Transcendental number

In mathematics, a transcendental number is a complex number that is not algebraic, that is, not a solution of a non-zero polynomial equation with rational coefficients. In other words, transcendental numbers are numbers that do not arise from Euclidean geometry or ordinary algebraic expressions.

The most prominent examples of transcendental numbers are π and e. Only a few classes of transcendental numbers are known, indicating that it can be extremely difficult to show that a given number is transcendental.

However, transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable, but the sets of real and complex numbers are uncountable. All transcendental numbers are irrational, since all rational numbers are algebraic. (The converse is not true: not all irrational numbers are transcendental.)