Transcendental numbers are irrational numbers which are not algebraic numbers, i.e. they are not a solution of some polynomial equation of any finite degree (they transcend the algebraic numbers, so to speak). “Most” irrational numbers are transcendental (an uncountable infinity) while “few” irrational numbers are algebraic numbers (a countable infinity).
The transcendence of a given number is much harder to prove than the irrationality of said number. In 1882, Ferdinand von Lindemann
published a proof that the number
is transcendental. He first showed that
to any nonzero algebraic power is transcendental, and since
is algebraic (see Euler’s identity
is therefore transcendental and
must be transcendental. Although the irrationality of Apéry’s constant
was proved in the late 20 th
century, its transcendence is still an open problem
. Whether or not the Euler–Mascheroni constant
is transcendental or at least irrational is another open problem.
- Ivan Niven, Numbers: Rational and Irrational. New York: Random House for Yale University (YEAR). (add YEAR)