Worlds of David Darling
Encyclopedia of Science
Home > Encyclopedia of Science

transcendental number

A 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 eπ 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 ab is transcendental. Using Euler's formula, eiπ= -1, and taking both sides to the power -i gives (-1)-i = (eiπ)-i = eπ. 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 x2 + x(e + π) + eπ = 0 would have roots e and π, making both numbers algebraic.) But although it is known that eπ is transcendental, the status of ee, πe, and ππ remains uncertain.

Related category