# Liouville's proof of the existence of transcendental numbers

The existence of transcendental numbers can be shown easily by considering the cardinality of the set of solutions to polynomials with integer cofficents and the cardinality of the real numbers.

It is written here that http://en.wikipedia.org/wiki/Transcendental_number Liouville first proved the existence of transcendental numbers in 1844. I doubt that he proved this result by a cardinality argument, as I think that these ideas were first brought by Cantor and this was well after 1844. How did Liouville prove the existence of transcendental numbers.

• Cantor's argument would only appear 15-20 years later. Read about Liouville numbers. Commented Jan 3, 2013 at 21:08
• See en.wikipedia.org/wiki/Liouville_number. Basically, the irrational algebraic numbers (non-transcendental numbers) "poorly approximate" rational numbers.
– A.S
Commented Jan 3, 2013 at 21:09
• Roughly speaking, irrational algebraic numbers are poorly approximable by rationals. Liouville showed that numbers whose decimal expansion involves only $0$'s and $1$'s, where the number of $0$'s between consecutive $1$'s grows rapidly, are well approximable by rationals. There is a hint of diagonalization in the construction. Commented Jan 3, 2013 at 21:15