Here are the top ten transcendental numbers, as put together by Dave Richeson.
1.
(Liouville, 1851): the first known transcendental number not expressed as a continued fraction.
2.
(Hermite, 1873): the first non-contrived example of a transcendental number.
3.
(Lindeman, 1882): use the Lindemann-Weierstrass theorem (below) and Euler’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
.
4.
use the Lindemann-Weierstrass theorem and the fact that
5.
use the Lindemann-Weierstrass theorem and the fact that is the inverse function for
.
Hilbert’s 7th problem (1900). If
and
are algebraic numbers with
and
not rational, then
is transcendental.
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