Chapter 1: Real Numbers (Introduction & Important Content)
1. Introduction
Real Numbers are the combination of Rational and Irrational numbers. In this chapter, we focus on the fundamental properties of integers: Euclid’s Division Lemma and the Fundamental Theorem of Arithmetic. These concepts help us understand HCF, LCM, and the irrationality of numbers like √2, √3, and √5.
2. Important Concepts
- Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes.
- HCF and LCM Relation: For any two positive integers $a$ and $b$, $HCF(a, b) \times LCM(a, b) = a \times b$.
- Irrationality: A number is irrational if it cannot be written in the form $p/q$.
- Rational Numbers: Their decimal expansion is either terminating or non-terminating repeating.
Important Questions & Answers
(1 Mark) Question: Find the HCF of 96 and 404 using prime factorization.
Answer: $96 = 2^5 \times 3$; $404 = 2^2 \times 101$. Common factor is $2^2$. So, HCF = 4.
Answer: $96 = 2^5 \times 3$; $404 = 2^2 \times 101$. Common factor is $2^2$. So, HCF = 4.
(2 Marks) Question: Check whether $6^n$ can end with the digit 0 for any natural number $n$.
Answer: For a number to end with 0, its prime factors must include both 2 and 5. Factors of $6^n = (2 \times 3)^n$. Since 5 is not a factor, $6^n$ cannot end with 0.
Answer: For a number to end with 0, its prime factors must include both 2 and 5. Factors of $6^n = (2 \times 3)^n$. Since 5 is not a factor, $6^n$ cannot end with 0.
(3 Marks) Question: Prove that $\sqrt{3}$ is an irrational number.
Answer: (Brief hint) Assume $\sqrt{3} = p/q$. Square both sides: $3 = p^2/q^2 \Rightarrow p^2 = 3q^2$. This means 3 divides $p$. Let $p = 3k$. Substituting leads to 3 dividing $q$. Since both have a common factor 3, our assumption is wrong. Hence, $\sqrt{3}$ is irrational.
Answer: (Brief hint) Assume $\sqrt{3} = p/q$. Square both sides: $3 = p^2/q^2 \Rightarrow p^2 = 3q^2$. This means 3 divides $p$. Let $p = 3k$. Substituting leads to 3 dividing $q$. Since both have a common factor 3, our assumption is wrong. Hence, $\sqrt{3}$ is irrational.
(4 Marks) Question: Find the HCF and LCM of 12, 15, and 21 by prime factorization and verify the relationship (if applicable).
Answer:
$12 = 2^2 \times 3$, $15 = 3 \times 5$, $21 = 3 \times 7$.
HCF = 3.
LCM = $2^2 \times 3 \times 5 \times 7 = 420$.
Note: The product formula only works for two numbers.
Answer:
$12 = 2^2 \times 3$, $15 = 3 \times 5$, $21 = 3 \times 7$.
HCF = 3.
LCM = $2^2 \times 3 \times 5 \times 7 = 420$.
Note: The product formula only works for two numbers.
(5 Marks) Question: An army contingent of 616 members is to march behind an army band of 32 members in a parade. What is the maximum number of columns in which they can march?
Answer: To find the maximum number of columns, we find the HCF(616, 32).
Using Euclid’s Algorithm:
$616 = 32 \times 19 + 8$
$32 = 8 \times 4 + 0$
Since the remainder is 0, the HCF is 8. Maximum columns = 8.
Answer: To find the maximum number of columns, we find the HCF(616, 32).
Using Euclid’s Algorithm:
$616 = 32 \times 19 + 8$
$32 = 8 \times 4 + 0$
Since the remainder is 0, the HCF is 8. Maximum columns = 8.
