Cbse Class 7 Maths Notes Chapter 8

Rational Numbers: 

• A number that can be expressed in the form of  ${\displaystyle \frac{p}{q}}$, where $p$ and $q$ are integers and $q\ne 0$, is called a rational number. 

• For e.g.,${\displaystyle \frac{5}{2}},{\displaystyle \frac{9}{7}},{\displaystyle \frac{3}{8}}$  etc.

Numerator and Denominator:

• All rational numbers are written in the form of ${\displaystyle \frac{p}{q}}$ , where $p$ is the numerator and $q$ is the denominator.

• All integers and fractions are rational numbers. 

• For e.g. – $2$ is an integer it can expressed in the form of rational number as ${\displaystyle \frac{4}{2}}$

Equivalent Rational Numbers: 

• If we multiply the same non-zero integer in the numerator and denominator of a given rational number then we obtain another rational number.

• The obtained rational number will be equivalent to the given rational number.

•  For e.g. – The equivalent form of  ${\displaystyle \frac{5}{2}}$ can be ${\displaystyle \frac{5\times 2}{2\times 2}}={\displaystyle \frac{10}{4}}$ , 

• A rational number can have an infinite number of equivalent forms.

Positive and Negative Rational Numbers

• If both the numerator and denominator of a rational number are positive, then such a rational number is called Positive Rational Number.

•  For e.g. - ${\displaystyle \frac{5}{4}},{\displaystyle \frac{9}{7}},{\displaystyle \frac{13}{7}}$ etc.

• If either numerator or denominator is negative, then such a rational number is called Negative Rational Number. 

• For e.g. - ${\displaystyle \frac{-5}{4}},{\displaystyle \frac{7}{-3}},{\displaystyle \frac{3}{-9}}$  etc.

• Zero is the only number which is neither negative nor positive rational number. 

Standard Form of Rational Numbers:

• If the denominator of the rational numbers is positive and the denominator and numerator have no common factor except $1$, then such a rational number is said to be in its standard form. 

• For e.g., ${\displaystyle \frac{3}{2}}$ is in standard form since the numerator and denominator do not have anything in common.

• Any rational number can be written into standard form by taking out the common factors.

Comparison of Rational Numbers:

• Two rational numbers can be compared only when they have the same denominator.

• So, to compare rational numbers we should make denominator of the rational numbers same and then compare numerator

•  For e.g., Compare ${\displaystyle \frac{2}{3}}$ and ${\displaystyle \frac{3}{2}}$ 

Making denominator same 

${\displaystyle \frac{2\times 2}{3\times 2}}={\displaystyle \frac{4}{6}}$ 

${\displaystyle \frac{3\times 3}{2\times 3}}={\displaystyle \frac{9}{6}}$ 

Now, in ${\displaystyle \frac{4}{6}}$ and ${\displaystyle \frac{9}{6}}$ numerator of  ${\displaystyle \frac{9}{6}}$ is greater therefore ${\displaystyle \frac{3}{2}}>{\displaystyle \frac{2}{3}}$ 

• In case of negative rational numbers follow the same step by ignoring the sign and reverse the comparison sign or order in the end.

Rational Numbers between two Rational Numbers: 

• If we want to find rational numbers between two rational numbers then firstly convert the given two rational numbers in the same denominator form, then many rational numbers can be found between two given rational numbers.

• There are infinite rational numbers between two rational numbers.

Operations on Rational Numbers: 

• Addition – If the denominators of the rational numbers are the same then we simply add the numerator like we add integers keeping their denominator same. 

• For e.g. –

      ${\displaystyle \frac{-2}{3}}+{\displaystyle \frac{4}{3}}={\displaystyle \frac{-2+4}{3}}$ 

       $={\displaystyle \frac{2}{3}}$ 

• If the denominators of rational numbers are different, then we take the LCM of rational numbers to make the denominator the same and then add them. 

• For e.g. -  If  we want to add ${\displaystyle \frac{-5}{2}}$  and ${\displaystyle \frac{6}{3}}$, first find their LCM of $2$ and $3$  which is $6$ , so ${\displaystyle \frac{-5\times 3}{2\times 3}}={\displaystyle \frac{-15}{6}}$  and $${\displaystyle \frac{6\times 2}{3\times 2}}={\displaystyle \frac{12}{6}}$$. Thus,${\displaystyle \frac{-15}{6}}+{\displaystyle \frac{12}{6}}={\displaystyle \frac{-3}{6}}$or ${\displaystyle \frac{-5}{2}}+{\displaystyle \frac{6}{3}}={\displaystyle \frac{-3}{6}}$.

• Subtraction – If the denominators of the rational numbers are the same then we simply subtract the numerator like we subtract integers keeping their denominator same. 

• For e.g. –

      ${\displaystyle \frac{-2}{3}}-{\displaystyle \frac{4}{3}}={\displaystyle \frac{-2-4}{3}}$ 

       $={\displaystyle \frac{-6}{3}}$ 

• If the denominators of rational numbers are different, then we take the LCM of rational numbers to make the denominator the same and then subtract them. 

• For e.g. -  If  we want to subtract ${\displaystyle \frac{-5}{2}}$  and ${\displaystyle \frac{6}{3}}$, first find their LCM of $2$ and $3$  which is $6$ , so ${\displaystyle \frac{-5\times 3}{2\times 3}}={\displaystyle \frac{-15}{6}}$  and $${\displaystyle \frac{6\times 2}{3\times 2}}={\displaystyle \frac{12}{6}}$$. Thus, ${\displaystyle \frac{-15}{6}}-{\displaystyle \frac{12}{6}}={\displaystyle \frac{-15-12}{6}}={\displaystyle \frac{-27}{6}}$or  ${\displaystyle \frac{-5}{2}}-{\displaystyle \frac{6}{3}}={\displaystyle \frac{-27}{6}}$.

• Multiplication – In multiplication of rational numbers, we multiply the numerator of one rational number with the numerator of another rational number and similarly do the same with the denominator. 

• For e.g. – 

  $${\displaystyle \frac{3}{5}}\times {\displaystyle \frac{-1}{2}}={\displaystyle \frac{3\times (-1)}{5\times 2}}$$  $$={\displaystyle \frac{-5}{10}}$$ 

• Division – In division of rational numbers, we simply multiply one rational number with the reciprocal of another rational number. 

• For e.g. – If we want to divide ${\displaystyle \frac{4}{9}}$ by ${\displaystyle \frac{5}{3}}$, then we just multiply ${\displaystyle \frac{4}{9}}$ with reciprocal of  ${\displaystyle \frac{5}{3}}$ which will be ${\displaystyle \frac{3}{5}}$ , we get  ${\displaystyle \frac{4}{9}}\times {\displaystyle \frac{3}{5}}={\displaystyle \frac{12}{45}}$.

Class 7 Maths Chapter 8 Revision Notes

Maths Class 7 Rational Numbers Notes Important Topics

Notes of Class 7 Revision Notes Chapter 8 is prepared by Vedantu to help you revise your questions in this chapter. The following chapter presents several new concepts relating to Rational Numbers. This chapter helps you create solid basics for higher education and to achieve good results in the examinations. So the Rational numbers class 7 notes are important topics for higher studies’ point of view. So let’s look into the main topics of Rational Numbers which are discussed in this chapter.

• Introduction

• What is a Rational Number?

• How to identify rational numbers?

• Types of Rational Numbers.

• Standard Form of Rational Numbers.

• Positive and Negative Rational Numbers.

• Arithmetic Operations on Rational Numbers.

• Multiplicative Inverse of Rational Numbers.

• Rational Numbers Properties.

• Rational Numbers and Irrational Numbers.

• How to Find the Rational Numbers between Two Rational Numbers?

• Frequently Asked Questions on Rational Numbers.

What is a Rational Number?

A rational number is defined as any number which can be presented in the form of p/q where q ≠ 0. Rational numbers are represented by the symbol Q. Rational numbers can be natural numbers, whole numbers, integers, or fractions.

Examples of rational numbers are ½, 5, 0.25, etc.

Rational Numbers for the Class 8 project covers the conditions for checking a rational number.

To check if a number is rational or not, verify the below conditions.

• The number should be represented in the form of p/q, where q ≠ 0.

• The ratio between p/q can be simplified and presented in decimal form.

Types of Rational Numbers

• Positive Rational Number

If both the numerator and denominator are of the same signs, then the rational numbers are called positive rational numbers. Example: ⅖,(-4)/(-5).

• Negative Rational Numbers

If numerator and denominator are of opposite signs, then the rational number is called a negative rational number. Example: (-2)/5.

Arithmetic Operations on Rational Numbers

Let us discuss how we can perform arithmetic operations on rational numbers, say a/b and c/d.

• Addition of Rational Number: When we add a/b and c/d, we need to make the denominator the same. Hence, we can write it as (ad+bc)/bd.

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

• Subtraction of Rational Number: Similar to addition, if we subtract a/b and c/d, then also, we need to make the denominator the same, first, and then do the subtraction.

Example: 1/2 – 3/4 = (2-3)/4 = -1/4

• Multiplication of Rational Numbers: When multiplying two rational numbers, the numerator and denominator are multiplied, respectively.

If a/b is multiplied by c/d, then we write it as  (a × c)/(b × d).

Example: 1/2 × 3/4 = (1 × 3)/(2 × 4) = 3/8

• Division of Rational Number: If a/b is divided by c/d, then it is represented as:

(a/b) ÷ (c/d) = ad/bc

Example: 1/2 ÷ 3/4 = (1×4)/(2×3) = 4/6 = ⅔

Multiplicative Inverse of Rational Numbers

The multiplicative inverse of the rational number is defined as the reciprocal of the given fraction.

If the number is a/b then its multiplicative inverse will be b/a.

Fact: If a number is multiplied by its multiplicative then the inverse of the resultant value is equal to 1.

For example, 3/7 is a rational number, then the multiplicative inverse of the rational number 3/7 is 7/3.

Properties of Rational Numbers

As we know a rational number is a subset of the real number, the rational number will obey all the properties of the real number system. Following are some of the important properties of rational numbers.

• If we add, subtract, or multiply any two rational numbers, the results are always a rational number.

• A rational number remains the same if we multiply or divide both the numerator and denominator with the same factor.

• If we add zero to a rational number resultant will be the same number itself.

• Rational numbers are closed under addition, subtraction, and multiplication operations.

Follow the given methods to find the Rational Numbers between two Rational Numbers

Between two rational numbers, there are “n” rational numbers. 

Now, let us have a look at the two different methods for finding rational numbers.

• Method 1: Find the equivalent fraction for the given rational numbers and then find out the rational numbers in between them. Those numbers should be the required rational numbers.

• Method 2: Find the mean value for the two given rational numbers. The mean value will be the required rational number. In we have to find more rational numbers, repeat the same process with the old and the newly obtained rational numbers.

Did You Know?

√2 cannot be written as a simple fraction. There are many such numbers, and as they are not rational they can be called irrational numbers.

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Mathematics can be a difficult subject for class 7 students to achieve good grades in, but if they prepare methodically by having revision notes, they can easily achieve more marks in their Maths exam.

• Class 7 Maths Chapter 8 Revision Notes will assist you in predicting the types of questions that could be asked during the examination.

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Tips on How to Prepare for Exams Using Class 7 Maths Notes of Rational Numbers

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• Every question should be carefully read before attempting. Since there are some difficult questions, there is a risk that we would give the incorrect answer if the questions are not fully understood.

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Conclusion

Vedantu Subject Experts have prepared Class 7 Maths Revision Notes Chapter 8 to help you sort out your doubts in the chapter's topics and questions. In this chapter, you'll learn about new topics relating to rational numbers, their properties and forms, the difference between rational and irrational numbers, and some solved examples. This chapter will assist you in laying a strong foundation in preparation for higher classes and exams. The experts ensured to provide comprehensive solutions to all varieties of problems in Ch 8 Class 7 Maths Revision Notes. Students can completely depend on this free PDF of revision notes to prepare for their exams.

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