Besides classifying the real numbers
into rational and
irrational numbers, we can also separate them into
*algebraic* and *transcendental*. If a real number
satisfies an algebraic equation, which is an equation of the form

where all of the `c`'s are integers, we say that it is an
*algebraic number*. Real numbers that satisfy no such equations
are called *transcendental numbers*. It is fairly easy to see
that every rational number ^{a}/_{b}
satisfies the equation `bx` − `a` = 0
and so all rational numbers are algebraic. Therefore, every transcendental
number must be irrational.

Some irrational numbers are algebraic though. For example, all square roots (for example, 2) satisfy quadratic equations, and other roots satisfy other equations of higher degree. On the other hand, numbers such as log 2 and π are transcendental.

The existence of transcendental numbers was not proved until 1840,
when Joseph Liouville proved that the number
0.1100010000000000000000010000..., where 1's appear in the `n`!^{th}
position for all natural numbers `n`
(! is the factorial symbol) is transcendental.
The proof is too complicated to present here, but suffice to say that
it can be shown that algebraic numbers can approximate this number
but can never equal it.

It was shown in 1882 by Ferdinand von Lindemann that π was transcendental. Showing that π was transcendental also proved that squaring the circle, one of the three famous construction problems of antiquity was impossible.

In 1934, Gelfond and Schneider
independently proved that `a ^{b}` is transcendental
if

In a sense, it can be said that almost all numbers are transcendental. In the 1870s, Georg Cantor showed that the real algebraic numbers are countably infinite, while the real numbers (which are comprised of the real algebraic numbers and real transcendental numbers) are uncountably infinite; thus, transcendental numbers are, in a sense, more numerous than numbers that are not transcendental.

There are still some gaps in our knowledge of what numbers are
transcendental. We know that π and `e` are both transcendental,
and, while we know that `e ^{π}` is transcendental,
we don't yet know if either π +

Something to think about: Can you find two transcendental numbers that add to an integer? It's not too difficult to find such numbers. π, for example is transcendental. Therefore, 10 − π must also be transcendental. However, if you add the two numbers together, the sum is 10.