REAL NUMBERS

Euclid’s Division Lemma And Algorithm

Here , Example

Euclid’s division Algorithm Euclid’s division Algorithm –

1. Apply Euclid’s division lemma to and . So , we find whole numbers , such that .

2. If , d is the HCF of and . If apply the division lemma to and . 3. Continue the process till the remainder is zero . The divisor at this stage

will be the required HCF .

Example :- Using Euclid’s division algorithm find the HCF of 12576 and 4052 .Since 12576 4052 we apply the division lemma to 12576 and 4052 to get Since the remainder 420 , we apply the division lemma to 4052 and 420 to get We consider the new divisor 420 and new remainder 272 apply the division lemma to get

Now we continue this process till remainder is zero . The remainder has now become 0 , so our procedure stops . Since the divisor at this stage is 4 , the HCF of 12576 and 4052 is 4 .

Fundamental Theorem of Arithmetic

Now factorise a large number say 32760 2 32760 2 16380 2 8190

3 4095 3 1365 5 455 7 91 13 13

Revisiting Irrational Numbers

Theorem - proofLet us assume on contrary that is rational where a and b are co-prime . () squaring on both sides

Here 2 divides , so it also divides . so we can write a=2c for some integer c .

Substituting for we get

Here 2 divides , so it also divides .

This creates a contradiction that a and b have no common factors other than 1 . This contradiction has arisen because of our wrong assumption .

So we conclude that is a irrational number .

Revisiting Rational numbers and their decimal expansions

Theorem

Example

Thank You Made by :- Amit Choube Class :- 10th B