Irrationality and Transcendence
Ω can be proven irrational from the fact that e is transcendental; if Ω were rational, then there would exist integers p and q such that
so that
and e would therefore be algebraic of degree p. However e is transcendental, so Ω must be irrational.
Ω is in fact transcendental as the direct consequence of Lindemann–Weierstrass theorem. If Ω were algebraic, exp(Ω) would be transcendental and so would be exp−1(Ω). But this contradicts the assumption that it was algebraic.
Read more about this topic: Omega Constant
Famous quotes containing the words irrationality and:
“Doubtless, we are as slow to conceive of Paradise as of Heaven, of a perfect natural as of a perfect spiritual world. We see how past ages have loitered and erred. “Is perhaps our generation free from irrationality and error? Have we perhaps reached now the summit of human wisdom, and need no more to look out for mental or physical improvement?” Undoubtedly, we are never so visionary as to be prepared for what the next hour may bring forth.”
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