Chapter 1 - Complex Numbers

— date: 03-07-2023 type: Lecture subject: MATH1051 tags: lecture Topic::

Chapter 1 - Complex Numbers

Lecture Notes 2008

1 Complex Numbers

1.1 - Notation

The symbol $\in$ means “is an element of”. Thus we write:

$$x\in \mathbb{R}$$

to mean $x$ is an element of $\mathbb{R}$. That is, $x$ is a real number. Hence, we can write, for example, $2\in \mathbb{R},\pi \in \mathbb{R},-\sqrt{3}\in \mathbb{R}$, and so forth.

1.2 - Complex Numbers

Complex numbers were introduced in the 16th century to obtain roots of polynomial equations. A complex number is of the form

$$z=x+iy$$

where $x,y\in \mathbb{R}$ and $i$ is (formally) a symbol satisfying $i^2=-1$. The quantity $x$ is called the real part of $z$ and $y$ is called the imaginary part of $z$. The set of all complex numbers is denoted $\mathbb{C}$. Thus $3-2i\in \mathbb{C}$.

1.2.1 Example

The real part of $3-2i$ is $3$ and the imaginary part is $-2$ (not $-2i$). Complex numbers can be added and multiplied by replacing $i^2$ everywhere with $-1$. For example $(2i{)}^2=4i^2=-4$.

1.2.2 Example

Simplify $(3-2i)(1+i)$.

$$\begin{array}{rl}(3-2i)(1+i)& =3+3i-2i-2i^2\\ & =3+i+2=5+i\end{array}$$

1.2.3 Example

Suppose $a,\in \mathbb{R}$. Simplify $(a+bi)(a-bi)$.

$$\begin{array}{rl}(a+bi)(a-bi)& a^2-abi+abi-b^2{i}^2\\ & =a^2+b^2\end{array}$$

1.2.4 Example

Simplify $\frac{3i-2i}{1-i}$.

$$\begin{array}{rl}\frac{3i-2i}{1-i}& =\frac{3i-2i}{1-i}\times \frac{1+i}{1+i}\\ & =\frac{(3-2i)(1+i)}{1^2+1^2}\\ & =\frac{1}{2}(3-2i)(1+i)\\ & =\frac{1}{2}(5+i)\\ & =\frac{5}{2}+\frac{1}{2}i\end{array}$$

It is a fact if we consider complex roots of polynomials and count them with their correct multiplicity, then a polynomial of degree $n$ always has $n$ roots. For example, every quadratic has two roots.

1.2.5 Example

Find the roots of $x^2+2x+2=0$.

$$\begin{array}{rl}{x}^2+2x+2& =(x+1{)}^2+1=0\\ ⟺(x+1{)}^2& =-1=i^2\\ ⟺x+1& \pm i\\ \therefore x& =-1\pm i\text{ are the roots.}\end{array}$$

Alternatively use the quadratic formula:

$$\begin{array}{rl}x& =\frac{1}{2}(-2\pm \sqrt{4-8})=\frac{1}{2}(-2\pm 2i)\\ & =-1\pm i.\end{array}$$

1.3 Polar Form

A complex number $z=x+iy$ may be represented by a point in the complex plane where the vertical axis is the imaginary axis and the horizontal axis is the real axis.

We can also specify $z$ by giving the length $r$ and the angle $\theta$ in figure 1. The quantity $r$ is called the modulus of $z$, denoted $|z|$. It measures the distance of $z$ from the origin. The angle $\theta$ is called the argument of $z$. We have:

1.3.1 Example

Write $z=1+i$ in polar form. First find the modulus:

$$|z|=\sqrt{1+1}=\sqrt{2}$$

We have for the argument $\tan \theta =\frac{y}{x}$, so we can take

$$\begin{array}{rl}\theta & =\arctan \frac{y}{x}\\ & =\arctan 1\\ & =\frac{\pi }{4}\end{array}$$ $$\begin{array}{rl}\therefore z& =\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)\end{array}$$

1.4 Euler’s Formula

Euler’s formula states for any real number $\theta$:

$$\cos \theta +i\sin \theta =e^{i\theta }$$

(To make sense of this, one has to define the exponential function for complex arguments. This may be done using a series.) Thus every complex number $z=x+iy$ can be represented in polar form

$$z=r{e}^{i\theta }$$