transcendental numberTranscedentalNumber2001-11-04 06:31:042008-01-17 10:41:50Definition\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\def\Q{\mathbb{Q}}
\def\Z{\mathbb{Z}}A \emph{transcendental number} is a complex number
that is not an algebraic number.
That is, it is a complex number that is transcendental over $\Q$
(or, equivalently, over $\Z$).
Well known transcendental numbers include $\pi$ and $e$
(the base of natural logarithms).
Cantor showed that, in a sense,
``almost all'' complex numbers are transcendental:
there are uncountably many complex numbers, but
\PMlinkname{only countably many algebraic numbers}{AlgebraicNumbersAreCountable}.