Transcendental Number

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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, ${\displaystyle \sqrt{2}}$ is irrational. However it is algebraic, since it is a root of the polynomial ${\displaystyle x^2-2}$. 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 ${\displaystyle \pi }$ and ${\displaystyle e}$.