Prove that 2

Waylon Mcbride

Waylon Mcbride

Answered question

2022-05-12

Prove that 2 3 is irrational or even it is transcendental?

Answer & Explanation

Skyler Barber

Skyler Barber

Beginner2022-05-13Added 17 answers

Gelfond-Schneider theorem: if a , b are algebraic, a 0 , 1, b irrational, then all values of a b are transcendental.
So: 2 3 is transcendental.
To show
(1) 5 ( 2 + 1 1 + 2 4 )
is transcendental, it suffices to show
(2) 2 + 1 1 + 2 4
is irrational. I leave that to you, using the rational root theorem.
Note: Numerically evaluating ( 2 ) we get 1.8709999 so we may mistakenly think it is rational. In fact with more digits it is 1.8709999455101501528.

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