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The square root of the Gelfond–Schneider constant is the transcendental number = 1.632 526 919 438 152 844 77.... This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence.
A more general proof shows that the mth root of an integer N is irrational, unless N is the mth power of an integer n. [7] That is, it is impossible to express the mth root of an integer N as the ratio a ⁄ b of two integers a and b, that share no common prime factor, except in cases in which b = 1.
For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x 2 − 2 = 0. The golden ratio (denoted or ) is another irrational number that is not transcendental, as it is a root of the polynomial equation x 2 − x − 1 = 0.
In mathematics, the exponential of pi e π, [1] also called Gelfond's constant, [2] is the real number e raised to the power π. Its decimal expansion is given by: e π = 23.140 692 632 779 269 005 72... (sequence A039661 in the OEIS) Like both e and π, this constant is both irrational and transcendental.
In the twentieth century work by Axel Thue, [6] Carl Siegel, [7] and Klaus Roth [8] reduced the exponent in Liouville's work from d + ε to d/2 + 1 + ε, and finally, in 1955, to 2 + ε. This result, known as the Thue–Siegel–Roth theorem , is ostensibly the best possible, since if the exponent 2 + ε is replaced by just 2 then the result is ...
Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...
Enjoy a classic game of Hearts and watch out for the Queen of Spades!
The following 1953 proof by Dov Jarden has been widely used as an example of a non-constructive proof since at least 1970: [4] [5] CURIOSA 339. A Simple Proof That a Power of an Irrational Number to an Irrational Exponent May Be Rational. is either rational or irrational. If it is rational, our statement is proved.