Rational numbers


In mathematics, numbers can be classified according to their characteristics and use. The rational numbers represent the set of numbers that can be split to discuss parts of a whole. These numbers are frequently used to represent measurements in different areas such as architecture, medicine, chemistry, biology, etc.


What are rational numbers?

Rational or fractional numbers are those that can be described through a fraction . They are represented with the consonant Q that comes from the Italian word “Quoziente” , which is translated as quotient. They are made up of whole numbers, zero, and fractional numbers .

  • What are the rational numbers
  • Classification
  • What are they for
  • Characteristics of rational numbers
  • History
  • How they are represented
  • Properties
  • Operations
  • Examples of rational numbers

What are the rational numbers

Rational numbers are: whole numbers and fractional numbers:

  • The integers are numbers that have decimals. Example: 3
  • The zero is a number of null value representing that there is no item number or count. In the case of rational numbers, zero can be accompanied by decimals. Example: 0.5
  • The fractional numbers are not integers, for example 2/6, 4/5, 6/9.


Rational numbers can be classified into:

  • Not null (Q *): they are all rational numbers discarding zero.
  • Non-negative (Q +): they are all positive rational numbers and zero.
  • Non-positive (Q-): they are all negative rational numbers and zero.
  • Positive (Q * +): are all positive numbers minus zero.
  • Negative (Q * -): are all negative numbers without including zero.
  • Decimal numbers: they are those that can be written in fractions.

Depending on their decimal expression, decimal numbers can be classified as limited or periodic rational numbers .

The limited ones are those that have a fixed decimal representation . Example ½ = 0.5

The newspapers are those with an unlimited number of figures . These can be pure newspapers or mixed newspapers. Pure newspapers have a pattern after the comma, Example: 5.333333,

Mixed repeating numbers have a pattern after the given number. Example: 5.5414727272727272

What are they for

Rational numbers are used to express measures in elements that we can divide . For example, if we talk about a cake that we divide into 4 pieces and we eat one of them, we can say that we ate ¼ (0.25) of the other and if we ate all four 4/4 (1) we will have eaten the entire cake .

In addition, thanks to this type of numbers, it is easier to learn the operations to divide.

Characteristics of rational numbers

Among the most outstanding characteristics of rational numbers, the following can be mentioned:

  • They are infinite .
  • They can be expressed in fractions or with decimals .
  • They represent one or more parts of a whole .


The history of rational numbers has an unknown origin. However, in ancient times, it is the Egyptians who made the most use of these numbers to solve their problems using fractions of a whole.

In Egypt , they were used to solve problems in the area of ​​construction .

In ancient Greece , they take that name because the word rational comes from the Latin ratio which means reason or separation. At this time, Pythagorean mathematics made use of these numbers to express measurable quantities in different disciplines such as construction, music , anatomy, etc.

In modern times , they are identified with the notation Q which is the initial of the Italian word “Quoziente” thanks to the work carried out by Italian mathematician Giuseppe Peano in 1895.

These numbers are widely used in the teaching of mathematics and all kinds of fractional operations . In the professional field, the rational numbers allow to obtain exact measurements in the construction of pieces of metal, wood and other materials; allow us to obtain the real weight of food or objects; They are represented in the tables of measurement of medical and chemical utensils among others.

How they are represented

Rational numbers are represented by the letter Q which is the first letter of the Italian word “Quoziente” , which is translated as quotient. This representation comes from the works of Giuseppe Peano in 1895 on this numerical set.


Rational numbers are used in mathematical operations such as addition, subtraction, multiplication and division, and in each of these, the numerical set has the following properties.

  • Properties of rational numbers for addition and subtraction:
    • Internal property
    • Associative property
    • Commutative property
    • Neutral element (the number 0)
    • Inverse additive or opposite element
  • Properties for multiplication and division:
    • Internal property
    • Associative property
    • Commutative property
    • Distributive property
    • Neutral element (the number 1)


With rational numbers we can perform operations such as addition, subtraction, multiplication and division .

Examples of rational numbers

Here are several examples of rational numbers.

Example of non-null (Q *)

They are all discarding zero.

½ = 0.5

Example of non-negatives (Q +)

They are all positives and zero

¼ = 0.25

Example of non-positives (Q-)

They are all negatives and zero.


Example of positives (Q * +)

They are all positives minus zero.


Example of negatives (Q * -)

They are all negatives not including zero.


Example of decimal numbers

They are the ones that can be written in fractions.


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