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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}\}. \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).

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Figure2.1.1Complex numbers as vectors 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. \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*}

SubsectionExercises

1

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

a. \(z = 5 + 2i\text{,}\) \(s = -4\text{,}\) \(w = -1 + 2i\text{.}\)

b. \(z = 3i\text{,}\) \(s = 1/2\text{,}\) \(w = -3 + 2i\text{.}\)

c. \(z = 1 + i\text{,}\) \(s = 0.6\text{,}\) \(w = 1 - i\text{.}\)

2

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.