What are real numbers? This is the set of numbers that includes naturals, integers, rationals and irrationals. Throughout this article we’ll look at what each one of them is. On the other hand, real numbers are represented by the letter “R” (ℜ).

In this article we will know the classification of the 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.

## What are the real numbers?

Real numbers can be represented on a number line , comprising both rational and irrational numbers.

That is, the classification of real numbers includes positive and negative numbers, 0 and numbers that cannot be expressed as fractions of two integers and that have non-null numbers as their denominator (i.e., that are not 0). Later on we will specify which type of number corresponds to each of these definitions.

Something that is also said about real numbers is that they are a subset of 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, real numbers are practically the majority of numbers that we deal with in our daily lives and beyond (when we study mathematics, especially at a more advanced level).

Examples of real numbers are: 5, 7, 19, -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 look at it in detail.

### 1. Natural numbers

As we saw, within the real numbers we found different types of numbers. In the case of natural numbers, these are the numbers we use to count (for example: I have 5 coins in my hand). That is: 1, 2, 3, 4, 5, 6… Natural numbers are always integers (that is, a natural number could not be “3.56”, for example) .

Natural numbers are expressed by the letter “N” in handwriting. This is a subset of the whole numbers.

Depending on the definition, we find that natural numbers either start from 0 or from 1. These types 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 “constructed” (they are the “base” of departure): integers, rational, real… Some of their properties are: addition, subtraction, division and multiplication; that is, these mathematical operations can be performed with them.

### 2. Whole numbers

Other numbers that are part of the classification of real numbers are integers, which are represented by the “Z” (Z).

They include: 0, natural numbers and natural numbers with a negative sign (0, 1, 2, 3, 4, -1, -2, -3, -4…). Integers are a subset of rational numbers.

Thus, it is about those numbers written without fraction, that is, “in full form”. They can be positive or negative (for example: 5, 8, -56, -90, etc.). On the other hand, numbers that include decimals (such as “8.90”) or that result from some square roots (for example √2), are not integers.

Integers also include 0. Actually, integers are part of natural numbers (they are a small group of these).

### 3. Rational numbers

The following numbers within the classification of the 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 its fraction .

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 a whole number, integers are rational numbers.

The set of this type of numbers, the rational ones, is expressed by a “Q” (capital).
Thus, decimal numbers, which are rational numbers, are of three types:

• Exact decimals: such as “3.45”.
• Pure decimals: like “5.161616…” (since the 16th is repeated indefinitely).
• Mixed periodic decimals: as for example “6,788888… (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. Irrational numbers are represented as: “R-Q”, which means: “the set of real numbers minus the set of rational ones” .

These types 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 they are not periodic.

Within the irrational numbers, we can find the number pi (expressed by π), which consists of the relationship between the length of a circle and its diameter. We also find some others, such as: the number of Euler (e), the golden number (φ), the roots of prime numbers (for example √2, √3, √5, √7…), etc.

Like the previous ones, since it is 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? In our everyday life, we constantly use mathematics: to calculate changes, 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, computers, new technologies, etc. From them we can manufacture products, calculate data that interest us, etc.

On the other hand, beyond the mathematical sciences, 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 “soaked” in mathematics.

So, you could practically say that… we live in numbers! There will be people who use them to work, and others to do simpler calculations of their day to day.

### Structuring the Mind

On the other hand, numbers and mathematics structure the mind; they allow us to create mental “boxes” where we can organize and incorporate information. So, in reality, mathematics not only serves to “add or subtract”, but also to compartmentalize our brain and our mental functions.

Finally, the good thing about understanding the different types of numbers, such as those included in the classification of real numbers, will help us to enhance our abstract reasoning, beyond mathematics.

#### Bibliographic references:

• Coriat, M. and Scaglia, S. (2000). Representation of the real numbers in the line. Teaching of science, 18(1): 25-34.

• Romero, I. (1995). The introduction of the real number in secondary education. Doctoral thesis. Granada: Departamento de Didáctica de la Matemática. University of Granada.

• Skemp, R.R. (1993). Psychology of learning mathematics. Morata, 3ª Ed. Madrid.