Real Numbers

In Class 9 you began exploring real numbers and met irrational numbers such as sqrt(2) and pi. In this chapter you will study the Fundamental Theorem of Arithmetic, which states that every composite number can be expressed (factorised) as a product of primes, and that this factorisation is unique apart from the order of the prime factors. For example, 3825 = 3^2 x 5^2 x 17 and no other set of primes multiplies to give 3825. You will use prime factorisation to find the HCF and LCM of two or three numbers and verify that for any two positive integers a and b, HCF(a, b) x LCM(a, b) = a x b. These ideas help solve real problems such as arranging items in equal stacks or finding when two traffic lights change together. Using the theorem that if a prime p divides a^2 then p divides a, you will prove by contradiction that sqrt(2), sqrt(3) and sqrt(5) are irrational, and that numbers such as 3 + 2sqrt(5) are irrational too.

Detailed Worksheet: Real Numbers Section A - Definitions (10 marks) 1. State the Fundamental Theorem of Arithmetic. (2 marks) 2. Define a rational number and an irrational number, giving two examples of each. (2 marks) 3. Define the HCF and the LCM of two positive integers. How are they found from prime factorisations? (2 marks) 4. State the relation between the HCF and LCM of two positive integers a and b. Does this relation hold for three numbers? (2 marks) 5. State the theorem used to prove the irrationality of sqrt(p) for a prime p. What is meant by a proof by contradiction? (2 marks) Section B - Calculations and Applications (15 marks) 6. Express 5005 and 7429 as a product of their prime factors. (3 marks) 7. Find the LCM and HCF of 510 and 92 and verify that LCM x HCF = product of the two numbers. (3 marks) 8. Find the LCM and HCF of 12, 15 and 21 by the prime factorisation method. (3 marks) 9. Given that HCF(306, 657) = 9, find LCM(306, 657). (3 marks) 10. Explain why 7 x 11 x 13 + 13 and 7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 are composite numbers. (3 marks) Section C - Diagrams (10 marks) 11. Draw a factor tree for 3825 and write its prime factorisation in exponential form. (4 marks) 12. Draw factor trees for 336 and 54. Use them to find the HCF and LCM of the two numbers. (3 marks) 13. Draw a Venn diagram showing the prime factors of 96 and 404, placing the common prime factors in the overlapping region. Use it to find the HCF and LCM. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Check whether 6^n can end with the digit 0 for any natural number n. Explain your answer using the Fundamental Theorem of Arithmetic, and state for which kind of numbers a^n can end in 0. (5 marks) 15. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point? Explain why LCM and not HCF is used here. (5 marks) 16. Prove that 3 + 2sqrt(5) is irrational, given that sqrt(5) is irrational. A student argues that the sum of two irrational numbers is always irrational. Give a counter-example. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show all steps of factorisation. In proofs, state clearly what is assumed and where the contradiction arises.