is ?6 a irrational number

Asked by himanish gothwal | 16th Apr, 2013, 10:29: PM

Expert Answer:

Yes, we prove the statement by contradiction i.e. Assume that sqrt(6) is a rational number. 
Then there exist positive integers p and q such that the hcf(p,q) = 1 and p/q = sqrt(6). 
Square both sides: p^2 / q^2 = 6, 
p^2 = 6q^2. 
Now as 2 is a divisor of the right-hand side (RHS), it implies that 2 is also a divisor of the left-hand side (LHS). 
This is only possible if 2 is a factor of p. 
Let p =2k. Then k is a positive integer as well. 
Thus, 4k^2 = 6q^2, 
2k^2 = 3q^2. 
As 2 is a factor of the LHS, 2 is also a factor of the RHS. 
But this is only possible if 2 is a factor of q. 
=> hcf(p,q) >= 2. Contradiction! 

Thus sqrt(6) is irrational. 

Answered by  | 17th Apr, 2013, 06:02: AM

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