Why is the square root of 5 an irrational number?
1 Answer
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Explanation:
Here's a sketch of a proof by contradiction:
Suppose
Without loss of generality, we may suppose that
Then by definition:
#5 = (p/q)^2 = p^2/q^2#
Multiply both ends by
#5 q^2 = p^2#
So
Then since
So
So we have:
#5 q^2 = p^2 = (5m)^2 = 5*5*m^2#
Divide both ends by
#q^2 = 5 m^2#
Divide both ends by
#5 = q^2/m^2 = (q/m)^2#
So
Now
So our hypothesis that