Chapter Notes: Rational Numbers

# Rational Numbers Class 7 Notes Maths Chapter 1

 Table of contents Introduction Need for Rational Numbers What are Rational Numbers? Properties of Rational Numbers Operations on Rational Numbers Addition of Rational Numbers Subtraction of Rational Numbers Multiplication of Rational Numbers Division of Rational Numbers Positive and Negative Rational Numbers Representation of Rational Numbers on the Number Line Rational Numbers between Two Rational Numbers

## Introduction

In your study of numbers, you started by counting objects around you. The numbers you use for counting are called counting numbers or natural numbers. They are 1, 2, 3, 4, and so on. When we include 0 with natural numbers, we get the whole numbers, which are 0, 1, 2, 3, and so on.
Then we combined the negatives of natural numbers with whole numbers to create integers. Integers are numbers like -3, -2, -1, 0, 1, 2, 3, and so on. So, we expanded the number system from natural numbers to whole numbers and from whole numbers to integers.

You also learned about fractions, which are numbers written in the form of numerator/denominator, where the numerator can be 0 or a positive whole number, and the denominator is a positive whole number.

Showing 1 part of a pizza in the form of a fraction
In this chapter, we will introduce the concept of rational numbers, along with exploring how to add, subtract, multiply, and divide them.

## Need for Rational Numbers

We already learned how integers can be used to represent opposite situations involving numbers. For example, we used positive integers to denote distances to the right of a place and negative integers to represent distances to the left. Similarly, we used positive integers to represent profits and negative integers for losses.

Now, some situations involve fractional numbers, similar to the examples mentioned earlier. For instance, we can represent a distance of 750m above sea level as 3/4 km. But can we represent 750m below sea level in km? Can we denote the distance of 3/4 km below sea level as -3/4? We observe that -3/4 is neither an integer nor a fractional number. Therefore, we need to expand our number system to include such numbers.

Pic depicting above and below sea level

## What are Rational Numbers?

A rational number is a number that can be written in the form p/q, where p and q are integers and q ≠ 0.

• The denominator of a rational number can never be zero.
• A rational number is positive if its numerator and denominator are both either positive integers or negative integers.   If either the numerator or the denominator of a rational number is a negative integer, then the rational number is called a negative rational number.

• The rational number zero is neither negative nor positive.
• On the number line:
• Positive rational numbers are represented to the right of 0.
• Negative rational numbers are represented to the left of 0.
• All numbers, including whole numbers, integers, fractions and decimal numbers, can be written in the numerator/denominator form.

### Equivalent Rational Number

By multiplying or dividing both the numerator and the denominator of a rational number by the same non-zero integer, we can get another rational number that is equivalent to the given rational number.

### The standard form of Rational Number

A rational number is said to be in its standard form if its numerator and denominator have no common factor other than 1, and its denominator is a positive integer.

To reduce a rational number to its standard form, divide its numerator and denominator by its Highest Common Factor (HCF). To find the standard form of a rational number with a negative integer as the denominator, divide its numerator and denominator by their HCF with a minus sign.

Question for Chapter Notes: Rational Numbers
Try yourself:Which of the following properties is NOT true for rational numbers?

## Properties of Rational Numbers

### (a) Closure Property:

(i) Whole Numbers:

• Addition: Whole numbers are closed under addition (e.g., 0+5=50+5=5), but not under subtraction or division.
• Subtraction: Whole numbers are not closed under subtraction (e.g., 5−7=−257=2).
• Multiplication: Whole numbers are closed under multiplication (e.g., 0×3=00×3=0).
• Division: Whole numbers are not closed under division.

(ii) Integers:

• Addition: Integers are closed under addition (e.g., −6+5=−16+5=1), subtraction (e.g., 7−5=275=2), and multiplication.
• Subtraction: Integers are closed under subtraction.
• Multiplication: Integers are closed under multiplication.
• Division: Integers are not closed under division.

(iii) Rational Numbers:

• Subtraction: Rational numbers are closed under subtraction  57−32=5⋅3−2⋅72
• Multiplication: Rational numbers are closed under multiplication  23×
• Division: Rational numbers are not closed under division, especially when dividing by zero.

Note: Rational numbers exclude zero in the denominator for division closure.
Division by zero is undefined in the set of rational numbers.

### (b) Commutativity Property:

(i) Whole Numbers:

• Subtraction: Not commutative.
• Multiplication: Commutative.
• Division: Not commutative.

(ii) Integers:

• Subtraction: Not commutative 5−
• Multiplication: Commutative.
• Division: Not commutative.

(iii) Rational Numbers:

• Subtraction: Subtraction is not commutative for rational numbers.
• Multiplication: Rational numbers follow commutativity in multiplication 73×65=
• Division: Division is not commutative for rational numbers.

### (c) Associativity Property:

(i) Whole Numbers:

• Subtraction: Not associative.
• Multiplication: Associative.
• Division: Not associative.

(ii) Integers:

• Subtraction: Not associative.
• Multiplication: Associative.
• Division: Not associative.

(iii) Rational Numbers:
23+(56+79)=(23+56)+7

• (a) Addition: Associative for rational numbers
• (b) Subtraction: Not associative for rational numbers.
• (c) Multiplication: Associative for rational numbers 75×(23×34)=(75×23)×3
• (d) Division: Not associative for rational numbers.

### => The Role of Zero (0):

• 2+0=0+2=22+0=0+2=2
• Adding zero to a whole number results in the same whole number.
• In general, �+0=0+�=�a+0=0+a=a, where a is a whole number.
• −5+0=0+(−5)=−55+0=0+(5)=5
• Adding zero to an integer yields the same integer.
• In general, �+0=0+�=�b+0=0+b=b, where b is an integer.
• 27+0=0+27=
• Adding zero to a rational number results in the same rational number.
• In general, �+0=0+�=�c+0=0+c=c, where c is a rational number.
• Zero is called the identity for the addition of rational numbers.
• It serves as the additive identity for integers and whole numbers as well.

### => The Role of 1:

1. Multiplication with Whole Numbers:
• 5×1=1×5=55×1=1×5=5
• Multiplying any whole number by 1 gives the same whole number.
• In general, �×1=1×�=�a×1=1×a=a for any whole number a.
2. Multiplication with Rational Numbers:
• 27×1=1×27=27Multiplying any rational number by 1 results in the same rational number.
• In general, �×1=1×�=�a×1=1×a=a for any rational number a.
3. Multiplicative Identity:
• 1 is called the multiplicative identity for rational numbers.
• It is also the multiplicative identity for integers and whole numbers.

### Negative of a Number:

=> For integers, the negative of a number is found by adding the additive inverse.

• 1+(−1)=(−1)+1=01+(1)=(1)+1=0, so the negative of 1 is -1.
• 2+(−2)=(−2)+2=02+(2)=(2)+2=0, making 2 the negative or additive inverse of -2, and vice versa.
• In general, for an integer a, �+(−�)=(−�)+�=0a+(a)=(a)+a=0.

=>For rational numbers, similar principles apply.

In general, for a rational number

### Reciprocal:

The reciprocal or multiplicative inverse of a rational number  is another rational number

For example, Zero has no reciprocal because there is no rational number that, when multiplied by 0, results in 1.
Zero has no reciprocal because there is no rational number that, when multiplied by 0, results in 1.
In general, ifis the reciprocal of  , then

• 23+(−23)=(−23)+23=0

### Comparing Two Rational Numbers

While comparing positive rational numbers with the same denominator, the number with the greatest numerator is the largest. It is easy to compare these numbers if their denominators are the same.

Eg:
A positive rational number is always greater than a negative rational number. While comparing negative rational numbers with the same denominator, compare their numerators ignoring the minus sign. The number with the greatest numerator is the smallest.

Positive rational numbers lie to the right of 0, while negative rational numbers lie to the left of 0 on the number line.

To compare rational numbers with different denominators, convert them into equivalent rational numbers with the same denominator, which is equal to the LCM of their denominators. You can find infinite rational numbers between any two given rational numbers.

Question for Chapter Notes: Rational Numbers
Try yourself: Which of the following rational numbers is the smallest?

## Operations on Rational Numbers

The following points are to be noted while finding the sum of rational numbers:

• If the denominators of the given rational numbers are the same, then the denominator of their sum will also be the same. The numerator of the sum of two rational numbers with the same denominator is the sum of the numerators of the given numbers.
• To add rational numbers with different denominators, we convert them into equivalent rational numbers with the same denominator.

Thus, the sum is .

• Two rational numbers whose sum is zero are called additive inverses of each other.
• The denominator of the difference of two rational numbers with the same denominator is the same as the common denominator of the given numbers.

Question for Chapter Notes: Rational Numbers
Try yourself:What is the sum of two rational numbers -3/7 and 5/7?

## Subtraction of Rational Numbers

• To subtract rational numbers with different denominators, we must convert them into equivalent rational numbers with the same denominator.

## Multiplication of Rational Numbers

• To multiply two rational numbers, we simply multiply their numerators and denominators with their correct signs.
• Two rational numbers whose product is 1 are called reciprocals of each other.
Eg.  (-13/6) * (6/-13) = 1
• A rational number and its reciprocal will always have the same sign.

## Division of Rational Numbers

To divide one rational number by another, we actually multiply the first number with the reciprocal of the second number.

### Solved Examples

Example 1: Simplify the rational number -16/(-24)

Ans:

Step 1: Identify the common factors of both numerator and denominator. In this case, the common factors are 2, 4, and 8.
Step 2: Divide both numerator and denominator by the greatest common factor, which is 8.
(-16) ÷ 8 = 2
(-24) ÷ 8 = 3
So, -16/(-24) simplifies to 2/3.

Example 2: Add the rational numbers 2/5 and 3/10
Ans:

Step 1: Find the least common denominator (LCD) of both denominators. In this case, the LCD is 10.
Step 2: Convert both fractions to equivalent fractions with the LCD as the new denominator.
2/5 = 4/10 (multiply both numerator and denominator by 2)
3/10 = 3/10 (no change needed)
Step 3: Add the equivalent fractions.
4/10 + 3/10 = (4 + 3) / 10 = 7/10
So, 2/5 + 3/10 = 7/10.

## Positive and Negative Rational Numbers

1. Positive Rational Numbers are the numbers whose both the numerator and denominator are positive.
Example: 3/4, 12/24 etc.

2. Negative Rational Numbers are the numbers whose one of the numerator or denominator is negative.
Example: (-2)/6, 36/(-3) etc.
Remark: The number 0 is neither a positive nor a negative rational number.

## Representation of Rational Numbers on the Number Line

Representation of whole numbers, natural numbers and integers on a number line is done as follows

Rational Numbers can also be represented on a number line like integers i.e. positive rational numbers are on the right to 0 and negative rational numbers are on the left of 0.

Representation of rational numbers can be done on a number line as follows:

Between any two rational numbers, there are infinitely many rational numbers.

## Rational Numbers between Two Rational Numbers

To find the rational numbers between two rational numbers, we have to make their denominator the same then we can find the rational numbers.

Example: Find three rational numbers between 1/4 and 1/2.

Ans:
(i) First, we find the mean of given numbers.
Mean is

(ii) Again we find another rational number between 1/4 and 3/8 .For this, again we calculate mean of 1/4 and 3/8.
Mean is

(iii) For the third rational number, we again find mean of 3/8 and 1/2.

Mean is

Hence, 5/16, 3/8, and 7/16 are 3 rational numbers between 1/4 and 1/2.

Example: Find rational numbers between 3/5 and 3/7.
Ans: To find the rational numbers between 3/5 and 3/7, we have to make their denominator same.
LCM of 5 and 7 is 35.
Hence the rational numbers between 3/5 and 3/7 are

These are not the only rational numbers between 3/5 and 3/7.
If we find the equivalent rational numbers of both 3/5 and 3/7 then we can find more rational numbers between them.

Hence we can find more rational numbers between 3/5 and 3/7.

Remark: Between any two given rational numbers, we need not necessarily get an integer but there are countless rational numbers between them.

The document Rational Numbers Class 7 Notes Maths Chapter 1 is a part of the Class 7 Course Mathematics (Maths) Class 7.
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## Mathematics (Maths) Class 7

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## FAQs on Rational Numbers Class 7 Notes Maths Chapter 1

 1. What is the need for understanding Rational Numbers in mathematics?
Ans. Rational numbers are essential in mathematics as they help us represent fractions, decimals, and integers. They are used in various mathematical operations such as addition, subtraction, multiplication, and division, making them a fundamental concept in mathematics.
 2. How are Rational Numbers defined and what sets them apart from other types of numbers?
Ans. Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not equal to zero. They include integers, fractions, and decimals that either terminate or repeat. This distinguishes them from irrational numbers, which cannot be expressed as a fraction of two integers.
 3. What are some properties of Rational Numbers that make them unique?
Ans. Rational numbers have properties such as closure under addition, subtraction, multiplication, and division. They also have the property of being able to be written in simplest form, where the numerator and denominator have no common factors other than 1.
 4. How do you perform operations on Rational Numbers, such as addition, subtraction, multiplication, and division?
Ans. Operations on Rational Numbers involve following the rules of arithmetic for fractions. Addition and subtraction require finding a common denominator, while multiplication involves multiplying the numerators and denominators. Division is done by multiplying by the reciprocal.
 5. Can Rational Numbers be both positive and negative, and how do we differentiate between the two?
Ans. Yes, Rational Numbers can be both positive and negative. Positive rational numbers have a positive sign (+) in front of them, while negative rational numbers have a negative sign (-) in front of them. The sign indicates the direction on the number line.

## Mathematics (Maths) Class 7

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