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}\}.
\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),
\end{equation*}
and if \(k\) is a real number, we define scalar multiplication by
\begin{equation*}
k\cdot(x,y) = (kx,ky).
\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.
\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. The complex number \(z-w\) can be represented by the vector from \(w\) to \(z\) in the plane (see Figure 2.1.1).

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.
\end{align*}

The modulus of \(z=x+yi\text{,}\) denoted \(|z|\text{,}\) is given by
\begin{equation*}
|z| = \sqrt{x^2 + y^2}.
\end{equation*}

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

One final bit of terminology: the conjugate of \(z = x+yi\text{,}\) denoted \(\overline{z}\text{,}\) is
\begin{equation*}
\overline{z} = x-yi.
\end{equation*}

Example2.1.2Arithmetic 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.
\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.
\end{align*}

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

3

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

a. \(|w\cdot z| = |w|\cdot |z|\text{.}\)

b. \(\overline{zw} = \overline{z}\cdot \overline{w}\text{.}\)

c. \(\overline{z + w} = \overline{z} + \overline{w}\text{.}\)

d. \(z + \overline{z} = 2\text{Re}(z)\text{.}\) (Hence, \(z + \overline{z}\) is a real number.)

e. \(z - \overline{z} = 2\text{Im}(z)i\text{.}\)

f. \(|z| = |\overline{z}|\text{.}\)

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*}
a. Prove that \(a^2 + b^2 = c^2\text{.}\)

b. Find the complex number \(z = x + yi\) that generates the famous triple (3,4,5).

c. Find the complex number that generates the triple (5,12,13).

d. 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.