## 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,

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,

and if \(k\) is a real number, we define scalar multiplication by

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

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 \(i^2 = -1\text{.}\)

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

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

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

A few other computations:

### Exercises Exercises

###### 1.

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

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

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 \(z\text{.}\)

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.

- \(|w\cdot z| = |w|\cdot |z|\text{.}\)
- \(\overline{zw} = \overline{z}\cdot \overline{w}\text{.}\)
- \(\overline{z + w} = \overline{z} + \overline{w}\text{.}\)
- \(z + \overline{z} = 2\text{Re}(z)\text{.}\) (Hence, \(z + \overline{z}\) is a real number.)
- \(z - \overline{z} = 2\text{Im}(z)i\text{.}\)
- \(|z| = |\overline{z}|\text{.}\)

###### 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

- Prove that \(a^2 + b^2 = c^2\text{.}\)
- Find the complex number \(z = x + yi\) that generates the famous triple (3,4,5).
- Find the complex number that generates the triple (5,12,13).
- 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.