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Question 1. Asked on :29 June 2021:11:08:46 AM

Prove that √5 is an irrational number.

-Added by Master Purushottam Mathematics » Real Numbers

Answer:

Master Purushottam

Let us assume that √5 is a rational number. 

 we can write it as √5 = pq 

Now we can reduce p and q and we get reduced value of p and q as a and b which are co-prime have no common factor other than 1. 

So we have √5 = ab 

⇒ √5b = a  

⇒ 5b2 = a2   [squaring both sides]  ............ (i) 

⇒ b2 = a25 

5 divides a2 so a is also divisible by 5   (theorem 1.3) ..... (ii) 

so 5 is a factor of a.

Now taking a = 5c for some integer c 

⇒ 5b2 = (5c)2  

⇒ 5b2 = 25 c2 

⇒ b2 = 5 c2 

⇒ c2 = b25 

5 divides b2 so b is also divisible by 5 (theorem 1.3)  ...... (iii) 

from (ii) and (iii) we get that 

a and b are divisible by 5 this implies that 5 is common factor of a and b, while we assume that a and b are co-prime having no common factor other than 1. 

This contradiction has risen due to our wrong assumption so √5 cann't be 

expressed as pq, Hence √5 is an irrational number. 

-Answered by Master Purushottam On 29 June 2021:11:10:35 AM(21404Average Rating Based on rating)

 

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