We define a number z to be algebraic if it satisfies a polynomial with integer coefficients. And we say that z is algebraic of degree n if n is the degree of the smallest polynomial satisfied by z. For example, the degree of 2 is 2, since it satisfies the polynomial x² - 2 = 0 and no polynomial of smaller degree. Similarly, 2 + 3 is algebraic of degree 4, as we invite the reader to show by inventing a 4th degree polynomial which this number satisfies, and also convincing himself that there is no smaller.

This raises the question: Are there numbers which are not algebraic, that is, which do not satisfy any polynomial with coefficients in Z?

This question was answered in the affirmative in 1844 by Joseph Liouville (1809-1882). He first proved that if a real number z is algebraic of degree n we can say about it that there exists a...