Question 708870: prove that root of three is irrational
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Assume 
Square both sides: 
Then we can say: 
If  is odd, then  is odd because the product of any two odd numbers is odd. Similarly,  must then also be odd. Consequently,  is odd, and since by definition an odd number has no even factor,  must be odd.
If is even, then , , and must be even. But if and are both even, then  is not irreducible. Consequently, and must both be odd.
If is the  th odd number,  . Likewise, can be represented as 
Go back to  and substitute:
^2\ =\ (2m\ +\ 1)^2)
A little algebra (verification left as an exercise for the student) gives us:
)
Since the LHS is clearly odd and the RHS is clearly even, this equation has no solutions over the integers. Therefore, reductio ad absurdum, the square root of 3 cannot be expressed as a rational number, hence  is irrational.
John
Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
|
|
|
