Real numbers


The real numbers are all those numbers that are included within the rational numbers . They can be positive , negative and include the number zero , as in the case of irrational numbers . These numbers can be written in different ways, some of them very simple, generally used in simple mathematical operations , and in more complex forms . This group of numbers also includes fractions of whole numbers that have non- zero numbers in their denominator .


What are real numbers?

Real numbers are all those numbers that are found on the number line and that make up the group of rational and irrational numbers , negative and positive, including the number zero.

  • What are the real numbers
  • What are real numbers for?
  • features
  • History
  • Classification
  • Properties of real numbers
  • Operations
  • Examples

What are the real numbers

The system of real numbers is formed mainly by two large groups, that of rational numbers , which are all those numbers that can be expressed as the division of two whole numbers , and the system of irrational numbers whose decimal representation is expansive, infinite and aperiodic. Rational numbers can also be divided into subgroups , among which we can mention: non-integer fractions with their negative notations; the integers including negative numbers and positive integers; the latter in turn include the natural numbers and zero .


What are real numbers for?

Real numbers are digits that help us to carry out all the mathematical operations necessary to solve a situation or problem. They also have the function of designating the number of elements that have a certain set. They are used to identify places or objects and to order and rank categories.


Among the main characteristics of real numbers we can mention the following:

  • It is formed by the union of rational and irrational numbers .
  • They are a complete set .
  • These types of numbers and the number line are closely related.
  • For each real number there is a point that represents it on the number line.
  • The natural numbers are complete and it is an ordered set .
  • They are numbers that have associativity and commutativity .
  • They all have an order and are written consecutively .
  • When they are used for counting , then we mean that they have a cardinal function .


They arose as a need that the settlers of the primitive era had to count and solve problems that were constantly presented to them. There is a record that the people of Babylon  used them to count, for example, their animals. The Egyptians first used fractions by means of mathematicians like Pythagoras . During the 19th centuries, constructions and systematizations of real numbers were carried out by means of two important mathematicians, Georg Cantor and Richard Dedekind.. These two historical figures in the world of mathematics managed to systematize real numbers using a series of advances invented by them.


Real numbers are classified or divided into the following groups:

  • Natural numbers: These are the numbers that we normally use to count . They can start with 0 or 1. They serve as a base to form larger numbers, and are numbers that have divisibility and distribution of numbers. With them you can add, subtract, multiply and divide.
  • Whole numbers: They are the numbers that can be written without using a fraction . They are complete numbers and are used to express quantities , depths , temperatures . Together, they make the smallest group of real numbers.
  • Rational numbers: They are the numbers that can be expressed as a fraction of two whole numbers, which have a numerator and a denominator . It is represented by the letter Q.  They can also be defined as types of integer pair equivalences .
  • Irrational numbers: They are the real numbers that are not rational numbers either. These numbers cannot be expressed as fractions . Among them we can mention the radius of a circle, the golden number and the square root .

Properties of real numbers

The properties of real numbers vary depending on the type of operation , this being the case, then we have the following:

  • Properties of Sum
    • Internal property: the result of adding two real numbers will result in another real number.
    • Associative property : if there are more than two addends, it does not matter which sum is done first, if the numbers are all real.
    • Commutative property : the order of the digits will not alter the sum.
    • Existence of the neutral element: any number that is added with 0 will result in the same number.
  • Subtraction properties
    • If the minuend or subtrahend are positive, the subtraction will give a positive result , otherwise the result will be negative.
    • If the minuend is negative and the subtrahend positive, the sum is done and the result has a minus sign .
    • Subtracting a positive number is the same as adding a negative number.
    • Subtracting a negative number is the same as adding a positive one.
  • Multiplication properties
    • Internal property
    • Associative property
    • Commutative property
    • Distributive property
    • Inverse or opposite element
    • Common factor


The mathematical operations that can be performed using real numbers are:  addition or addition, multiplication , subtraction , division, and power .


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  • 0.1234512345…
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  • √5

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