Transcendental Numbers

Besides classifying the real numbers (or, for that matter, the complex numbers) into rational and irrational, 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 (for !, read "factorial") 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 one of the three famous construction problems of antiquity was impossible. It was not until 1934, when Gelfond and Schneider independently proved that a ^b is transcendental if a is algebraic (but not equal to 0 or 1), and b is irrational and algebraic. Since 10 ^{log 2} is equal to 2, 2 would be transcendental if log 2 were algebraic. Therefore, log 2 is either rational or transcendental. It is pretty easy to show it isn't rational, so it must be transcendental.

There are still some gaps in our knowledge of what numbers are transcendental. We know that π and e are both transcendental, we don't yet know if either π + e or π ^e are transcendental, let alone irrational.