Rational Numbers Class 8 NCERT: Complete Notes, Examples, HOTS & Practice Questions
Introduction | परिचय ( Rational Numbers Class 8 )
Mathematics is not only about counting numbers but also understanding different types of numbers used in daily life. One such important concept is Rational Numbers. Rational numbers help us represent fractions, ratios, percentages, and many real-life quantities accurately. ( Rational Numbers Class 8 )
गणित केवल संख्याओं की गिनती तक सीमित नहीं है, बल्कि विभिन्न प्रकार की संख्याओं को समझना भी आवश्यक है। Rational Numbers (परिमेय संख्याएँ) गणित का एक महत्वपूर्ण अध्याय है जो हमें भिन्न, अनुपात, प्रतिशत और दैनिक जीवन की कई समस्याओं को हल करने में मदद करता है।
Table of Contents
- What are Rational Numbers?
- Representation of Rational Numbers
- Positive and Negative Rational Numbers
- Standard Form of Rational Numbers
- Equivalent Rational Numbers
- Comparison of Rational Numbers
- Properties of Rational Numbers
- Rational Numbers Between Two Numbers
- Real-Life Applications
- HOTS Questions
- Tricky Questions
- MCQs
- Important Formulae
- Key Points
- FAQs
What are Rational Numbers? ( Rational Numbers Class 8 )
A rational number is a number that can be expressed in the form:
where:
- p = Integer
- q = Non-zero Integer (q ≠ 0)
Examples
- 1/2
- 3/4
- -5/7
- 8/3
- -9/11
Hindi Explanation
यदि किसी संख्या को p/q के रूप में लिखा जा सके जहाँ p और q पूर्णांक हों तथा q शून्य न हो, तो ऐसी संख्या को परिमेय संख्या (Rational Number) कहते हैं।
उदाहरण:
- 2/5
- -7/9
- 11/4
- -13/8
Positive and Negative Rational Numbers
Positive Rational Numbers
Numbers greater than zero.
Examples:
- 2/3
- 5/8
- 7/9
Negative Rational Numbers
Numbers less than zero.
Examples:
- -2/3
- -5/8
- -7/9
Hindi
जो परिमेय संख्याएँ शून्य से बड़ी होती हैं वे धनात्मक परिमेय संख्याएँ कहलाती हैं जबकि जो शून्य से छोटी होती हैं वे ऋणात्मक परिमेय संख्याएँ कहलाती हैं।
Standard Form of Rational Numbers
A rational number is said to be in standard form when:
- Numerator and denominator have no common factor except 1.
- Denominator is positive.
Example
12/18
Divide both by 6:
= 2/3
Therefore, 2/3 is the standard form.
Hindi
जब अंश और हर का HCF = 1 हो तथा हर धनात्मक हो, तब संख्या को मानक रूप (Standard Form) कहा जाता है।
Equivalent Rational Numbers
Rational numbers having the same value are called equivalent rational numbers.
Example:
1/2 = 2/4 = 3/6 = 4/8
Hindi
जिन परिमेय संख्याओं का मान समान हो उन्हें समतुल्य परिमेय संख्याएँ कहते हैं।
Comparison of Rational Numbers
Method 1: Same Denominator
Compare numerators.
Example:
5/9 and 7/9
Since 7 > 5
Therefore:
7/9 > 5/9
Method 2: Cross Multiplication
Compare:
3/5 and 4/7
Cross multiply:
3 × 7 = 21
4 × 5 = 20
Therefore:
3/5 > 4/7
Properties of Rational Numbers
1. Closure Property
The sum, difference, and product of two rational numbers is always a rational number.
Example
1/2 + 3/4 = 5/4
2. Commutative Property
For Addition:
a + b = b + a
For Multiplication:
a × b = b × a
3. Associative Property
(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)
4. Distributive Property
a × (b + c)
= ab + ac
Hindi
वितरण नियम के अनुसार गुणा को जोड़ पर वितरित किया जा सकता है।
Rational Numbers Between Two Rational Numbers
Between any two rational numbers there are infinitely many rational numbers.
Example:
Between 1/2 and 3/4
Add numerator and denominator:
(1+3)/(2+4)
= 4/6
= 2/3
Hence 2/3 lies between 1/2 and 3/4.
Real-Life Applications
English
Rational numbers are used in:
- Banking
- Shopping discounts
- Measurement
- Cooking recipes
- Sports statistics
- Engineering calculations
Hindi
परिमेय संख्याओं का उपयोग:
- बैंकिंग
- खरीदारी
- प्रतिशत निकालने में
- खेल आँकड़ों में
- इंजीनियरिंग गणनाओं में
- रसोई और माप-तौल में
HOTS (Higher Order Thinking Skills) Questions
Q1.
Find five rational numbers between 2/5 and 3/5.
Answer:
21/50, 22/50, 23/50, 24/50, 25/50
Q2.
Is every integer a rational number?
Answer:
Yes.
Because:
5 = 5/1
-3 = -3/1
Hence every integer is a rational number.
Q3.
Can zero be a rational number?
Answer:
Yes.
0 = 0/1
Therefore zero is a rational number.
Tricky Questions
Q1.
Which is greater?
-2/3 or -5/6
Answer:
-2/3 = -4/6
Since -4/6 > -5/6
Answer: -2/3
Q2.
Can the denominator of a rational number be zero?
Answer:
No.
Division by zero is not defined.
Q3.
Is 7 a rational number?
Answer:
Yes.
7 = 7/1
Multiple Choice Questions (MCQs)
1. Which of the following is a rational number?
A) √2
B) π
C) 3/5
D) √7
✅ Answer: C
2. Standard form of 18/24 is:
A) 9/12
B) 3/4
C) 6/8
D) 12/16
✅ Answer: B
3. Which property is represented by:
2/3 + 5/7 = 5/7 + 2/3
A) Closure
B) Associative
C) Commutative
D) Distributive
✅ Answer: C
4. Zero is:
A) Natural Number only
B) Rational Number
C) Irrational Number
D) None
✅ Answer: B
Important Formulae
| Concept | Formula |
|---|---|
| Rational Number | p/q |
| Reciprocal | q/p |
| Standard Form | HCF = 1 |
| Mean Rational Number | (a+b)/2 |
Key Points for Revision
| Rational number = p/q, q ≠ 0 |
| Every integer is a rational number |
| Zero is a rational number. |
| Denominator can never be zero. |
| Infinite rational numbers exist between two rational numbers. |
| Rational numbers follow closure, commutative, associative, and distributive properties. |
Frequently Asked Questions (FAQs)
What is a rational number?
A number that can be written in the form p/q where q ≠ 0.
Is 0 a rational number?
Yes, because 0 = 0/1.
Is every fraction a rational number?
Yes, provided the denominator is not zero.
Why are rational numbers important?
They help represent fractions, ratios, percentages, and real-life measurements accurately.
Conclusion
Rational Numbers form the foundation of higher mathematics. Understanding their properties, comparison methods, and applications helps students solve complex mathematical problems with ease. For Class 8 students, mastering this chapter is essential for future topics such as algebra, percentages, and advanced number systems.
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