The transcendental functions are functions that let you leave
the algebraic world. That is, let **K** be a field (think
of **Q**, the rational numbers) embedded inside a much larger
field **F** such that **F**/**K** is *not* an
algebraic field extension (think of **R**, the real numbers).
A function `f`:**K**→**F** (or
even **K**^{n}→**F**) is
called *transcendental* if there exist values of `x`
for which `f(x)` is *not* algebraic over **K**.

In fact, it's probably more convenient to introduce a
topology and consider only (piecewise) continuous functions;
those are the only ones considered in practice. Even better is to
consider the field of rational functions **K**(`x`),
and to define a function as transcendental if it is not algebraic
over **K**(`x`). This means that
values `f(x)` of an algebraic (non-transcendental)
function will satisfy *the same* algebraic equation
(involving also `x`) for all values of `x`. The
end result is the same for practical purposes, though.

For instance, the square root function sqrt is
algebraic and *not* transcendental: the square root of any
number is algebraic, as it satisfies (duh!)

sqrt(`x`)^{2} = `x`

But the exponential function `exp` is transcendental.
For instance, `e`=`exp`(1) is not an algebraic
number. Sine is also transcendental, a consequence of the same
proof that we cannot trisect an angle with compass and
straightedge.