Real Numbers (वास्तविक संख्याएँ)
Table of Contents
- Introduction to Real Numbers
- Types of Real Numbers
- Rational Numbers
- Irrational Numbers
- Fundamental Theorem of Arithmetic
- HCF and LCM
- Decimal Expansion of Rational Numbers
- Important Formulae
- Key Points for Board Exams
- NCERT Questions and Answers
- Multiple Choice Questions (MCQs)
- Assertion and Reason Questions
- Previous Year Questions (PYQ)
- HOTS Questions
- Expected Questions for Board Exam 2027
- One Shot Revision
- FAQs
Abbreviations
| Abbreviation | Full Form |
|---|---|
| HCF | Highest Common Factor |
| LCM | Least Common Multiple |
| PYQ | Previous Year Question |
| HOTS | Higher Order Thinking Skills |
| MCQ | Multiple Choice Question |
| CBSE | Central Board of Secondary Education |
| NCERT | National Council of Educational Research and Training |
Introduction to Real Numbers
English
Real Numbers are all numbers that can be represented on a number line. They include rational numbers and irrational numbers. ( Class 10 Real Numbers )
हिंदी
वास्तविक संख्याएँ (Real Numbers) वे सभी संख्याएँ हैं जिन्हें संख्या रेखा पर प्रदर्शित किया जा सकता है। इसमें परिमेय तथा अपरिमेय दोनों संख्याएँ शामिल होती हैं।
Types of Real Numbers
Real Numbers=Rational Numbers+Irrational Numbers
Rational Numbers (परिमेय संख्याएँ)
वे संख्याएँ जिन्हें p/q के रूप में लिखा जा सके, जहाँ q ≠ 0 हो।
Examples:
- 1/2
- 3/5
- -7/8
- 4
Irrational Numbers (अपरिमेय संख्याएँ)
वे संख्याएँ जिन्हें p/q के रूप में नहीं लिखा जा सकता।
Examples:
- √2
- √3
- √5
- π
Fundamental Theorem of Arithmetic
Statement
Every composite number can be expressed as a product of prime numbers in a unique way.
Example
360 = 2 × 2 × 2 × 3 × 3 × 5
or
360 = 2³ × 3² × 5
हिंदी
प्रत्येक संयुक्त संख्या (Composite Number) को अभाज्य गुणनखंडों (Prime Factors) के गुणनफल के रूप में केवल एक ही तरीके से लिखा जा सकता है।
HCF and LCM ( Class 10 Real Numbers )
Prime Factorization Method
Example
Find HCF and LCM of 72 and 120.
72 = 2³ × 3²
120 = 2³ × 3 × 5
HCF
= 2³ × 3
= 24
LCM
= 2³ × 3² × 5
= 360
Verification
HCF×LCM=Product of Two Numbers
24 × 360 = 8640
72 × 120 = 8640
Verified.
Decimal Expansion of Rational Numbers ( Class 10 Real Numbers )
Important Rule
A rational number p/q will have a terminating decimal expansion if the prime factorization of q contains only 2 and/or 5.
Examples of Terminating Decimals
1/8 = 0.125
3/20 = 0.15
7/125 = 0.056
Examples of Non-Terminating Recurring Decimals
1/3 = 0.3333…
5/11 = 0.454545…
13/7 = 1.857142857…
Irrational Numbers and Their Properties ( Class 10 Real Numbers )
Important Properties
English
- Sum of a rational and irrational number is irrational.
- Product of a non-zero rational number and irrational number is irrational.
- Irrational numbers have non-terminating and non-recurring decimal expansions.
हिंदी
- परिमेय + अपरिमेय = अपरिमेय
- शून्य के अतिरिक्त परिमेय × अपरिमेय = अपरिमेय
- अपरिमेय संख्याओं का दशमलव प्रसार अनंत तथा असांत आवर्ती होता है।
Important Formulae ( Class 10 Real Numbers )
| Formula | Application |
|---|---|
| HCF × LCM = Product of Numbers | Verification |
| Prime Factorization | HCF and LCM |
| p/q | Rational Numbers |
Key Points for Board Exam 2027 ( Class 10 Real Numbers )
| Irrational Number Proofs अत्यंत महत्वपूर्ण हैं। |
| Decimal Expansion से हर वर्ष प्रश्न पूछे जाते हैं। |
| HCF और LCM आधारित प्रश्न बोर्ड परीक्षा में नियमित आते हैं। |
| Prime Factorization अच्छे से तैयार करें। |
| NCERT के सभी उदाहरण अवश्य हल करें। |
| Assertion-Reason और Competency Based Questions का अभ्यास करें। |
NCERT Important Questions with Solutions
Question 1
Find the HCF of 96 and 404.
Solution
96 = 2⁵ × 3
404 = 2² × 101
HCF = 2²
= 4
Answer = 4
Question 2
Find HCF and LCM of 12, 15 and 21.
Solution
12 = 2² × 3
15 = 3 × 5
21 = 3 × 7
HCF = 3
LCM = 2² × 3 × 5 × 7
= 420
Answer:
HCF = 3
LCM = 420
Question 3
Show that √3 is irrational.
Solution
Assume √3 is rational.
√3 = p/q
Squaring both sides:
3q² = p²
Therefore p is divisible by 3.
Let p = 3k
Substituting:
q² = 3k²
Hence q is also divisible by 3.
Both p and q are divisible by 3, which contradicts the condition that they are co-prime.
Therefore,
√3 is irrational.
Question 4
Show that 5√2 is irrational.
Solution
√2 is irrational.
5 is a non-zero rational number.
Therefore,
5√2 is irrational.
MCQ Practice Questions
1. Which of the following is irrational?
A. 3/5
B. 0.25
C. √7
D. -8
✅ Answer: C
2. Which number has terminating decimal expansion?
A. 3/7
B. 5/11
C. 7/125
D. 2/9
✅ Answer: C
3. HCF of 18 and 24 is
A. 4
B. 6
C. 8
D. 12
✅ Answer: B
4. LCM of 12 and 15 is
A. 45
B. 50
C. 60
D. 90
✅ Answer: C
5. Which of the following is rational?
A. π
B. √5
C. √11
D. 0.75
✅ Answer: D
Assertion and Reason Questions
Question 1
Assertion (A): √2 is irrational.
Reason (R): √2 cannot be written in p/q form.
Answer:
✅ Both A and R are true and R is the correct explanation.
Question 2
Assertion (A): 5/16 has terminating decimal expansion.
Reason (R): Denominator contains only factor 2.
Answer:
✅ Both A and R are true.
Previous Year Questions (PYQ)
PYQ 1
Find HCF and LCM of 26 and 91.
Answer:
HCF = 13
LCM = 182
PYQ 2
Show that √5 is irrational.
Answer:
Use contradiction method.
Hence proved irrational.
PYQ 3
Determine whether 77/2100 has terminating decimal expansion.
Solution
77/2100
= 11/300
300 = 2² × 3 × 5²
Contains factor 3.
Therefore,
Non-terminating recurring decimal.
HOTS Questions
HOTS 1
Can two irrational numbers have a rational sum?
Answer
Yes.
√2 + (-√2) = 0
0 is rational.
HOTS 2
Can two irrational numbers have a rational product?
Answer
Yes.
√3 × √3 = 3
3 is rational.
HOTS 3
Find a rational and an irrational number between 2 and 3.
Answer
Rational = 2.5
Irrational = √5
HOTS 4
If HCF = 18 and LCM = 540 and one number is 90, find the other.
Solution
18 × 540 = 90 × x
9720 = 90x
x = 108
Answer = 108
Expected Questions for Board Exam 2027
1 Mark Questions
- Define irrational number.
- Give one example of a rational number.
- Is √13 irrational?
2 Marks Questions
- Find HCF of 84 and 126.
- Determine decimal expansion of 7/200.
3 Marks Questions
- Prove √7 is irrational.
- Find HCF and LCM of 36 and 90.
4 Marks Questions
- Determine whether 343/3125 has terminating decimal expansion.
- Show that 6√5 is irrational.
Competency-Based Question
A school has 540 boys and 450 girls. They are to be arranged in equal groups with maximum students in each group.
Find the maximum number of groups.
(Hint: Use HCF)
Answer = 90
One Shot Revision
Last Minute Revision
- Real Numbers = Rational + Irrational
- √2, √3, √5, √7 are irrational.
- HCF and LCM via prime factorization.
- HCF × LCM = Product.
- Denominator containing only 2 and 5 gives terminating decimal.
- Decimal expansion questions are very important.
- Irrational number proofs are favorite board questions.
- NCERT examples must be practiced.
- PYQ and competency questions are highly important.
- This chapter can easily fetch full marks.
Frequently Asked Questions (FAQ)
Q1. Most Important Topic?
Irrational Numbers, Decimal Expansion and HCF-LCM.
Q2. Is Euclid Division Lemma included for 2027 Board Exam?
No. In the latest reduced syllabus followed by many boards, Euclid Division Lemma has been removed. Always verify with your board’s latest syllabus notification before the exam.
Q3. How many marks can come from Real Numbers?
Generally 4–6 marks questions are asked directly or indirectly from this chapter.
Q4. Best Preparation Strategy?
- Complete NCERT Exercise thoroughly.
- Practice at least 10 irrational proof questions.
- Solve previous 5 years PYQs.
- Focus on decimal expansion and HCF-LCM applications.
Conclusion
Real Numbers is one of the easiest and highest-scoring chapters in Class 10 Mathematics. A strong command over Rational Numbers, Irrational Numbers, Prime Factorization, HCF-LCM and Decimal Expansion can help students secure excellent marks in the 2027 Board Examination. Practice NCERT questions, PYQs and HOTS regularly to strengthen your concepts and boost confidence.
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