Introduction to Complex Numbers
Complex numbers emerged as an essential mathematical tool, akin to how negative numbers extend the realm of positive numbers. Initially conceived to address the unsolvable problems (which we will discuss here for motivation) posed by cubic equations, complex numbers were introduced as a natural expansion of the real number system. Few could have predicted the profound impact they would have, transforming the scientific and engineering domains with their extensive applications. Although termed 'imaginary' in academic texts, complex numbers have proven to be instrumental in depicting the intricacies of the natural world with greater fidelity than real numbers alone.
Bombelli's Cubic Equation
Bombelli's cubic equation is famous in the history of mathematics for its role in the development of complex numbers. The specific equation that Rafael Bombelli studied in the 16th century is:
This equation is notable because it led to the introduction and understanding of complex numbers. At the time, mathematicians were exploring solutions to cubic equations, and they encountered cases where the solutions involved what are now known as complex numbers, though they didn't have a clear understanding of these numbers at the time.
Bombelli was the first to address the arithmetic of complex numbers. In dealing with the above equation, he encountered expressions involving the square root of negative numbers. While initially these expressions seemed nonsensical, Bombelli found a way to manipulate them and correctly deduced the real solution to the equation, which is \( x = 4 \). His work laid the groundwork for the later formal development of complex numbers, which became a fundamental concept in mathematics and its applications.
Let us see what and how he did it.
Cardano's Formula
We all know how to find roots of a quadratic equation. Is it possible to find roots of a cubic equation? Indeed it is possible. There is a neat formula for this. Girolamo Cardano (15011576) and others (thre is a very interesting story on the discovery of roots of cubic equations formula "Read Here on Wikipedia").
Cardano's formula for the real roots of a depressed cubic equation \( x^3 + px + q = 0 \) (where the quadratic term is missing) is a bit complex compared to formula for roots of quadratic equation.
In this formula:
 \( \sqrt[3]{} \) represents the cube root.
 The expression inside the square root, \( \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3 \), is the discriminant \( \Delta \).
Now apply Cardano's formula to the Bombelli's cubic equation \( x^3 = 15x + 4 \), we have \( p = 15 \) and \( q = 4 \).
Let's plug in the values:
 Calculate the discriminant part \( \Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3 = \left(\frac{4}{2}\right)^2 + \left(\frac{15}{3}\right)^3 \).
 Simplify the discriminant: \( \Delta = 4 + (5)^3 = 4  125 = 121 \).
 Now apply the formula: \( x = \sqrt[3]{2 + \sqrt{121}} + \sqrt[3]{2  \sqrt{121}} \).
Since \( \sqrt{121} \) involves the square root of a negative number, it would typically be addressed using complex numbers. However, since complex numbers were not discovered back then, mathmaticians got stuck at this point. It can be seen by hit and trial that \(4\) is the root of this equation. Bombelli accepted \(\sqrt{1}\) and did algebra that we will learn soon in this chapter. He was able to prove that this number is equal to \(4\). We will again come back to this equation.
What is a Complex Number?
It is quite interesting that mathematicians discovered complex numbers while dealing with cubic equations. On the other hand, we will begin with quadratic equations to understand complex numbers. To motivate the introduction of complex numbers, consider the quadratic equation \( x^2  4x + 9 = 0 \). When we attempt to solve this equation by the method of completing the square, we encounter an apparent contradiction:
\( x^2  4x + 9 = 0 \)
\( \implies (x  2)^2 + 5 = 0 \)
The contradiction arises because for any real number \( x \), the square of \( x  2 \) is nonnegative, and thus \( (x  2)^2 + 5 \) is always greater than 0. This implies that our equation has no solution within the real numbers.
To resolve this and find solutions to such equations, we need to extend the real number system. The extension involves defining a solution for the square root of a negative number. Since no real number squared gives a negative result, we introduce a new kind of number for which this is possible.
We define that for any negative number \( a \), the square root \( \sqrt{a} \) is equal to \( \sqrt{1}\sqrt{a} \). The term \( \sqrt{1} \) does not correspond to any real number, so we deem it an imaginary number. We represent \( \sqrt{1} \) with the symbol \( i \), sometimes called iota. By definition, we set \( i^2 = 1 \) axiomatically.
With this new definition, a number like \( \sqrt{5} \) can be expressed as \( \sqrt{1}\sqrt{5} \), or \( i\sqrt{5} \). Now, our original equation has solutions in this extended system of numbers, which we call the complex numbers. Each complex number has a real part and an imaginary part and is generally expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit. Using this framework, the solutions to our original equation are:
Definition
A complex number is defined as any number of the form \( a + bi \), where \( a \) and \( b \) are real numbers (\( a, b \in \mathbb{R} \)), and \( i \) is the imaginary unit, defined as \( i = \sqrt{1} \), characterized by the property \( i^2 = 1 \). The value \( a \) is termed the real part of the complex number, denoted as \( \text{Re}(a + bi) = a \), while \( b \) is the imaginary part, denoted as \( \text{Im}(a + bi) = b \).
Complex numbers are inherently twodimensional in nature, as they are defined by an ordered pair of real numbers. The variable \( z \) is commonly used to represent a complex number, with the standard expression being \( z = a + bi \).
The set of all complex numbers is represented by the symbol \( \mathbb{C} \), and it includes the set of all real numbers \( \mathbb{R} \) as a subset, where the imaginary part is zero (\( \mathbb{R} \subseteq \mathbb{C} \)). A complex number whose imaginary part is zero is a real number, and conversely, a complex number with a zero real part is called a purely imaginary number.
Here are some examples to illustrate complex numbers:

Standard Complex Number:

\( z = 3 + 4i \)
 Here, \( \text{Re}(z) = 3 \) is the real part and \( \text{Im}(z) = 4 \) is the imaginary part.

\( z = 6  7i \)
 \( z = \sqrt{2} + \sqrt{3}i \)
 \( z = 4  i \)


Real Number as a Complex Number:
 \( z = 5 + 0i \) or simply \( z = 5 \)
 This is a real number because the imaginary part is zero. It is a complex number with \( \text{Im}(z) = 0 \).
 \( z = 5 + 0i \) or simply \( z = 5 \)

Purely Imaginary Number:
 \( z = 0 + 2i \) or simply \( z = 2i \)
 This is a purely imaginary number because the real part is zero. It is a complex number with \( \text{Re}(z) = 0 \).
 \( z = 0 + 2i \) or simply \( z = 2i \)
Each of these examples adheres to the form \( a + bi \) and fits within the complex number system \( \mathbb{C} \).
Example 1
a. \( \sqrt{8} = 2\sqrt{2}i \)
b. \( \sqrt{15} = \sqrt{15}i \)
c. \( \sqrt{64} = 8i \)
d. \( \sqrt{90} = 3\sqrt{10}i \)
A Fallacy
We know that \(\sqrt{25}=5\). But look at the following steps of reasoning.
\( \sqrt{25} = \sqrt{5 \times 5} \)
\( = \sqrt{5} \cdot \sqrt{5} \)
\( = \sqrt{5}i \cdot \sqrt{5}i \)
\( = 5i^2 \)
\( = 5(1) \)
\( = 5 \) (which is incorrect)
The fallacy lies in the step from 2 to 3, where the square root of a product of negative numbers is wrongly broken down into the product of square roots of individual negative numbers. The principal square root of a positive number is always positive in the real number system, and the property \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \) holds for nonnegative real numbers \( a \) and \( b \) but does not generally hold for negative numbers in the context of real numbers.
Correctly, \( \sqrt{25} \) is \( 5 \), and the equation \( \sqrt{5} \cdot \sqrt{5} \) cannot be simplified as \( \sqrt{5}i \cdot \sqrt{5}i \) within the system of real numbers. The use of complex numbers (i.e., \( i = \sqrt{1} \)) should be applied with care, keeping in mind the fundamental differences between the real and complex number systems.
Example 2
Solve the quadratic equation \( 8x^2  4x + 1 = 0 \).
Solution:
Applying the quadratic formula:
\( x = \frac{b \pm \sqrt{b^2  4ac}}{2a} \)
\( \implies x = \frac{4 \pm \sqrt{(4)^2  4 \cdot 8 \cdot 1}}{2 \cdot 8} \)
\( \implies x = \frac{4 \pm \sqrt{16  32}}{16} \)
\( \implies x = \frac{4 \pm \sqrt{16}}{16} \)
\( \implies x = \frac{4 \pm 4i}{16} \)
\( \implies x = \frac{1 \pm i}{4} \)
Thus, the solutions to the equation \( 8x^2  4x + 1 = 0 \) are \( x = \frac{1}{4} + \frac{i}{4} \) and \( x = \frac{1}{4}  \frac{i}{4} \).
Example 3
Solve the quadratic equation \( 3x^2  x + 4 = 0 \).
Solution:
Applying the quadratic formula to the equation:
\( x = \frac{b \pm \sqrt{b^2  4ac}}{2a} \)
\( \implies x = \frac{1 \pm \sqrt{(1)^2  4 \cdot 3 \cdot 4}}{2 \cdot 3} \)
\( \implies x = \frac{1 \pm \sqrt{1  48}}{6} \)
\( \implies x = \frac{1 \pm \sqrt{47}}{6} \)
\( \implies x = \frac{1 \pm \sqrt{47}i}{6} \)
Thus, the solutions to the equation \( 3x^2  x + 4 = 0 \) are \( x = \frac{1}{6} + \frac{\sqrt{47}}{6}i \) and \( x = \frac{1}{6}  \frac{\sqrt{47}}{6}i \).