Classification of Real Numbers

What are the real numbers? It is the set of numbers that include natural numbers, integers, rational numbers, 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 know the classification of real numbers, formed by the different types of numbers mentioned at the beginning. We will see what its fundamental characteristics are, as well as examples. Finally, we will talk about the importance of mathematics and its meaning and benefits.

• Recommended article: "How to calculate percentiles? Formula and procedure "

What are the real numbers?

Real numbers can be represented on a number line, understanding this the rational and irrational numbers.

That is, the classification of real numbers includes positive and negative numbers, 0, and numbers that are not can be expressed by fractions of two integers and that have nonzero numbers as denominators (that is, they are not 0). Later we will specify what type of number corresponds to each of these definitions.

Something that is also said about real numbers is that it is a subset of complex or imaginary numbers (these are represented by the letter "i").

Classification of 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 day to day and beyond it (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 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 we use to count (for example: I have 5 coins in my hand). That is to say: the 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 handwritten letter "N". It is a subset of the whole numbers.

Depending on the definition, we find that the 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 the natural numbers, other types of numbers are “built” (they are the starting “base”): integers, rational, real... Some of its properties are: addition, subtraction, division and multiplication; that is, you can perform these mathematical operations with them.

2. Integer numbers

Other numbers that are part of the classification of real numbers are whole numbers, which are represented by "Z" (Z).

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

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

Whole numbers also include 0. Actually, the whole numbers are part of the natural numbers (they are a small group of these).

3. Rational numbers

The following numbers within the classification of real numbers are rational numbers. In this case, rational numbers are any number that can be expressed as the component of two whole numbers, or as their fraction.

For example 7/9 (it is 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, the whole numbers are rational numbers.

The set of this type of numbers, the rational numbers, is expressed by a "Q" (capital letter). Thus, decimal numbers that are rational numbers are of three types:

• Exact decimals: such as "3.45".
• Pure repeating decimals: such as "5,161616 ..." (since 16 is repeated indefinitely).
• Mixed repeating decimals: such as “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. Irrational numbers are represented as: "R-Q", which means: "the set of reals minus the set of rationals".

These types of numbers are all those real numbers that are not rational. Thus, these cannot be expressed as fractions. These are numbers that have infinite decimal places, and that 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 Euler number (e), the golden number (φ), the roots of prime numbers (for example √2, √3, √5, √7…), etc.

Like the previous ones, as it is part of the classification of real numbers, it is a subset of the latter.

The sense of numbers and mathematics

What good are mathematics and the concept of numbers? What can we use mathematics for? Without going any further, in our day to day we constantly use mathematics: to calculate changes, to pay, to calculate expenses, to calculate times (of trips, for example), to compare schedules, etc.

Logically, beyond the day, mathematics and numbers have infinite applications, especially in the field of engineering, computer science, new technologies, etc. From them we can manufacture products, calculate data that interest 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 important sciences or careers such as medicine or biology are also “drenched” in math.

So, you can practically say that... We live among numbers! There will be people who use them to work, and others to perform simpler calculations of their day to day.

Structure the mind

On the other hand, numbers and mathematics structure the mind; They allow us to create mental "drawers" where we can organize and incorporate information. So actually 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, as in this case those included in the classification of real numbers, will help us to enhance our abstract reasoning, beyond the math.

Bibliographic references:

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

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

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