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The most beautiful solution ever!

First of all I want to say that this is not the namesake competition . We are not " you against me " but " infinity against infinity". This post is for all those who have either created or have perceived " the most beautiful solution ever " .I created this for the dreamers who may want to wander through the realms of mathematics in one alley having many shops . Please provide the link to that particular solution(s).

Note by Raven Herd
1 year, 1 month ago

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Claim: An irrational raised to the power of an irrational can be a rational.

Proof: Either $$\sqrt{2}^\sqrt{2}$$ is irrational or rational.

If it is rational, we are done.

Otherwise, $${\sqrt{2}^\sqrt{2}}^\sqrt{2} =\sqrt{2}^2 = 2$$ is rational. · 1 year, 1 month ago

Nice example, but I think that the last equation should be written as

$$\large (\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = \sqrt{2}^{(\sqrt{2}\times \sqrt{2})} = \sqrt{2}^{2} = 2,$$

since $$\large \sqrt{2}^{\sqrt{2}^{\sqrt{2}}} = 1.7608.....$$ is irrational. · 1 year, 1 month ago

Nice : ) · 1 year, 1 month ago