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Section 2.1 Basic 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.

complex numbers represented by vectors
Figure 2.1.1. Complex 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\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.

Example 2.1.2. 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*}

Exercises Exercises

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

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.