Here are the top ten transcendental numbers, as put together by Dave Richeson.
(Liouville, 1851): the first known transcendental number not expressed as a continued fraction.
(Hermite, 1873): the first non-contrived example of a transcendental number.
(Lindeman, 1882): use the Lindemann-Weierstrass theorem (below) and Eulerâ€™s identity,
This showed that it is impossible to square the circle.
Lindemann-Weierstrass Theorem (1882/1885). If are distinct algebraic numbers and are nonzero algebraic numbers, then .
use the Lindemann-Weierstrass theorem and the fact that
use the Lindemann-Weierstrass theorem and the fact that is the inverse function for .
Hilbertâ€™s 7th problem (1900). If and are algebraic numbers with and not rational, then is transcendental.
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