Rational Numbers

The word rationale is derived from the ratio which means it is very much related to the ratio concept. The number is expressed as p/q where numerator and denominator are integers. The questions related to rational numbers are going to be discussed in this section such as the definition of rational numbers, examples of rational numbers, types of rational numbers, and the way of identifying the specific number.

Definition

The number can be defined as a number that is represented as p/q form and numerator and denominator are integers and the denominator is not equal to zero. The alphabet q is used for the denotation of the set of rational numbers and the rational number can also be defined as the number expressed in the sub-divisional form and the numerator and denominator of the fraction are integers and the value of the denominator is not zero then the subdivision is called the rational number. A rational number is expressed as a fraction but it is not considered infraction because the fraction is made up of whole numbers and the rational number is made up of integers as its numerator and denominator.

Examples of Rational Numbers

The number that is represented as a fraction form and the denominator and numerator of that fraction are integers then the number is rational. Examples of rational numbers are 5/3 -4/6 0.4 Or 4/10 -0.6 or -6/10

Types of Rational Numbers

Rational numbers are of different types and only fractions that have their numerator and denominator don’t need to be integers are only rational numbers. Rational numbers can be expressed in various ways and also denoted in various forms of numbers. The different types of rational numbers are

• Integers such as -2, 0, 5, etc are rational numbers
• Fractions having numerator and denominator are integers such as 4/5, -8/6 are rational numbers
• The decimals that are terminating such as 0.45, 0.765, and 0.2546 are rational numbers
• Non-terminating decimals are also considered as rational numbers which means the repeating pattern after the decimal point such as 0.222, 0. 1212… etc are rational numbers
•  The decimals having a repetition of some special pattern after the decimal point are known as non-terminating decimals

Properties of the rational number

The number that is stated in the fractional form and having its numerator and denominator are positive or negative and the denominator is greater or less than zero is known as a rational number. The properties of the following numbers are as follows:

Property of closure

The closure property of rational numbers shows that in case the operations such as addition, subtraction, and multiplication operations are performed on the two rational numbers then the obtained result will also be a rational number. The numbers are nearer during the operation of adding, subtracting and multiplying numbers. The division operation of rational is not considered in closure property because the number divisible by 0 is not defined.

Commutative property

The function of “addition” and “multiplication” of “rational numbers” are included in the commutative property because the change in the order of numbers during the functioning does not affect the result of the operation. The subtraction or the division of rational numbers is not considered in the following property because the change in the order of the numbers changes the result of the operation. For example Law of commutative property of addition=> a+b = b+a

Associative property

The associative property can be defined as during the operation of the rational numbers in the form of a group and the change in the arrangement of the numbers will not affect the result of the performed actions on the numbers. Operations such as the addition and multiplication of rational numbers follow the associative law and the division and subtraction operation of rational numbers do not follow the associative law because of the change in the arrangement of numbers during these operations, the result of the operation is getting affected.

Distributive property

The distributive property of rational numbers can be defined as the operation such as multiplying a number with the value of sum in any arrangement that the output of the calculation is not affected.

Identity property

Identity property is the property of numbers that shows that any number multiplied by 1 remains the same. In the case of “rational numbers” 0 is the identity of addition and 1 is an identity for multiplication which means any rational number added to 0 remains the same and any rational number multiplied by 1 remains the same after the operation.

Inverse property

The inverse property says that the multiplication of a number such as x/y with its reciprocal then the obtained result will be 1. In the case of rational numbers, -x/y is additive inverse, and y/x is its multiplicative inverse.

How to identify Rational Numbers

There are rules of identifying the rational numbers that mean a fractional number can be a rational number in case that fraction meets the criteria of rational numbers and the criteria are

• The number should be a fraction of integers
• Every whole number is a rational number

Conclusion

This section concludes that the number is represented as a fraction having an integer numerator and the denominator is known as a rational number. The examples of rational numbers and their types are discussed in this section.