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Chapter Analysis
Intermediate9 pages • EnglishQuick Summary
The chapter on Real Numbers begins with exploring the properties of positive integers, including Euclid's division algorithm and the Fundamental Theorem of Arithmetic. Euclid's division algorithm helps in computing the HCF of two integers, while the Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of primes. The chapter also discusses the proof of irrational numbers using these concepts and how they affect the decimal expansion of rational numbers.
Key Topics
- •Euclid's Division Algorithm
- •Fundamental Theorem of Arithmetic
- •Prime Factorization
- •Irrational Numbers
- •Proof by Contradiction
- •Terminating and Non-Terminating Decimals
- •HCF and LCM
- •Properties of Real Numbers
Learning Objectives
- ✓Understand and apply Euclid's Division Algorithm for finding HCF.
- ✓Use the Fundamental Theorem of Arithmetic to decompose numbers into prime factors.
- ✓Prove the irrationality of certain numbers using mathematical concepts.
- ✓Analyze conditions for decimal expansions to be terminating.
- ✓Compute the LCM and HCF using prime factorization.
- ✓Develop skills in mathematical reasoning and proof techniques.
Questions in Chapter
Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
Page 6
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54
Page 6
Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25
Page 6
Given that HCF (306, 657) = 9, find LCM (306, 657).
Page 6
Check whether 6^n can end with the digit 0 for any natural number n.
Page 6
Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
Page 6
Sonia takes 18 minutes to drive one round of a field, while Ravi takes 12 minutes for the same. After how many minutes will they meet again at the starting point?
Page 6
Prove that √5 is irrational.
Page 9
Prove that 3√2 + 5 is irrational.
Page 9
Prove that the following are irrationals: (i) 1/√2 (ii) √7 - √5 (iii) √6 + √2
Page 9
Additional Practice Questions
What is the Fundamental Theorem of Arithmetic and its significance?
mediumAnswer: The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes uniquely, apart from the order of the factors. It is significant as it lays the foundation for various mathematical proofs and applications involving integers.
Explain with an example how Euclid's Division Algorithm works.
easyAnswer: Euclid's Division Algorithm states that for any two integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. For example, 15 divided by 6 gives quotient 2 and remainder 3, hence 15 = 6 × 2 + 3.
Why is the number π considered irrational?
mediumAnswer: The number π is considered irrational because it cannot be expressed as a fraction of two integers. Its decimal expansion is non-terminating and non-repeating, which is a characteristic of irrational numbers.
Discuss the method to determine if a decimal is terminating or non-terminating using prime factorization.
mediumAnswer: A fractional decimal p/q is terminating only if the prime factorization of q has no prime factors other than 2 or 5. Otherwise, the decimal is non-terminating.
Prove that √2 is irrational using the Fundamental Theorem of Arithmetic.
hardAnswer: Assume √2 is rational, meaning √2 = a/b for coprime integers a and b. This implies 2b² = a², meaning a² is even, hence a is even, say a = 2c. Then b² = 2c² implies b is even, contradicting a and b being coprime.