Real numbers


Real numbers are any number that corresponds to a point on the real line and can be classified into natural, integer, rational, and irrational numbers.

In other words, any real number is between minus infinity and plus infinity and we can represent it on the real line.

Real numbers are all the numbers that we find most frequently since complex numbers are not found accidentally, but have to be specifically searched for.

Real numbers are represented by the letter R ↓

Domain of real numbers

So, as we have said, the real numbers are the numbers between the infinite extremes. That is, we will not include these infinities in the set.

Domain of real numbers.

Real numbers on the real line

This line is called a real line since we can represent all real numbers on it.

Royal line.

The real numbers and the Matrioshka

We have to understand the set of reals as the Matrioshka, that is, as the set of traditional Russian dolls organized from largest to smallest.

The series of dolls would be such that the largest doll contains the next smallest dolls. This set of dolls collected within the largest doll is called Matrioshka. Schematically:

(Doll A> Doll B> Doll C) = Matrioshka

Martioshka scheme

We can see the Matrioshka from the side (figure to the left of the equal) and also from above or below (figure to the right of the equal). Of the two ways we can clearly see the hierarchy of dimensions that the series follows.

So, in the same way that we collect the Russian dolls we can also organize the real numbers following the same method.

Scheme of the real numbers

In this scheme we can clearly see that the organization of the real numbers is similar to the Russian doll game seen from above or below.

Classification of real numbers

As we have seen, real numbers can be classified into natural, integer, rational and irrational numbers.

  • Natural numbers

Natural numbers are the first set of numbers that we learn as children. This set does not take into account the number zero unless otherwise specified (neutral zero).


Hint → We can remember the natural numbers thinking that they are the numbers that we use “naturally” to count. When we have our hand we ignore zero, the same for natural numbers.

First elements of the set of natural numbers.

  • Integer numbers

Whole numbers are all natural numbers and include zero and all negative numbers.


Example of some of the elements of the set of integers.

Hint: → We can remember the whole numbers thinking that they are all the numbers that we naturally use to count together with their opposites and including zero. Unlike rational numbers, integers represent "entirely" their value.

  • Rational numbers

Rational numbers are the fractions that can be formed from whole and natural numbers. We understand fractions as quotients of whole numbers.


Hint → We can remember rational numbers thinking that being fractions of whole numbers, it is “rational” for the result to be a whole number or a finite or semi-periodic decimal number.

Example of some of the elements of the set of rational numbers.

  • Irrational numbers

Irrational numbers are decimal numbers that cannot be expressed either exactly or periodically.


Hint → We can remember the irrational numbers thinking that they are all the numbers that do not fit in the previous classifications and that also belong to the real line.

Example of some elements of the set of irrational numbers.

Examples of real numbers

In the following example about real numbers, check that the following numbers correspond to points on the real line.

  • Natural numbers: 1,2,3,4 ...
  • Whole numbers:…, -4, -3, -2, -1, 0, 1, 2, 3, 4…
  • Rational numbers: any fraction of whole numbers.
  • Irrational numbers:

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