COMPLEX NUMBERS: PROPERTIES, EXAMPLES, OPERATIONS - MATHS - 2025

The complex numbers are the numerical set covering real numbers and all the roots of the polynomials including pairs roots of negative numbers. These roots do not exist in the set of real numbers, but in complex numbers there is the solution.

A complex number consists of a real part and a part called "imaginary". The real part is called a, for example, and the imaginary part ib, with a and b real numbers and "i" as the imaginary unit. In this way the complex number takes the form:

Figure 1.- Binomial representation of a complex number in terms of real part and imaginary part. Source: Pixabay.

Examples of complex numbers are 2 - 3i, -πi, 1 + (1/2) i. But before operating with them, let's see where the imaginary unit i originates from, considering this quadratic equation:

x ² - 10x + 34 = 0

In which a = 1, b = -10 and c = 34.

When applying the resolving formula to determine the solution, we find the following:

How to determine the value of √-36? There is no real number that squared produces a negative quantity. Then it is concluded that this equation has no real solutions.

However, we can write this:

√-36 = √-6 ² = √6 ² (-1) = 6√-1

If we define a certain value x such that:

x ² = -1

So:

x = ± √-1

And the above equation would have a solution. Therefore, the imaginary unit was defined as:

i = √-1

And so:

√-36 = 6i

Many mathematicians of antiquity worked on solving similar problems, notably the Renaissance Girolamo Cardano (1501-1576), Nicolo Fontana (1501-1557) and Raffaele Bombelli (1526-1572).

Years later René Descartes (1596-1650) called the quantities “imaginary” like √-36 in the example. For this reason √-1 is known as the imaginary unit.

Properties of complex numbers

-The set of complex numbers is denoted as C and includes the real numbers R and the imaginary numbers Im. The number sets are represented in a Venn diagram, as shown in the following figure:

Figure 2. Venn diagram of number sets. Source: F. Zapata.

-All complex number consists of a real part and an imaginary part.

-When the imaginary part of a complex number is 0, it is a pure real number.

-If the real part of a complex number is 0, then the number is pure imaginary.

-Two complex numbers are equal if their respective real part and imaginary part are the same.

-With complex numbers, the known operations of addition, subtraction, multiplication, product and enhancement are carried out, resulting in another complex number.

Representation of complex numbers

Complex numbers can be represented in various ways. Here are the main ones:

- Binomial form

It is the form given at the beginning, where z is the complex number, a is the real part, b is the imaginary part and i is the imaginary unit:

Or also:

One way to graph the complex number is through the complex plane shown in this figure. The imaginary axis Im is vertical, while the real axis is horizontal and is denoted as Re.

The complex number z is represented in this plane as a point of coordinates (x, y) or (a, b), as it is done with the points of the real plane.

The distance from the origin to the point z is the modulus of the complex number, denoted as r, while φ is the angle that r makes with the real axis.

Figure 3. Representation of a complex number in the complex plane. Source: Wikimedia Commons.

This representation is closely related to that of vectors in the real plane. The value of r corresponds to the modulus of the complex number.

- Polar shape

The polar form consists of expressing the complex number by giving the values ​​of r and of φ. If we look at the figure, the value of r corresponds to the hypotenuse of a right triangle. The legs are worth a and b, or x and y.

From the binomial or binomial form, we can move to the polar form by:

The angle φ is the one formed by the segment r with the horizontal axis or imaginary axis. It is known as the complex number argument. In this way:

The argument has infinite values, taking into account that every time a turn is turned, which is worth 2π radians, r occupies the same position again. In this general way, the argument of z, denoted Arg (z), is expressed like this:

Where k is an integer and is used to indicate the number of turns turned: 2, 3, 4…. The sign indicates the direction of rotation, if it is clockwise or counterclockwise.

Figure 4. Polar representation of a complex number in the complex plane. Source: Wikimedia Commons.

And if we want to go from the polar form to the binomial form, we use the trigonometric ratios. From the previous figure we can see that:

x = r cos φ

y = r sin φ

In this way z = r (cos φ + i sin φ)

Which is abbreviated like this:

z = r cis φ

Examples of complex numbers

The following complex numbers are given in binomial form:

a) 3 + i

b) 4

d) -6i

And these in the form of an ordered pair:

a) (-5, -3)

b) (0, 9)

c) (7.0)

Finally, this group is given in polar or trigonometric form:

a) √2 cis 45º

b) √3 cis 30º

c) 2 cis 315º

What are they for?

The usefulness of complex numbers goes beyond solving the quadratic equation shown at the beginning, since they are essential in the field of engineering and physics, especially in:

-The study of electromagnetic waves

-Analysis of alternating current and voltage

-The modeling of all kinds of signals

-Theory of relativity, where time is assumed as an imaginary magnitude.

Complex number operations

With complex numbers we can perform all the operations that are done with real ones. Some are easier to do if the numbers come in binomial form, such as addition and subtraction. In contrast, multiplication and division are simpler if they are carried out with the polar form.

Let's see some examples:

- Example 1

Add z ₁ = 2 + 5i and z ₂ = -3 -8i

Solution

The real parts are added separately from the imaginary parts:

z ₁ + z ₂ = (2 + 5i) + (-3 -8i) = -1 -3i

- Example 2

Multiply z ₁ = 4 cis 45º and z ₂ = 5 cis 120º

Solution

It can be shown that the product of two complex numbers in polar or trigonometric form is given by:

z ₁. z ₂ = r ₁.r ₂ cis (φ ₁ + φ ₂)

According to this:

z ₁. z ₂ = (4 × 5) cis (45 + 120) = 20 cis 165º

Application

A simple application of complex numbers is to find all the roots of a polynomial equation like the one shown at the beginning of the article.

In the case of the equation x ² - 10x + 34 = 0, applying the resolving formula we obtain:

Therefore the solutions are:

x ₁ = 5 + 3i

x ₂ = 5 - 3i

References

1. Earl, R. Complex numbers. Recovered from: maths.ox.ac.uk.
2. Figuera, J. 2000. Mathematics 1st. Diversified. CO-BO editions.
3. Hoffmann, J. 2005. Selection of Mathematics topics. Monfort Publications.
4. Jiménez, R. 2008. Algebra. Prentice Hall.
5. Wikipedia. Complex numbers. Recovered from: en.wikipedia.org