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CBSE Notes
Class 7
Maths
Chapter 8 Rational Numbers

CBSE Notes Class 7 Maths Chapter 8 Rational Numbers

A rational number is any number that can be expressed in the form qp​ , where p and q are integers, and q=0 . Rational numbers include both positive and negative numbers as well as zero. They can be represented on a number line, with positive values to the right of zero and negative values to the left. Every fraction is a rational number, though not all rational numbers are positive. Rational numbers follow properties like closure, commutativity, and associativity under addition, subtraction, and multiplication. They also have additive and multiplicative identities: 0 and 1, respectively.

1.0Definitions

  1. Natural Numbers (N): Counting numbers like 1, 2, 3, 4, etc., are called natural numbers, represented as N = {1, 2, 3, 4, ...}.
  2. Whole Numbers (W): Adding 0 to the set of natural numbers gives us whole numbers: W = {0, 1, 2, 3, 4, ...}. Every natural number is a whole number, but 0 is the only whole number not classified as a natural number.
  3. Integers (I or Z): Integers include all positive counting numbers, their negatives, and zero: Z = { ..., -3, -2, -1, 0, 1, 2, 3, ... }.
  4. Rational Numbers (Q) : A rational number can be written as qp​, where p and q are integers, and q=0. For example, −85​ is a rational number. Here, p is the numerator and q is the denominator.
  • Positive Rational Numbers: If both numerator and denominator are either positive or negative, the rational number is positive, e.g., 53​,−9−5​ .
  • Negative Rational Numbers: If either the numerator or the denominator is negative, the rational number is negative, e.g., −53​,4−7​.

2.0Properties of Rational Numbers

  1. Equivalent Rational Numbers: For qp​  and any non-zero integer m, qp​=q×mp×m​.
  2. Reducing to Simplest Form: Simplify qp​ by dividing both p and q by their HCF. qp​=q÷mp÷m​.
  3. Standard Form: A rational number qp​ is in standard form when p and q have no common factors other than 1 (i.e., they are co-prime), and q is positive. This form ensures the simplest representation of the number.

3.0Rational Numbers on a Number Line

Rational numbers can be placed on a number line just like integers. Positive rational numbers lie to the right of zero, while negative rational numbers are to the left. Each rational number corresponds to a unique point, allowing us to visualize their relative values easily.

rational Numbers on a number line

4.0Comparison of Two Rational Numbers

To compare two rational numbers, ba​ and dc​ , follow these steps:

  1. Make the Denominators Equal: Find a common denominator, usually the least common multiple (LCM) of b and d. Rewrite each fraction with this common denominator:

\frac{a}{b}=\frac{a \times d}{b \times d} \text { and } \frac{c}{d}=\frac{c \times b}{d \times b}

  1. Compare the Numerators: With the same denominator, compare the numerators. If a × d < c × b, then ba​>dc​ ; otherwise, ba​<dc​.

This method allows for direct comparison of the two rational numbers.

5.0Rules for Inserting Rational Numbers

Case I: Same Denominators

  1. Compare the numerators if the denominators are the same.
  2. If numerators differ significantly, insert numbers by increasing the numerator step-by-step while keeping the denominator constant.
  3. If the difference between numerators is small, multiply both rational numbers by multiples of 10 to create space for more numbers.

Case II: Different Denominators

  1. To find rational numbers between two with different denominators, make the denominators equal.
  2. This is done by finding the LCM of the denominators and rewriting the numbers accordingly.

6.0Addition of Rational Numbers

Case I: Same Denominator To add qp​ and qr​ , add the numerators and keep the common denominator:

qp​+qr​=qp+r​  

Case II: Different Denominators

  1. Find the LCM of the denominators.
  2. Rewrite the rational numbers with the LCM as their common denominator.
  3. Add the resulting rational numbers.

7.0Subtraction of Rational Numbers

Subtraction is the opposite of addition. For rational numbers x and y, subtracting y from x means adding the negative of y:

x – y = x + (–y)

For two rational numbers ba​ and dc​:

ba​−dc​=ba​+(−dc​)

This involves adding the additive inverse of dc​ .

8.0Multiplication of Rational Numbers

To multiply two rational numbers, multiply their numerators and their denominators:

ba​×dc​=b×da×c​

This gives the product of ba​ and dc​

9.0Reciprocal of Rational Numbers

A rational number y is the reciprocal of x if x×y=1 . For example, the reciprocal of 187​ is 718​ because:187​×718​=1

The reciprocal of 187​  can also be expressed as (187​)−1 .

10.0Division of Rational Numbers

Division is the inverse of multiplication. For integers a and b (b=0) , division is defined as:

a÷b=a×b1​

To divide a rational number x by a non-zero rational number y, multiply x by the reciprocal of y.

11.0Absolute Value of a Rational Number

The absolute value of a rational number is its distance from zero on the number line, regardless of its sign.  

For example:

​21​​=21​,​−21​​=21​

The absolute value is denoted by vertical bars around the number: ​qp​​=∣q∣∣p∣​.

12.0Sample Questions on Rational Numbers

  1. What do we mean by rational numbers?

Ans: Numbers that can be expressed as the ratio of 2 integers, with the denominator not equal to zero is called Rational Numbers. (e.g., 32​,7−5​).

  1. How can rational numbers with the same denominator be added?

Ans: Add the numerators and keep the denominator the same:

qp​+qr​=qp+r​

  1. What is the reciprocal of a rational number?

Ans: The reciprocal of ba​ is ab​ , as long as neither a nor b is zero.

13.0Key Features of Class 7 Maths Chapter 8 Rational Numbers

Clear explanation of what rational numbers are, with simple examples.

Content prepared strictly according to the latest CBSE Syllabus for Class 7 and guidelines.

Detailed discussion of closure, commutative, associative, and distributive properties.

Simple rules and examples for addition, subtraction, multiplication, and division of rational numbers.

Quick summaries, key points, and formulas to make last-minute revision easy and effective.

Chapter-wise CBSE Notes for Class 7 Maths:

Class 7 Maths Chapter 1 - Integers Notes

Class 7 Maths Chapter 2 - Fractions and Decimals Notes

Class 7 Maths Chapter 3 - Data Handling Notes

Class 7 Maths Chapter 4 - Simple Equations Notes

Class 7 Maths Chapter 5 - Lines And Angles Notes

Class 7 Maths Chapter 6 - The Triangles and its Properties Notes

Class 7 Maths Chapter 7 - Comparing Quantities Notes

Class 7 Maths Chapter 8 - Rational Numbers Notes

Class 7 Maths Chapter 9 - Perimeter And Area Notes

Class 7 Maths Chapter 10 - Algebraic Expressions Notes

Class 7 Maths Chapter 11 - Exponents And Powers Notes

Class 7 Maths Chapter 12 - Symmetry Notes

Class 7 Maths Chapter 13 - Visualising Solid Shapes Notes


Chapter-wise NCERT Solutions for Class 7 Maths:-

Chapter 1: Integers

Chapter 2: Fractions and Decimals

Chapter 3: Data Handling

Chapter 4: Simple Equations

Chapter 5: Lines and Angles

Chapter 6: The Triangle and its Properties

Chapter 7: Comparing Quantities

Chapter 8: Rational Numbers

Chapter 9: Perimeter and Area

Chapter 10: Algebraic Expressions

Chapter 11: Exponents and Powers

Chapter 12: Symmetry

Chapter 13: Visualising Solid Shapes

Frequently Asked Questions

Rational numbers are marked based on their value. Positive and negative fractions are spaced evenly on the number line.

Natural numbers: Counting numbers starting from 1. Whole numbers: Natural numbers including 0. Integers: Positive and negative whole numbers, including 0.

Convert them to have the same denominator or use cross-multiplication to compare the numerators.

A rational number is in standard form when the numerator and denominator are coprime, and the denominator is positive.

Find the least common denominator (LCD), rewrite the numbers with the LCD, and then add the numerators.

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