Chapter 1: Real Numbers
Euclid's division lemma
For two positive integers a and b, a = bq + r, where 0 ≤ r < b. This gives a method to find the HCF of two numbers (Euclid's division algorithm).
The Fundamental Theorem of Arithmetic
Every composite number can be written as a product of primes in exactly one way, apart from the order. HCF = product of the smallest powers of the common primes. LCM = product of the greatest powers of all the primes. For two numbers, HCF × LCM = first number × second number.
Irrational numbers
A number that cannot be written as p/q (q ≠ 0) is irrational, like √2, √3 and √5. The sum of a rational and an irrational number, and the product of a non-zero rational and an irrational number, are both irrational.
Decimal expansions
A rational number p/q (in simplest form) has a terminating decimal expansion if q = 2m×5n. Otherwise it is non-terminating and repeating. An irrational number has a non-terminating, non-repeating expansion.
Important questions
Q1. State Euclid's division lemma.
For any two positive integers a and b, there exist integers q and r such that a = bq + r, where 0 ≤ r < b.
Q2. Find the HCF of 225 and 135 using Euclid's algorithm.
225 = 135 × 1 + 90; 135 = 90 × 1 + 45; 90 = 45 × 2 + 0. So the HCF = 45.
Q3. What does the Fundamental Theorem of Arithmetic say?
Every composite number can be written as a product of primes in exactly one way, apart from the order of the factors.
Q4. Express 140 as a product of its prime factors.
140 = 2 × 2 × 5 × 7 = 2² × 5 × 7.
Q5. Find the HCF and LCM of 12 and 18 by the prime factorisation method.
12 = 2² × 3 and 18 = 2 × 3². HCF = 2 × 3 = 6. LCM = 2² × 3² = 36.
Q6. Find the HCF and LCM of 26 and 91 and verify that HCF × LCM = product of the two numbers.
26 = 2 × 13 and 91 = 7 × 13. HCF = 13, LCM = 2 × 7 × 13 = 182. Check: 13 × 182 = 2366 = 26 × 91.
Q7. The HCF of two numbers is 9 and their LCM is 90. If one number is 18, find the other.
HCF × LCM = 9 × 90 = 810. The other number = 810 ÷ 18 = 45.
Q8. Explain why 7 × 11 × 13 + 13 is a composite number.
It equals 13 × (7 × 11 + 1) = 13 × 78. It has factors other than 1 and itself, so it is composite.
Q9. Explain why 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 is a composite number.
It equals 5 × (7 × 6 × 4 × 3 × 2 × 1 + 1) = 5 × 1009. It has the factor 5, so it is composite.
Q10. Prove that √2 is irrational.
Suppose √2 = p/q is rational, with p and q having no common factor. Then p² = 2q², so p is even. Put p = 2k, then q² = 2k², so q is even too. This is a contradiction, so √2 is irrational.
Q11. Prove that 3 + 2√5 is irrational.
Suppose 3 + 2√5 = r is rational. Then √5 = (r − 3)/2 would be rational, which is false since √5 is irrational. So 3 + 2√5 is irrational.
Q12. What is the difference between rational and irrational numbers?
A rational number can be written as p/q (q ≠ 0), an irrational number cannot. For example, 3/4 is rational and √3 is irrational.
Q13. Without long division, say whether 13/3125 has a terminating decimal expansion.
3125 = 5⁵, so the denominator is of the form 2^m × 5^n. So the expansion terminates.
Q14. Does 17/8 have a terminating decimal expansion?
Yes, because 8 = 2³, which is of the form 2^m × 5^n.
Q15. What kind of decimal expansion does 64/455 have?
455 = 5 × 7 × 13 has factors 7 and 13, which are not 2 or 5. So the expansion is non-terminating and repeating.
Q16. Can 6ⁿ end with the digit 0 for any natural number n?
No. 6ⁿ = 2ⁿ × 3ⁿ has no factor 5, but ending in 0 needs both 2 and 5 as factors.
Q17. Write Euclid's division lemma for a = 17 and b = 5.
17 = 5 × 3 + 2. Here q = 3 and r = 2.
Q18. What is the decimal expansion of an irrational number like?
Non-terminating and non-repeating.
Q19. Which is the smallest prime number and the smallest composite number?
The smallest prime number is 2 and the smallest composite number is 4.
Q20. What is the HCF of two co-prime numbers? Give an example.
1. For example, 8 and 15 have no common prime factor, so HCF(8, 15) = 1.
Quiz
1. What is the HCF of 96 and 404?
96 = 2⁵ × 3 and 404 = 2² × 101, so the HCF = 4.
2. What kind of decimal expansion does 7/(2³ × 5) have?
The denominator is of the form 2^m × 5^n, so it terminates.
3. What kind of number is √3?
√3 cannot be written in the form p/q.
4. The HCF of two numbers is 9 and the LCM is 90. If one number is 18, what is the other?
HCF × LCM = 810 and 810 ÷ 18 = 45.
5. What is 2 + √5?
The sum of a rational and an irrational number is irrational.