What is rational number give example?

Aug 09, 2022, 16:45 IST

Rational Numbers:

A rational number is a number that can be written as a ratio (p/q form). That means it can be written as a fraction, in which both the numerator (p) and the denominator (q) are integers and q not zero.

For Example:-

• The number 8 is a rational number because it can be written as the fraction 8/1.
• The number 0 is a rational number if it can be represented in p/q form where q is not equal to zero
• Likewise, 3/4 is a rational number because it can be written as a fraction.
• Even a big, clunky fraction like 7,324,908/56,003,492 is rational, simply because it can be written as a fraction.

What is Rational Number

In Maths, a rational number is a type of real numbers, which is in the form of p/q where q is not equal to zero. Any fraction with non-zero denominators is a rational number. Some of the examples of rational number are 1/2, 1/5, 3/4, and so on. The number “0” is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc. are not rational, since they give us infinite values. 

Wh​at is Irrational Number

Irrational numbers are those real numbers that can't be represented in the form of a ratio. In different words, those real numbers that are not rational numbers are  called as irrational numbers.

In this article, we will learn about what is a rational number, the properties of rational numbers along with its types, the difference between rational and irrational numbers, and solved examples. It helps to understand the concepts in a better way. Also, learn the various rational number examples and learn how to find rational numbers in a better way. To represent rational numbers on a number line, we need to simplify and write in the decimal form first.

Table of Content for Rational Number

• Defination of Rational Number
• How to identify Rational Number
• Example of Rational Number
• Types of Rational Number

1. Standard form of Rational Number
2. Positive and Negative Rational Number

• Some Example of Positive and Negative Number
• Arithmetic Operations on Rational Numbers
• Multiplicative Inverse of Rational Numbers
• Rational Numbers Properties
• How to Find the Rational Numbers between Two Rational Numbers
• Solved Example

What is a Rational Number?

A rational number, in Mathematics, are often defined as any number which may be represented within the sort of p/q where q ≠ 0. Also, we will say that any fraction fits under the category of rational numbers, where the denominator and numerator are integers and therefore the denominator isn't adequate to zero.

When the real number (i.e., fraction) is split, the result are going to be in decimal form, which can be either terminating decimal or the circulating decimal. the way to identify rational numbers?

How to identify rational numbers?

To identify if variety is rational or not, check the below conditions.

• It is represented within the sort of p/q, where q≠0.
• The ratio p/q are often further simplified and represented in decimal form.

The set of rational numerals:

• Include positive, negative numbers, and zero
• Can be expressed as a fraction

Example​s of Rational Numbers:

P = 20

Q = 10

PQ = 2

It is Rational Number

Types of Rational Numbers

A number is rational if we can write it as a fraction, where both denominator and numerator are integers and the denominator is a non-zero number

It is of two types:

1. Standard Form
2. Positive and Negative

Standard Form of Rational Numbers

The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore the divisor is positive. For example, 12/36 is a rational number. But it can be simplified as 1/3; common factors between the divisor and dividend are only one. So we can say that rational number ⅓ is in standard form.

Positive and Negative Rational Number

As we know that the rational number is in the form of p/q, where p and q are integers. Also, q should be a non-zero integer. The rational number can be either positive or negative. If the rational number is positive, both p and q are positive integers. If the rational number is in  the form of (p/q), then either p or q has negative value. It means that
-(p/q) = (-p)/q = p/(-q).

Some Example of Positive and Negative Number

Positive Number:

• In Positive Rational Number both the numerator and denominator are of same sign.
• Greater than 0
• Examples of positive rational numbers: 11/16, 9/5 and 3/9

Negative Rational Number

• In Negative Rational Number both numerator and denominator are of opposite sign.
• Less than 0
• Examples of negative rational numbers: -2/5, 9/-12 and -1/4.

Arithmetic Operations on Rational Numbers

In Maths, arithmetic operations are the basic operations we perform on integers. Let us discuss here how we can perform these operations on rational numbers, say a/b and p/q.

Addition: When we add p/q and a/b, we need to make the denominator the same. Hence, we get (pb+qa)/qb.

Example: 1/4+5/4 = (1+5)/4 = 6/4 = 3/2

Subtraction: Similarly, if we subtract a/b and p/q, then also, we need to make the denominator same, first, and then do the subtraction.

Example: 5/4- 1/4 = (5-1)/4 = 4/4 = 1

Multiplication: In case of multiplication, while multiplying two rational numbers, the numerator and denominators of the rational numbers are multiplied, respectively. If a/b is multiplied by p/q, then we get (p×a)/(q×b).

Example: 5/4 x 1/4 = 5/16

Division: If a/b is divided by p/q, then the result is: (a/b)÷(p/q) = aq/bp

Example: 5/4÷ 1/4 = (5×4)/(4×1) = 20/4 = 5

Multiplicative Inverse of Rational Numbers

As the rational number is represented in the form a/b, which is a fraction, then the multiplicative inverse of the rational number is the reciprocal of the given fraction.

For example, 3/4 is a rational number, then the multiplicative inverse of the rational number 3/4 is 4/3, such that (3/4)x(4/3) = 1

Rational Numbers Properties

Since a real number may be a subset of the important number, the real number will obey all the properties of the important numeration system. a number of the important properties of the real numbers are as follows:

• The results will be rational number if we multiply, add, or subtract any two rational numbers.
• A rational number remains an equivalent if we divide or multiply both the numerator and denominator with an equivalent factor.
• If we add zero to a real number then we'll get an equivalent number itself.
• Rational numbers are closed under addition, subtraction, and multiplication.

How to Find the Rational Numbers between Two Rational Numbers?

There are “n” numbers of rational numbers between two rational numbers. The rational numbers between two rational numbers can be found easily using two different methods. Now, let us have a look at the two different methods.

Method 1: 

Find out the equivalent fraction for the given rational numbers and find out the rational numbers in between them. Those numbers should be the required rational numbers.

Method 2: 

Find out the mean value for the two given rational numbers. The mean value should be the required rational number. In order to find more rational numbers, repeat the same process with the old and the newly obtained rational numbers.

Solved Example:

Q1. Identify whether mixed fraction, 1³/₂ is a rational number.

Ans: The Simplest form of 1³/₂ is 5/2

Numerator = 5, which is an integer

Denominator = 2, is an integer and not equal to zero.

So, yes, 5/2 is a rational number.

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Abou​t Rational Number and its Example Pdf