Proof of Impossibility - The Existence of Transcendental Numbers

The Existence of Transcendental Numbers

There exists at least one number for which it is impossible to find any algebraic equation that this number satisfies (i.e. you plug the number into the equation wherever X occurs and the equation equals zero). Stated another way: There exists at least one number which does not satisfy any equations of the form called an algebraic equation:

where are integers, not all zero. Numbers that do not satisfy any equation of this form are called transcendental numbers.

"It is not immediately obvious that there are any transcendental numbers, though actually, as we shall see in a moment, almost all real numbers are transcendental" (Hardy and Wright, p. 160)

Hardy and Wright (p. 160) offer theorems to show that:

1. The aggregate of algebraic numbers is enumerable.
2. Almost all real numbers are transcendental.
3. A real algebraic number of degree n is not approximable to any order greater than n (Liouville's theorem)

This last theorem "enables us to produce as many examples of transcendental numbers as we please" (Hardy and Wright p. 161).

Hardy and Wright go on to prove that pi and the "exponential" e are transcendental. Hermite offered the first proof that e is transcendental (cf Notes in Hardy and Wright p. 177).

In a footnote p. 190 Hardy and Wright discuss the diagonal method of Cantor that demonstrates the existence of transcendental numbers.