## transcendental numberA number that can't be expressed as the root of a polynomial equation with integer coefficients. Transcendental numbers are one of the two types of irrational number, the other being algebraic numbers. Their existence was proved in 1844 by the French mathematician Joseph Liouville (1809–1882). Georg Cantor showed (1874) that there are more transcendental than algebraic numbers; the set of all algebraic numbers is enumerable (like the integers); the set of transcendental numbers is not. Although transcendentals make up the vast majority of real numbers, it is often surprisingly hard, and may even be impossible, to tell whether a certain number is transcendental or algebraic. For example it is known that both π and e
are transcendental and also that at least one of π + e and π
× e must be transcendental, but it is not known which. It is also
known that e^{π} is transcendental. This follows from
the Gelfond-Schneider theorem, which says that if a
and b are algebraic, a is not 0 or 1, and b is not
rational, then a is transcendental. Using Euler's formula,
^{b}e^{iπ}= -1, and taking both sides to the power
-i gives (-1)^{-i} = (e^{iπ})^{-i}
= e^{π}. Since the theorem tells us that the left hand
side is transcendental, it follows that the right hand side is too. (It
also follows that e × π and e + π are not both algebraic,
because if they were then the equation x^{2} + x(e
+ π) + eπ = 0 would have roots e and π, making both numbers
algebraic.) But although it is known that e^{π} is transcendental,
the status of e, π^{e}^{e}, and π^{π}
remains uncertain. ## Related category• TYPES OF NUMBERS | |||||

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