>>11494966>>11494970>the sum of pure irrationals (irrationals that are not the sum of a rational and an irrational) ... is always irrationalTo begin, there is no such thing as a "pure irrational number" in the first place: any irrational number is the sum of (rational) and (irrational). Your lemma is unfortunately dead on arrival.
However, you would be interested in transcendental numbers.
https://en.wikipedia.org/wiki/Transcendental_numberA complex number is an algebraic number if it is a root of some nonzero polynomial with rational coefficients; else, it is a transcendental number. Since any rational number is algebraic, even though some irrationals are also algebraic, ALL TRANSCENDENTALS ARE IRRATIONALS.
So it's just a walk in the park to prove that is irrational, right? Just prove it's transcendental.
Good luck; it is /exceptionally difficult/ to prove that a number is transcendental. Very few examples are known to us humans.
https://en.wikipedia.org/wiki/Baker%27s_theoremThis one theorem is essentially the current limit of our ability to prove transcendence, generalizing the two biggest results in the field, the Lindemann–Weierstrass theorem and the Gelfond–Schneider theorem.
We have that is transcendental, that is transcendental, and even that at least one of is transcendental, but not that is transcendental.
https://en.wikipedia.org/wiki/Schanuel%27s_conjectureThis hypothesis is quite similar to your lemma: if true, is transcendental (and therefore irrational) as are many other numbers like it. But the conjecture was posed almost 60 years ago and no progress toward a proof has been made since, so good luck!
No, seriously. I'm too much of a freshman brainlet to even attempt a solution (haven't even taken Linear Algebra yet), but I want to see this solved in my lifetime. Please.
