Complete notes with examples and important points.
Euclid's Division Lemma states that for any two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that a = bq + r, where 0 ≤ r < b. This lemma is used to find the HCF of two positive integers.
Step 1: Since 225 > 135, we apply the division lemma to 225 and 135 to get 225 = 135 × 1 + 90.
Step 2: Since the remainder 90 ≠ 0, we apply the division lemma to 135 and 90 to get 135 = 90 × 1 + 45.
Step 3: We consider the new divisor 90 and new remainder 45, and apply the division lemma to get 90 = 2 × 45 + 0.
Since the remainder is zero, the process stops. The divisor at this stage is 45. Therefore, the HCF of 135 and 225 is 45.
Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
140 = 2 × 70 = 2 × 2 × 35 = 2 × 2 × 5 × 7 = 2² × 5 × 7.
An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q, where q ≠ 0. In this section, we prove the irrationality of numbers like √2, √3, √5.
Proof: Let us assume, to the contrary, that √2 is rational. So, we can find integers r and s (≠ 0) such that √2 = r/s. Suppose r and s have a common factor other than 1. Then, we divide by the common factor to get √2 = a/b, where a and b are coprime. So, b√2 = a. Squaring on both sides, we get 2b² = a². Therefore, 2 divides a². This implies that 2 divides a. So, we can write a = 2c for some integer c. Substituting for a, we get 2b² = (2c)² = 4c², which means b² = 2c². This means that 2 divides b², and so 2 divides b. Therefore, a and b have at least 2 as a common factor. But this contradicts the fact that a and b have no common factors other than 1. This contradiction has arisen because of our incorrect assumption that √2 is rational. So, we conclude that √2 is irrational.
Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p/q, where p and q are coprime, and the prime factorisation of q is of the form 2ⁿ5ᵐ, where n, m are non-negative integers.