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Questions and Answers
What is the assumption made in the proof that 2 is irrational?
What is the assumption made in the proof that 2 is irrational?
What does the equation 2b2 = 4c2 imply?
What does the equation 2b2 = 4c2 imply?
Why is the assumption that 3 is rational led to a contradiction?
Why is the assumption that 3 is rational led to a contradiction?
What is the result of squaring both sides of the equation b3 = a)
What is the result of squaring both sides of the equation b3 = a)
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What can be said about the sum or difference of a rational and an irrational number?
What can be said about the sum or difference of a rational and an irrational number?
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What is the purpose of Theorem 1.3 in the proofs of irrationality?
What is the purpose of Theorem 1.3 in the proofs of irrationality?
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What is the assumption made in the proof that 5 – 3 is irrational?
What is the assumption made in the proof that 5 – 3 is irrational?
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What can be said about the product of a non-zero rational and irrational number?
What can be said about the product of a non-zero rational and irrational number?
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Why is the proof of irrationality of 2 and 3 similar?
Why is the proof of irrationality of 2 and 3 similar?
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What is the role of Theorem 1.3 in the proof of irrationality of 3?
What is the role of Theorem 1.3 in the proof of irrationality of 3?
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Study Notes
Fundamental Theorem of Arithmetic
- Any natural number can be written as a product of its prime factors.
- Any natural number can be obtained by multiplying prime numbers, allowing them to repeat as many times as we wish.
- There are infinitely many primes, and if we combine all these primes in all possible ways, we will get an infinite collection of numbers, which include all the primes and all possible products of primes.
LCM and HCF of Positive Integers
- HCF (a, b) × LCM (a, b) = a × b for any two positive integers a and b.
- HCF (a, b) = Product of the smallest power of each common prime factor in the numbers.
- LCM (a, b) = Product of the greatest power of each prime factor involved in the numbers.
Examples and Exercises
- Example 2: Find the LCM and HCF of 6 and 20 by the prime factorisation method.
- HCF (6, 20) = 2 and LCM (6, 20) = 2 × 2 × 3 × 5 = 60.
- Example 3: Find the HCF of 96 and 404 by the prime factorisation method.
- HCF of these two integers is 22 = 4, and LCM (96, 404) = 9696.
- Example 4: Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
- HCF (6, 72, 120) = 21 × 31 = 2 × 3 = 6, and LCM (6, 72, 120) = 23 × 32 × 51 = 360.
Irrational Numbers
- A number 's' is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.
- Examples of irrational numbers include 2, 3, π, - 0.10110111011110..., etc.
- Theorem 1.2: Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.
- Theorem 1.3: 2 is irrational.
Proof of Irrationality
- The proof of irrationality is based on the Fundamental Theorem of Arithmetic and the technique of 'proof by contradiction'.
- The proof involves assuming that the number is rational and then showing that it leads to a contradiction.
Examples of Irrational Numbers
- Example 5: Prove that 3 is irrational.
- Similar proof to that of 2, using Theorem 1.3 with p = 3.
- Example 6: Show that 5 – 3 is irrational.
- Assume, to the contrary, that 5 – 3 is rational, and then show that it leads to a contradiction.
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Description
Explore the concept of prime factorization and its relation to decimal expansions. Learn how to apply the Fundamental Theorem of Arithmetic to understand the nature of decimal expansions.
