## Rational Numbers

Rational numbers are the set of whole numbers and fractional numbers represented by fractions. This set is located on the real number line but they are unlike natural numbers which are consecutive. For example, the number 4 is followed by the number 5 which is followed by 6. Negative numbers also have succession such as the number -9 is followed by -8, which is followed by -7. Rational numbers have no such succession. Between each rational number as there are infinitely many numbers that can be written for all eternity.

All fractional numbers are rational numbers, used to represent measurements. Sometimes it is more convenient to express numbers in this way rather than to convert them to a terminating or repeating decimal due to the endless number of decimal places that exist.

## Definition of Rational Numbers

So, **what are rational numbers?** We can start by saying that a rational number is a number or value that can be referred to as the quotient of two integers, or more precisely, an integer and a positive natural number. That is to say that is a rational number is a number that is written as a fraction.

The rational numbers are fractional numbers; however integers can also be expressed as a fraction. Thus they also can be considered as rational numbers by simply being written as a ratio between the whole number and the number 1 as the denominator.

The set of rational numbers is denoted by the letter **ℚ**, which comes from the Anglo-Saxon word “quotient” and forms a subset of real numbers along with integers whose denotation is the letter Z.

A rational number can be expressed in different ways, without altering the quantity, by equivalent fractions, i.e. ½ can be expressed as 2/4 or 4/8, because these fractions are reducible. There are also two classifications of rational numbers depending on their decimal expression, which are:

*Terminating rational numbers*, whose decimal representation has a determined and fixed number of digits, i.e. 1/8 is equal to 0.125.

*Repeating rational numbers*, which have unlimited decimal numbers and are divided into two groups: pure repeating decimals, whose pattern is found immediately after the comma, for example 0.6363636363, and mixed repeating decimals, of which the pattern is found after a certain number of figures, for example 5.48176363636363.

Repeating rational numbers, whose decimal numbers form a pattern, differ from irrational numbers, whose decimal numbers are non-terminating and non-repeating.

## Properties of Rational Numbers

There are different properties of rational numbers. The properties for addition and subtraction are:

*Internal Property* – When adding two rational numbers, the result will always be another rational number, although this result may be reduced to a minimum if necessary.

*Associative Property* – If different rational numbers are grouped together, the result does not change and will remain a rational number. For example:

*Commutative Property* – If the order of the numbers in the operation varies, the result does not change.

*Neutral Element* – If the number zero is added to any rational number, the answer will be the same rational number.

*Additive Inverse* or *Opposite Element* – When a negative element cancels the existence of another element, meaning that the result is zero when they are added together.

In addition, there are also the properties of rational numbers for multiplication and division, which are explained as follows:

*Internal Property* – When multiplying rational numbers, the result is also a rational number.

This also applies to the division.

*Associative Property* – When grouping different factors in an operation, the form of the grouping does not alter the result.

*Commutative Property* – The order of the factors does not alter the result with rational numbers.

*Distributive Property* – When combining addition and multiplication, the result is equal to the sum of the factors multiplied by each of the summands, for example:

*Neutral Element* – In multiplication and division of rational numbers, there is a neutral element that is the number one, which when it is the product or quotient with another rational number, the result will be the same number.

## Examples of Rational Numbers

Rational numbers are fractional numbers, meaning they can be written as a ratio of any two integers. For example:

Although it could also be expressed in this way:

However, integers may also be included within the rational numbers to form a quotient with a neutral number. For example:

However, we could also express the integer 3 as a fraction if it is needed in any mathematical operation. When reducing this fraction, the same result is obtained:

There are also negative rational integers, for example:

0.2424242424 can be considered as a rational number with repeating decimals and it can be expressed as a fraction. Thus: