Complex numbers

In this lesson, I will explain what complex numbers are.

Complex numbers are ordered pairs of real numbers a and b that satisfy certain rules in addition and multiplication operations. $$ z = (a,b) $$

The real part of a complex number is defined as the first element of the ordered pair (a,b).

$$ a = Re(z) $$

The imaginary part of a complex number is defined as the second element in the ordered pair (a,b).

$$ b = Im(z) $$

In mathematics, the set of complex numbers, denoted by the letter C.

It's a fundamental concept used in a wide range of fields, including algebra, geometry, calculus, and physics."

Why are complex numbers studied? Complex numbers form a much broader set than real numbers and offer a powerful mathematical tool that extends beyond the constraints of real numbers. Complex numbers allow for the performance of calculations and operations that are impossible using real numbers alone. One significant example is the calculation of the square root of a negative number, which is made possible by introducing the concept of imaginary numbers.

In this introductory lesson, I will provide the first useful concepts to understand what complex numbers are.

The Complex Plane 

The complex plane is a fundamental tool for understanding the properties and behavior of complex numbers.

By representing each complex number as a point (x,y) in the Cartesian plane, we can visualize the relationships between these numbers and gain insights into their algebraic and geometric properties.

Likewise, we can also represent any point (x,y) on the plane as a complex number z, thereby establishing a one-to-one correspondence between points in the plane and complex numbers.

In the complex plane, the horizontal axis corresponds to the real part (Re) and the vertical axis corresponds to the imaginary part (Im) of the complex number z=(a,b).

Note. The Cartesian plane is a fundamental tool for representing complex numbers, which are composed of both real and imaginary parts. This plane is commonly referred to as the complex plane or the Gauss plane, in honor of mathematician Carl Friedrich Gauss who made significant contributions to the field of complex numbers.

Let me give you a practical example.

The point (2,3) is a complex number z=(2,3).

In this case, the real component of the complex number z=(2,3) is equal to 2.

$$ Re(z) = 2 $$

The imaginary component of the complex number z=(2,3) is equal to 3.

$$ Im(z) = 3 $$

I'll give you another example.

The point (-1,2) is another complex number represented as z=(-1,2).

In this case, the real part of the complex number is -1, while the imaginary part is 2.

$$ Re(z) = -1 $$

$$ Im(z) = 2 $$

Operations with complex numbers

Addition and multiplication operations in the complex number set differ from those in the real number set.

Addition of complex numbers

To add two complex numbers, you need to add their components.

$$ (a,b)+(c,d)=(a+c,b+d) $$

For example, the sum of the complex numbers (2,3) and (-1,4) is as follows:

$$ (2,3)+(-1,4) = (2-1,3+4) = (1,7) $$

Multiplication of complex numbers

To multiply two complex numbers, you need to apply the following particular rule.

$$ (a,b) \cdot (c,d)=(ac-bd,ad+bc) $$

For example, the product of the complex numbers (2,3) and (-1,4) is (-14,5)

$$ (2,3) \cdot (-1,4) $$ $$ = (2 \cdot (-1))- 3 \cdot 4 \ , \ 2 \cdot 4 + 3 \cdot (-1)) $$ $$ = (-2- 12 \ , \ 8 -3) $$ $$ = (-14,5) $$

You can also divide complex numbers.

Complex numbers in algebraic form

Complex numbers have different representations: algebraic, trigonometric, exponential.

For now, I will only discuss the complex number in algebraic form.

$$ z = (a,b) = a + bi $$

The letter "i" represents the imaginary unit and is defined as the square root of -1.

It is expressed as a pair of real numbers in the form (0,1).

$$ i = (0,1) $$

Note. The concept of imaginary numbers, including "i", is essential in the study of complex numbers, which are expressed in the form a + bi, where a and b are real numbers.

Any point located on the plane (a,b) can be represented in algebraic form

For instance, the complex number z=(2,3) can be expressed in algebraic form as z=2+3i.

$$ z = (2,3) = 2 + 3i $$

This representation denotes the same complex number, albeit conveyed differently.

The complex number z=(-1,2) can be written in algebraic form as z=-1+2i.

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