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Prove that is an irrational number.

Asked by Topperlearning User 12th December 2013, 1:47 AM
Answered by Expert
Answer:

Let us assume that   is a rational;

,                                                                                    

So,

squaring both sides, we get

                                                                                 

Therefore  is divisible by 5 and also a is divisible by 5.   

 

Now we can write   for some integer c.             

Substituting this value of for a, in (1) we get


This means that  is divisible by 5 and also b is divisible by 5.  Therefore a and b have 5 as their common factor, but this contradicts the fact that a and b are co prime.  The contradiction arises because of our wrong assumption that  is a rational.                

 So, we conclude that  is an irrational.

 

Answered by Expert 12th December 2013, 3:47 AM
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