In our previous post, we discussed how to prove that the square root of 2 is irrational, using a proof by contradiction.
There is a less well-known proof that is a direct constructive approach to proving that the square root of 2 is irrational!
We consider an arbitrary rational number , and show that the difference between and cannot be zero. Hence, the square root of 2 cannot be rational.
Firstly, we have:
(Rationalizing the numerator)
Now, we analyse the numerator. We can write ,
, where are odd.
Since the largest power of two dividing is an odd power, whilst for the largest power of two dividing it is an even power, and cannot be the same number. Hence we have .
Now, we analyse the denominator. Firstly, we can consider just the rationals . Because if , it is clear that is not going to be .
Rearranging, we have: .
Multiplying throughout by , .
Going back to the original equation (boxed), we can conclude that:
We have shown constructively that is not a rational number!
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