Namely why?
Because it has such beautiful one-tweet Proofs? 
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Are there non-trivial examples of transcendental fractions like this that end up being rational?
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Pi/squareroot(x) being x natural has the same proof no?
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Since the algebraic numbers form a field, they are closed under multiplication and thus π being transcendental implies π/√2 is transcendental and hence irrational.
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My academic 3rd great grandfather C. L. Ferdinand Lindemann proved that Pi was transcendental putting an end to the ancient dream of squaring the circle.
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You've left all the hard work for later -- "π is transcendental" isn't just a triviality.
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Transcendental times algebraic is always transcendental, too.
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Almost. It is true except for a single case when the algebraic number is 0.

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@mondelimon al fin veo otra prueba aparte del clásico 2p/2m
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No sé si es muy obvia pero como que no me convence.
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