A complex number can be tested to see if it is transcendental using the Wolfram Language command Not[Element[x,
Algebraics]].
Transcendental numbers are important in the history of mathematics because their investigation provided the first proof that circle
squaring, one of the geometric problems
of antiquity that had baffled mathematicians for more than 2000 years was, in
fact, insoluble. Specifically, in order for a number to be produced by a geometric
construction using the ancient Greek rules, it must be either rational
or a very special kind of algebraic number known
as a Euclidean number. Because the number is transcendental, the construction
cannot be done according to the Greek rules.
Liouville showed how to construct special cases (such as Liouville's constant) using Liouville's approximation
theorem. In particular, he showed that any number that has a rapidly converging
sequence of rational approximations must be transcendental. For many years, it was
only known how to determine if special classes of numbers were transcendental. The
determination of the status of more general numbers was considered an important enough
unsolved problem that it was one of Hilbert's problems.
Great progress was subsequently made by Gelfond's theorem, which gives a general rule for determining if special cases of numbers
of the form are transcendental. Baker produced a further revolution
by proving the transcendence of sums of numbers of the
form
for algebraic numbers and .
The number e was proven to be transcendental by Hermite in 1873, and pi () by Lindemann in 1882. Gelfond's
constant
is transcendental by Gelfond's theorem since
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