Math Concepts

Rational Numbers

Hey readers! Today we have come up with the topic of rational numbers. In this topic we will discuss what rational numbers are, their properties and some examples . 
So let’s begin:  

Table of Contents

1. Introduction & Understanding of Rational Numbers
2. Properties of Rational Numbers
3. Irrational & Rational Numbers
4. Solved examples and Practice examples
5. Summary
6. Frequently Asked Questions (FAQs)

10th September 2020

Reading Time: 4 Minutes

Introduction and Understanding of Rational Numbers

Before beginning with the topic, let’s brush up with what we learnt in the previous topics. 1,2,3,4….are called natural numbers, and when we include 0 to natural numbers, then we call them whole numbers; zero is neither positive nor negative. Whole numbers and negative numbers like -1,-2,-3,-4….. are called integers. 
Now tell me, are there any other numbers between two consecutive integers??

Yes, there are; numbers like ½,⅓,¼…lie between two consecutive integers. These are called fractions. Similarly there are also numbers like 0.7,0.8,0.5...which lie between two integers. These are called decimal numbers. So rational numbers are nothing but collective form of integers, fractions, and terminating or repeating decimal numbers

Definition of Rational Numbers

Rational numbers are those numbers, which can be represented as fractions p/q, where p and q are integers, and q is not equal to 0 (as p/0 is undefined).
Rational numbers are denoted by Q, i.e., Q=p/q.

The word ‘rational’ is derived from ‘ratio’, which is a comparison of two or more values which is also known as a fraction. In simple words, it is the ratio of two integers.
Example: 3/2 is a rational number. It means that the integer 3 is divided by the integer 2.

Examples of Rational Numbers

  • Number 9 can be written as 9/1 where 9 and 1 both are integers.
  • 0.5 can be written as ½, 5/10 or 10/20 and in the form of all terminating decimals.
  • √81 is a rational number, as it can be simplified to 9 and can be expressed as 9/1.
  • 0.7777777 is a recurring decimal and is a rational number

What is an Irrational Number?

The numbers which are not   rational numbers are called irrational numbers. Irrational numbers could be written in decimals but not in the form of fractions which means it cannot be written as the ratio of two integers.
Irrational numbers have endless non-repeating digits after the decimal point. Such decimals are called non-terminating decimals. Below is an example of the irrational number:
Example: √8=2.82842712…

Examples of Irrational Numbers

  • 5/0 is an irrational number, with the denominator as zero.
  • π is an irrational number which has value 3.142…and is a non-terminating and non-repeating number.
  • √2 is an irrational number, as it cannot be simplified.
  • 0.212112111…is an irrational number as it is non-recurring and non-terminating.

Properties of Rational Numbers

Here are some properties based on arithmetic operations such as addition and multiplication when performed on the rational irrational numbers.

  • Closure Property:  This property states that when any two rational numbers are added, the result is also a rational number.

Example: 1/2 + 1/3 = (3 + 2)/6 = 5/6
So it is closed under addition, the same way for other operations also it remains closed.
Example: 1/2 -1/3=1/6
1/2* 1/3=1/6
1/2/1/3=3/2

  • Commutative Property: The sum of two rational numbers are commutative, as they can be added in any order.

For example:1/2+1/3=5/6;   or 1/3+1/2=5/6;
The product of two rational numbers is a rational number.
Example: 1/2 x 1/3 = 1/6;  or 1/3*1/2=1/6;
This property holds true for addition and multiplication , but it does not hold true for Subtraction and division of rational numbers.

  • Associative Property: In this property, when we Take any three rational numbers a, b and c. 

(a + b) + c and a + (b + c) is the same .
It states that you can add or multiply numbers regardless of how they are grouped.

For example, given numbers are 5, -6 and 2/3;  ( 5 – 6 ) + 2/3= -1+2/3=-1/3; Now, 5 + ( -6 + 2/3)=-1/3.
But again this property does not hold true for subtraction and division.

  • Distributive Property: This  property says that when we multiply a sum of integers by another integer it  equals to the same result when we multiply each integer by another  integer and then add the products together. In other words, it is the distributive property of multiplication by making use of integers.

Distributive property states that for any three integers  x, y and z we have
             x * ( y + z ) = (x * y) +( x * z);
            2*(3+4)=(2*3)+(2*4)  = 14


Irrational and Rational Numbers

The numbers which are NOT rational numbers are called irrational numbers.

The set of irrational numbers is represented by \(I\).

The differences between rational and irrational numbers are as follows:

Rational Numbers Irrational Numbers

These are numbers that can be expressed as fractions of integers.

Examples: \(0.75, \dfrac{-31}{5}\), etc

These are numbers that CANNOT be expressed as fractions of integers.

Examples: \(\sqrt{5}, \sqrt[3]{-2}\), etc.

They can be terminating decimals. They are NEVER terminating decimals.

They can be non-terminating decimals with repetitive patterns of decimals.

Example: \(1.414 \, 414 \, 414 ...\) has repeating patterns of decimals where \(414\) is repeating.

They should be non-terminating decimals with NO repetitive patterns of decimals.

Example: \( \sqrt{5} = 2.236067977499789696409173....\) has no repeating patterns of decimals

The set of rational numbers contains all natural numbers, all whole numbers, and all integers. The set of irrational numbers is a separate set and it does NOT contain any of the other sets of numbers.

Examples of rational and irrational numbers are shown. Rational numbers include natural numbers, whole numbers and integers.

Challenge your math skills
Challenging Questions
  1. Is \( \sqrt{4}\) rational or irrational?
  2. Is \( \dfrac{\sqrt{7}}{2} \) rational or irrational?
  3. Is \(\pi\) rational or irrational?
    Hint: Note that the exact value of \(\pi\) is NOT \(\dfrac{22}{7}\). Its value is a decimal \(3.141592653589793238...\), which has no repeating patterns of decimals.

Solved Examples

Example 1

 

 

Identify the rational numbers among the following:

\[ \sqrt{4},\, \sqrt{3},\, \dfrac{\sqrt{5}}{2},\, \dfrac{-4}{5},\, \pi,\, 1.41421356237309504...\]

Solution:

A rational number when simplified should either be a terminating decimal or a non-terminating decimal with repeating patterns of decimals.

Thus, the rational numbers among the given numbers are:

\[\mathbf{ \sqrt{4} \text{ and } \dfrac{-4}{5} }\]
Example 2

 

 

Find a rational number that lies between the following rational numbers:

\[ \dfrac{1}{2} \text{ and } \dfrac{2}{3}\]

Solution:

We know that the average of any two numbers lies between the two numbers.

Let's find the average of the given two rational numbers.

\[ \begin{aligned} \dfrac{ \dfrac{1}{2}+ \dfrac{2}{3}}{2} &= \dfrac{\dfrac{3}{6}+ \dfrac{4}{6}}{2}\\[0.3cm] &= \dfrac{ \left(\dfrac{7}{6} \right)}{2}\\[0.3cm] &= \dfrac{ \left(\dfrac{7}{6} \right)}{ \left(\dfrac{2}{1} \right)}\\[0.3cm] &= \dfrac{7}{6} \times \dfrac{1}{2}\\[0.3cm] &= \dfrac{7}{12} \end{aligned} \]

Thus, the rational number is:

\[\mathbf{\dfrac{7}{12}}\]

Practice questions


1. The sum of the rational numbers – 8/19 and -4/57 is _____
2. What number should be added to 3/8 to get -1/24? 
3. Which of the rational numbers 4/9, -5/6, -7/-12 and 11/-24 is the smallest?
4. Which of the rational numbers -4/9, 5/-12, 7/-18, 2/-3 is the greatest?
5. Simplify: 2/3 + -4/5 + 7/15 + -11/20
6. What number should be subtracted from -3/4 so as to get 5/6?
7. Which of the following rational numbers is in the standard form?
8. The sum of two rational numbers is -7. If one of the numbers is –15/19, the other number is _____
9. Which of the following forms a pair of equivalent rational numbers?
10. The value of {-8/13 × 26/-3} is _____
11. The reciprocal of a negative rational number _____
12. The value of (- 16/21 ÷ -4/3) is _____
13. Fill in the blanks: 5/12 ÷ (_____) = -35/18
14. The product of two numbers is -20/9. If one of the numbers is 4, find the other.
 
 
 
Answers:
1. -28/57    ,  2. -5/12    ,  3. -5/6     ,4. 7/-18   , 5. -13/60   , 6. -19/12   , 7. -9/28
8. -118/19   , 9.  -25/35 and 55/-77  , 10. 16/3 , 11.  is a negative rational number 
12.  4/7   ,  13. -3/14   , 14. -5/9.


How do we use rational numbers in our day to day life?

  • We use taxes in the form of fractions.
  • When you share a pizza or anything.
  • When you completed homework half portion,you say that you completed 50% i.e.,1/2
  • Interest rates on loans and mortgages.
  • Interest on savings accounts.

Summary

So here we come to the end of this topic. Let’s have a quick recap. Rational numbers are a set of integers, fractions and decimal numbers, which can be represented as fractions in the form of p/q where p and q are integers and q is not equal to 0. They are denoted as Q. 
If you have any questions or doubts, feel free to post in the comments section.


Frequently Asked Questions (FAQs)

1. What are rational and irrational numbers?

Rational numbers are the numbers that can be expressed in the form of a ratio (p/q=Q; where q is not equal to 0 )and irrational numbers cannot be expressed as a fraction. But both the numbers are real numbers and can be represented in a number line.
They can be differentiated as:
Rational numbers are finite or repeating decimals which can be represented as the ratio of two integers, whereas irrational numbers are infinite and non-repeating decimal numbers.
Examples of rational numbers are ½, ¾, 7/4, 1/100, etc.
Examples of irrational numbers are √2, √3, pi(π), etc.

2. Is pi a real number?

Pi (π) is an irrational number and hence it is a real number. The value of π is 22/7 or         3.14285714286.

3. If a decimal number is represented by a bar, then it is rational or irrational?

A decimal number with a bar represents that the number after the decimal is repeating, hence  it is a rational number.

4. 3.605551275…… is rational or irrational?

The ellipsis (…) after 3.605551275 shows that the number is non-terminating and also there is no repeated pattern here. Therefore, it is irrational.

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