Question

prove that √5 is an irrational number

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Answer 1

Question:

prove that √5 is an irrational number.

To prove:

• Prove that √5 is irrational number.

Solution:

Let us assume that √5 is rational 

Then √5 =  

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

→ √5 =  

(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