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 .