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Question 333527: i have understood the proof " square root of 2 is irrational" by contradiction method" but i have a quesiton in my mind:
In that proof, we supposed sqrt(2) was rational with p/q in its simplest form and when it was proved that our supposition is wrong, it was automatically proved that square root 2 is irrational.... i tried the same with square root of 4 and supposed that it WAS a rational with p/q in its simplest form.. this is what happened:
p/q = sqrt(4)
p² / q² = 4 --> equation 1
p² = 4q²
Now:
p² / q² = 4q² / q² (since p² = 4q²)
or, p / q = 4q / q
but here, p/q is not in tis simplest form, so our supposition is wrong and square root of 4 is irrationa;... lol ... whats wrong with my solution???????
Answer by Jk22(389)   (Show Source):
You can put this solution on YOUR website! Now:
p² / q² = 4q² / q² (since p² = 4q²)
or, p / q = 4q / q
the last line should be p/q=2 (sqrt(p^2/q^2)=p/q=sqrt(4q^2/q^2)=2q/q=2) this we knew from the beginning
maybe like p^2/q^2=4 => p^2=4q^2 : p^2 is even, hence p is even, so we can write p in the form p=2m.
so : 4m^2=4q^2, but this leads no information about q (q can be odd)
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reminder :
(in the case of sqrt(2), we get p^2=2q^2, the same reasoning : p^2 even, so p even, so p=2n, hence 2q^2=4n^2, so in this case we can deduce q is even too, which contradicts the simplification rule)
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