Chapter 2 Introduction

2.1 History, the imaginary unit

Complex numbers first appeared in writing in the 1500s, although they didn’t become widely accepted until the 1800s. The main motivation for their definition comes from finding solutions to quadratic and cubic equations.
One of the easiest examples is probably \(x^2 + x + 1 = 0\). We cannot factorize it using real numbers, so we need to apply the quadratic formula \[x_{1, 2} = \frac{- b \pm \sqrt{b^2 - 4 ac}}{2 a}\] with \(a = b = c = 1\). Plugging the numbers in we get \(x_{1, 2} \frac{- 1 \pm \sqrt{- 3}}{2}\).
The problematic part is \(\sqrt{- 3}\), since the square root was defined only for non-negative real numbers. We introduce a special unit, called the imaginary unit, denoted \(i\) (NOTE that in some disciplines the symbol \(j\) is used).

Definition: \[i^2 = - 1\]

 

Then we can write \(\sqrt{- 3} = \sqrt{- 1 \cdot 3} = \sqrt{- 1} \cdot \sqrt{3} = i \sqrt{3}\) to find the square root. The roots of the polynomial are therefore \(x_{1, 2} = \frac{- 1 \pm i \sqrt{3}}{2}\).

Since \(i\) is defined such that \(i^2 = - 1\), it follows by the rules of algebra that \((- i)^2 = - 1\) too. This raises questions about different powers of \(i\). We have

  • \(i^0 = 1\),

  • \(i^1 = i\),

  • \(i^2 = - 1\),

  • \(i^3 = i \cdot i^2 = i (- 1) = - i\),

  • \(i^4 = i^2 \cdot i^2 = (- 1) (- 1) = 1 = i^0\).

We notice that we circled around and \(i^0 = i^4\).

2.2 The rectangular form, Argand diagrams

In general, a complex number \(z\) is written as \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. \(a\) and \(b\) are called the real and imaginary component of the complex number, respectively, which we write as \(Re (z) = a\) and \(Im (z) = b\). We can describe the set of complex numbers, denoted \(\mathbb{C}\), as the set of all numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. In set notation, we write this as \[\mathbb{C}= \{a + bi \hspace{0.27em} | \hspace{0.27em} a, b \in \mathbb{R}, \hspace{0.27em} i^2 = - 1\}\] If \(b = 0\), \(z = a + 0 i = a\) is a real number (so the real numbers are a subset of the complex numbers). If \(a = 0\), \(z = 0 + bi = bi\) is an number.
If we take a standard Cartesian plane, with the x axis being the real numbers, and the y axis being the imaginary numbers, we get the . We can then represent complex numbers as points on the plane. A complex plane with complex numbers plotted on it is referred to as an .