Classification of Real Numbers
We explain what a real number is and what types of real numbers exist.
What are real numbers? It is the set of numbers that includes natural, integers, rational and irrational numbers. Throughout this article we will see what each of them consists of. On the other hand, real numbers are represented by the letter "R" (ℜ).
In this article we will learn about the classification of real numbers, formed by the different types of numbers mentioned at the beginning. We will see what their fundamental characteristics are, as well as examples. Finally, we will talk about the importance of mathematics and its meaning and benefits.
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What are real numbers?
The real numbers can be represented on a number line, including rational numbers.The real numbers can be represented on a number line, including rational and irrational numbers.
That is, the classification of real numbers includes positive and negative numbers, 0 and numbers that cannot be expressed by fractions of two integers and that have nonzero numbers as denominator (i.e., that are not 0). Later we will specify which type of number corresponds to each of these definitions.
Something that is also said of the real numbers is that they are a subset of the complex or imaginary numbers (these are represented by the letter "i").
Classification of the real numbers
In short, and to put it in a more understandable way, the real numbers are practically the majority of real numbers, the real numbers are practically the majority of numbers with which we deal in our day to day lives and beyond (when we study mathematics, especially at a more advanced level).
Examples of real numbers are: 5, 7, 19, 9, 9, 65, 90. √6, √9, √10, the number pi (π), etc. However, this classification, as we have already said, is divided into: natural numbers, integers, rational numbers and irrational numbers. What characterizes each of these numbers? Let's see it in detail.
1. Natural numbers
As we saw, within the real numbers we find different types of numbers. In the case of natural numbers, these are the numbers that we use to count (for example: I have 5 coins in my hand). That is to say: 1, 2, 3, 4, 5, 6... Natural numbers are always integers (i.e. a natural number could not be "3.56", for example)..
Natural numbers are expressed by the handwritten letter "N". It is a subset of the integers.
Depending on the definition, we find that natural numbers either start from 0 or from 1. This type of numbers are used as ordinals (for example, I am the second) or as cardinals (I have 2 pants).
From natural numbers, other types of numbers are "built" (they are the "base" of departure): integers, rational, real... Some of their properties are: addition, subtraction, division and multiplication; that is to say, these mathematical operations can be performed with them.
2. Integers
Other numbers that are part of the classification of real numbers are the integers, which are represented by the "Z" (Z).
They include: the 0, the natural numbers and the natural numbers with negative sign (0, 1, 2, 3, 4, 1, 2, 3, 4...). Integers are a subset of rational numbers.
Thus, they are those numbers written without a fraction, i.e., "in integer form". They can be positive or negative (for example: 5, 8, 56, 90, etc.). On the other hand, numbers that include decimals (e.g. "8.90") or that result from some square roots (e.g. √2), are not integers.
Integers also include 0. Actually, integers are part of the natural numbers (they are a small group of natural numbers).
Rational numbers
The next numbers in the classification of real numbers are the rational numbers. In this case, rational numbers are any number that can be expressed as the component of two integers, or as a fraction of two integers..
For example 7/9 (usually expressed by "p/q", where "p" is the numerator and "q" is the denominator). Since the result of these fractions can be an integer, integers are rational numbers.
The set of this type of numbers, the rational numbers, is expressed by a "Q" (capital letter). Thus, decimal numbers, which are rational numbers, are of three types:
 Exact decimals: as for example "3.45".
 Pure periodic decimals: as for example "5,161616..." (since 16 is repeated indefinitely).
 Mixed periodic decimals: as for example "6.788888... (the 8 is repeated indefinitely).
The fact that rational numbers are part of the classification of real numbers implies that they are a subset of this type of numbers.
4. Irrational numbers
Finally, in the classification of the real numbers we also find the irrational numbers. The irrational numbers are represented as: "RQ", which means: "the set of reals minus the set of rationals"..
This type of numbers are all those real numbers that are not rational. Thus, they cannot be expressed as fractions. These are numbers that have infinite decimals, and that are not periodic.
Among the irrational numbers, we can find the number pi (expressed by π), which is the ratio between the length of a circle and its diameter. We also find some others, such as: Euler's number (e), the golden number (φ), the roots of prime numbers (e.g. √2, √3, √5, √7...), etc.
Just like the previous ones, being part of the classification of real numbers, it is a subset of the latter.
The meaning of numbers and mathematics
What is the use of mathematics and the concept of numbers? What can we use mathematics for? Without going any further, we constantly use mathematics in our daily lives: to calculate change, to pay, to calculate expenses, to calculate times (of journeys, for example), to compare schedules, etc.
Logically, beyond the day, mathematics and numbers have an infinite number of applications, especially in the fields of engineering, computing, new technologies, etc. From them, we can manufacture products, calculate data of interest to us, etc.
On the other hand, beyond the sciences of mathematics, there are other sciences that are actually applied mathematics, such as physics, astronomy and chemistry. Other sciences or careers as important as medicine or biology are also "steeped" in mathematics.
So, it can practically be said that... we live among numbers! There will be people who use them for work, and others for simpler daytoday calculations.
Structuring the mind
On the other hand, numbers and mathematics structure the mind; they allow us to create mental "drawers" in which to organize and incorporate information. So, in reality, Mathematics is not only for "adding or subtracting", but also for compartmentalizing our brains and our mental functions. and our mental functions.
Finally, the good thing about understanding the different types of numbers, as in this case those included in the classification of real numbers, will help us to enhance our abstract reasoning, beyond mathematics.
Bibliographical references:

Coriat, M. and Scaglia, S. (2000). Representation of real numbers on the straight line. Science Education, 18(1): 2534.

Romero, I. (1995). The introduction of the real number in secondary education. Doctoral dissertation. Granada: Department of Didactics of Mathematics. University of Granada.

Skemp, R.R. (1993). Psicología del aprendizaje de las matemáticas. Morata, 3rd Ed.
(Updated at Apr 13 / 2024)