## Section2.1Basic Notions

The set of complex numbers is obtained algebraically by adjoining the number $i$ to the set $\mathbb{R}$ of real numbers, where $i$ is defined by the property that $i^2 = -1$. We will take a geometric approach and define a complex number to be an ordered pair $(x,y)$ of real numbers. We let $\mathbb{C}$ denote the set of all complex numbers,

\begin{equation*} \mathbb{C} = \{ (x,y) ~|~ x, y \in \mathbb{R}\}\text{.} \end{equation*}

Given the complex number $z = (x,y)\text{,}$ $x$ is called the real part of $z$, denoted Re$(z)\text{;}$ and $y$ is called the imaginary part of $z$, denoted Im$(z)\text{.}$ The set of real numbers is a subset of $\mathbb{C}$ under the identification $x \leftrightarrow (x,0)\text{,}$ for any real number $x\text{.}$

Addition in $\mathbb{C}$ is componentwise,

\begin{equation*} (x,y) + (s,t) = (x+s,y+t)\text{,} \end{equation*}

and if $k$ is a real number, we define scalar multiplication by

\begin{equation*} k\cdot(x,y) = (kx,ky)\text{.} \end{equation*}

Within this framework, $i = (0,1)\text{,}$ meaning that any complex number $(x,y)$ can be expressed as $x+yi$ as suggested here:

\begin{align*} (x,y)\amp = (x,0) + (0,y)\\ \amp = x(1,0) + y(0,1)\\ \amp = x + yi\text{.} \end{align*}

The expression $x + yi$ is called the Cartesian form of the complex number. This form can be helpful when doing arithmetic of complex numbers, but it can also be a bit gangly. We often let a single letter such as $z$ or $w$ represent a complex number. So, $z = x + yi$ means that the complex number we're calling $z$ corresponds to the point $(x,y)$ in the plane.

It is sometimes helpful to view a complex number as a vector, and complex addition corresponds to vector addition in the plane. The same holds for scalar multiplication. For instance, in Figure 2.1.1 we have represented $z = 2 + i\text{,}$ $w = -1 + 1.5i\text{,}$ as well as $z + w = 1 + 2.5i\text{,}$ as vectors from the origin to these points in $\mathbb{C}\text{.}$ The complex number $z-w$ can be represented by the vector from $w$ to $z$ in the plane.

We define complex multiplication using the fact that $i^2 = -1\text{.}$

\begin{align*} (x+yi)\cdot(s+ti) \amp = xs+ysi+xti+yti^2\\ \amp = (xs-yt) + (ys+xt)i\text{.} \end{align*}

The modulus of $z=x+yi\text{,}$ denoted $|z|\text{,}$ is given by

\begin{equation*} |z| = \sqrt{x^2 + y^2}\text{.} \end{equation*}

Note that $|z|$ gives the Euclidean distance of $z$ to the point (0,0).

The conjugate of $z = x+yi\text{,}$ denoted $\overline{z}\text{,}$ is

\begin{equation*} \overline{z} = x-yi\text{.} \end{equation*}

In the exercises the reader is asked to prove various useful properties of the modulus and conjugate.

###### Example2.1.2.Arithmetic of complex numbers.

Suppose $z = 3 - 4i$ and $w = 2 + 7i\text{.}$

Then $z+w = 5 +3i\text{,}$ and

\begin{align*} z\cdot w \amp = (3-4i)(2+7i)\\ \amp = 6 + 28 - 8i + 21i\\ \amp = 34 + 13i\text{.} \end{align*}

A few other computations:

\begin{align*} 4z \amp = 12 - 16i\\ |z| \amp = \sqrt{3^2 + (-4)^2} = 5\\ \overline{zw}\amp =34-13i\text{.} \end{align*}

### ExercisesExercises

###### 1.

In each case, determine $z + w\text{,}$ $sz\text{,}$ $|z|\text{,}$ and $z\cdot w\text{.}$

1. $z = 5 + 2i\text{,}$ $s = -4\text{,}$ $w = -1 + 2i$
2. $z = 3i\text{,}$ $s = 1/2\text{,}$ $w = -3 + 2i$
3. $z = 1 + i\text{,}$ $s = 0.6\text{,}$ $w = 1 - i$

a. $z + w = 4 + 4i, sz = -20 - 8i, |z| = \sqrt{29},$ and $zw = -9 + 8i\text{.}$

###### 2.

Show that $z\cdot \overline{z} = |z|^2\text{,}$ where $\overline{z}$ is the conjugate of $z\text{.}$

Hint

Let $z = a + bi\text{,}$ and show that the two sides of the equation agree.

###### 3.

Suppose $z=x+yi$ and $w=s+ti$ are two complex numbers. Prove the following properties of the conjugate and the modulus.

1. $|w\cdot z| = |w|\cdot |z|\text{.}$
2. $\overline{zw} = \overline{z}\cdot \overline{w}\text{.}$
3. $\overline{z + w} = \overline{z} + \overline{w}\text{.}$
4. $z + \overline{z} = 2\text{Re}(z)\text{.}$ (Hence, $z + \overline{z}$ is a real number.)
5. $z - \overline{z} = 2\text{Im}(z)i\text{.}$
6. $|z| = |\overline{z}|\text{.}$
###### 4.

A Pythagorean triple consists of three integers $(a,b,c)$ such that $a^2 + b^2 = c^2\text{.}$ We can use complex numbers to generate Pythagorean triples. Suppose $z = x + yi$ where $x$ and $y$ are positive integers. Let

\begin{equation*} a = { Re}(z^2)~~~~~b = {Im}(z^2)~~~~~ c = z\overline{z}. \end{equation*}
1. Prove that $a^2 + b^2 = c^2\text{.}$
2. Find the complex number $z = x + yi$ that generates the famous triple (3,4,5).
3. Find the complex number that generates the triple (5,12,13).
4. Find five other Pythagorean triples, generated using complex numbers of the form $z = x + yi\text{,}$ where $x$ and $y$ are positive integers with no common divisors.