Liouville numbers

Liouville numbers are special real numbers that can be approximated extremely closely by rational numbers (fractions). Specifically, for any very high level of precision, there's a rational number with a relatively small denominator that gets very close to the Liouville number, closer than typical algebraic numbers allow. They are important because they were the first numbers proven to be transcendental—that is, not roots of any polynomial with rational coefficients. These numbers highlight the fascinating complexity of real numbers and the limits of algebraic approximation.