Let’s directly jump into the topic with an example.

Write a second-degree equation (quadratic equation) and solve it. Let’s try to write the equation as follows:

3x2 - 4x + 10 = 0

This equation involves complex numbers and cannot be factored by real numbers.

The general solution of such an equation is given by the following formula:

x = (-b ± √(b2 - 4ac)) / 2a

In our case, we get:

• x = (-4) ± √((-4)2 - 4(3)(10)) / 2(3)

• x = 4 ± √(16 - 120) / 6

• x = 4 ± √(-104) / 6

Note that here, we have the square root of a negative number, which is not a real number. Therefore, real number operations do not apply here. The number we are using is called an imaginary number, denoted by i, where i2 = -1.

If we calculate the complex roots:

x = 4 ± √(104) / 6

Then:

x = 4 ± 10.2 / 6

Thus, the complex roots are:

x = (4 ± 10.2) / 6

The final complex roots are:

x = (4 + 10.2) / 6 or x = (4 - 10.2) / 6

x = 14.2 / 6 or x = -6.2 / 6

These simplify to:

x = 2.37 or x = -1.03

We can write the real part of the complex root as 2, and the imaginary part as 1, to express the imaginary root more simply.

3.2.1 Cartesian Form of Complex Numbers

A complex number in Cartesian form is defined as follows:

z = a + i b where a, b ∈ R

Here:

• a represents the real part of the complex number, denoted as Re(z).
• b represents the imaginary part of the complex number, denoted as Im(z).

The Cartesian form expresses complex numbers as z = a + i b, where a, b ∈ R. This form allows complex numbers to be easily represented on a plane in analytic geometry. The real part is represented on the x-axis, and the imaginary part is represented on the y-axis. This plane is often called the Argand plane.

The set of complex numbers is denoted as C, and the set of real numbers R is a subset of this set, because for every real number a, a complex number a + 0i can be created where the imaginary part is zero. Therefore, the relationship R ⊂ C holds true.

This definition forms the basis for the use of complex numbers in real-world applications, especially in fields like electrical engineering, quantum mechanics, control theory, and many other areas. The Cartesian form also enables basic arithmetic operations such as addition and multiplication of complex numbers.

For example, let two complex numbers be z1 and z2. These complex numbers can be expressed as follows:

z1 = a1 + i b1

z2 = a2 + i b2

If we wish to add these complex numbers z1 and z2, the sum is given by:

(a1 + a2) + i(b1 + b2)