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From Citizendium, the Citizens' Compendium

In mathematics, a transcendental number is any complex number that is not algebraic, i.e. it is not a root of any polynomial whose coefficients are integers, or, equivalently, it is not a root of any polynomial whose coefficients are rational.

Transcendental numbers are necessarily irrational, but there are many irrational numbers that are not transcendental. For instance, is irrational. However it is algebraic, since it is a root of the polynomial x² − 2. It is thus irrational but not transcendental.

Proving a number to be transcendental is generally much more difficult than just proving it is irrational. Examples of real numbers known to be transcendental are π and e.