Question

prove that√5 is an irrational number ​

Answer 1

Answer:

To prove that √5 is irrational number 

Let us assume that √5 is rational 

Then √5 =  $\frac{a}{b}$

(a and b are co primes, with only 1 common factor and b≠0) 

⇒ √5 =  $\frac{a}{b}$

(cross multiply) 

⇒ a = √5b 

⇒ a² = 5b² -------> α

⇒ 5/a² 

(by theorem if p divides q then p can also divide q²) 

⇒ 5/a ----> 1 

⇒ a = 5c 

(squaring on both sides) 

⇒ a² = 25c² ----> β 

From equations α and β 

⇒ 5b² = 25c²

⇒ b² = 5c² 

⇒ 5/b² 

(again by theorem) 

⇒ 5/b-------> 2 

we know that a and b are co-primes having only 1 common factor but from 1 and 2 we can that it is wrong. 

This contradiction arises because we assumed that √5 is a rational number 

∴ our assumption is wrong 

∴ √5 is irrational number

Answer 2

Answer:

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