us way too far afield, so we will draw a line in the sand right here and just agree that these nine fundamental behaviors are true. OK?

Mostly we have stated these nine properties carefully so that we can make reference to them later in other proofs. So we will be linking back here often. $■$

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

(This definition contains Notation CCN.) $△$

Example  CSCN
Conjugate of some complex numbers

$\begin{array}{lllllllllllllll}\hfill \overline{2+3i}=2-3i& & \hfill \overline{5-4i}=5+4i& & \hfill \overline{-3+0i}=-3+0i& & \hfill \overline{0+0i}=0+0i& & \hfill & & \hfill & & \hfill & & \hfill \end{array}$

$⊠$

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 $\overline{\alpha +\beta }=\overline{\alpha }+\overline{\beta }$ . $\square$

Proof   Let $\alpha =a+bi$ and $\beta =r+si$ . Then

$■$

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

Proof   Let $\alpha =a+bi$ and $\beta =r+si$ . Then

$\begin{array}{llll}\hfill \overline{\alpha \beta }& =\overline{\left(ar-bs\right)+\left(as+br\right)i}=\left(ar-bs\right)-\left(as+br\right)i& \hfill & \\ \hfill & =\left(ar-\left(-b\right)\left(-s\right)\right)+\left(a\left(-s\right)+\left(-b\right)r\right)i=\left(a-bi\right)\left(r-si\right)=\overline{\alpha }\overline{\beta }& \hfill & \end{array}$

$■$

Theorem  CCT
Complex Conjugation Twice
Suppose that $\alpha$ is a complex number. Then $\overline{\overline{\alpha }}=\alpha$ . $\square$

Proof   Let $\alpha =a+bi$ . Then

$■$

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 $\alpha =a+bi\in {ℂ}^{}$ , is the nonnegative real number

$△$

Example  MSCN
Modulus of some complex numbers

$\begin{array}{lllllllllllllll}\hfill \left|2+3i\right|=\sqrt{13}& & \hfill \left|5-4i\right|=\sqrt{41}& & \hfill \left|-3+0i\right|=3& & \hfill \left|0+0i\right|=0& & \hfill & & \hfill & & \hfill & & \hfill \end{array}$

$⊠$

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 $\left|-3\right|=\left|-3+0i\right|=3$ . Notice too how the modulus of the complex zero, $0+0i$ , has value $0$ .