1.1 Complex Numbers

1.1 Complex Numbers

Before we explore complex numbers, let's first consider the question: "Is there a real number whose square is negative?" To answer this question, we can examine a few simple examples. Let's take a look at equations 1 and 2 for better understanding.

Equation 1: $x^2 - 1 = 0$

[Graph of $f(x) = x^2 - 1$ crossing x-axis at $-1$ and $1$]
Figure 1.1

Equation 1 has two real solutions $x = -1, 1$. The graph of $f(x) = x^2 - 1$ crosses the $x$-axis at $(-1, 0)$ and $(1, 0)$.

Equation 2: $x^2 + 1 = 0$

[Graph of $f(x) = x^2 + 1$ not crossing x-axis]
Figure 1.2

Equation 2 has no real solution, since the graph of $f(x) = x^2 + 1$ does not cross the $x$-axis.

This is because squaring any real number never results in a negative value. Therefore, if equation 2 has a solution, we need to introduce a new kind of number, an imaginary number, defined as the square root of $-1$. This imaginary unit is represented by the symbol $i$ (iota). The imaginary number $i$ tells us that $i^2 = -1$.

Example 1

Simplify the following:
(i) $i^7$ (ii) $i^8$ (iii) $i^{17}$ (iv) $i^{-25}$

Solution

(i) $i^7 = i^6 \times i = (i^2)^3 \times i = (-1)^3 \times i = -i$

(ii) $i^8 = (i^2)^4 = (-1)^4 = 1$

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Complex numbers are essential for many technologies like smartphone signal processing and MRI imaging.

(iii) $i^{17} = i^{16} \times i = (i^2)^8 \times i = (-1)^8 \times i = i$

(iv) $i^{-25} = \dfrac{1}{i^{25}} = \dfrac{1}{i^{24} \times i} = \dfrac{1}{(i^2)^{12} \times i} = \dfrac{1}{(-1)^{12} \times i} = \dfrac{1}{i} = \dfrac{1}{i} \times \dfrac{i}{i} = \dfrac{i}{i^2} = \dfrac{i}{-1} = -i$

We have observed that the equation $x^2 + 1 = 0$ has no solution within the real number system. To address equations like this, we expand the real number system to include new types of numbers, leading to the development of the complex number system in rectangular form.

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History!
The idea of complex numbers first emerged in $16^{\text{th}}$ century, when Italian Mathematician Gerolamo Cardano discovered that equations could still be solved when they involved the square root of negative number. In the $18^{\text{th}}$ century, Carl Friedrich Gauss (German Mathematician and astronomer) expanded this early idea.
[Image of Gerolamo Cardano (1501 - 1576)]

A complex number is a number expressed in the form $x + iy$, where $x$ and $y$ are real numbers and $i$ is the imaginary unit, defined by $i^2 = -1$.