SQRT(2) … the second aproach

Without the geometry, the most simple way is this. Let b^2 be the square of the smallest possible natural b, such that it is twice as big as another square of a natural, a^2!

2*a^2=b^2 (1)

2*a^2=4*(b/2)^2 (2)

a^2=2*(b/2)^2 (3)

Now, if the b/2 is a natural, even a smaller pair of naturals satisfies the initial condition. And b must indeed be even, since b^2 is even.

(1) is a direct consequence, or another form of:  a/b=SQRT(2).


This one has been known since long ago, nothing new here in this post. Just a small clarification of the previous, much more ambitious post.

Done with that.


One thought on “SQRT(2) … the second aproach

  1. Oscar Cunningham says:

    Actually I think using mod 3 might be even simpler. Squares are 0 or 1 mod 3, so b^2 is 0 or 1 and 2a^2 is 0 or 2. Hence a and b divide by 3, contradiction.

    This is faster than the mod 2 proof above because you don’t have to rewrite the equation after you discover b is even.

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