Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways.

The fundamental theorem of algebra tells us that if we have a non-constant polynomial with rational coefficients (or equivalently, by clearing denominators, with integer coefficients) then that polynomial will have a root in the complex numbers. That is, for any non-constant polynomial ${\displaystyle P}$ with rational coefficients there will be a complex number ${\displaystyle \alpha }$ such that ${\displaystyle P(\alpha )=0}$. Transcendence theory is concerned with the converse question: given a complex number ${\displaystyle \alpha }$, is there a polynomial ${\displaystyle P}$ with rational coefficients such that ${\displaystyle P(\alpha )=0?}$ If no such polynomial exists then the number is called transcendental.

More generally the theory deals with algebraic independence of numbers. A set of numbers {α₁, α₂, …, α_n} is called algebraically independent over a field K if there is no non-zero polynomial P in n variables with coefficients in K such that P(α₁, α₂, …, α_n) = 0. So working out if a given number is transcendental is really a special case of algebraic independence where n = 1 and the field K is the field of rational numbers.

A related notion is whether there is a closed-form expression for a number, including exponentials and logarithms as well as algebraic operations. There are various definitions of "closed-form", and questions about closed-form can often be reduced to questions about transcendence.

Use of the term transcendental to refer to an object that is not algebraic dates back to the seventeenth century, when Gottfried Leibniz proved that the sine function was not an algebraic function. The question of whether certain classes of numbers could be transcendental dates back to 1748 when Euler asserted that the number log_ab was not algebraic for rational numbers a and b provided b is not of the form b = a^c for some rational c.

Euler's assertion was not proved until the twentieth century, but almost a hundred years after his claim Joseph Liouville did manage to prove the existence of numbers that are not algebraic, something that until then had not been known for sure. His original papers on the matter in the 1840s sketched out arguments using simple continued fractions to construct transcendental numbers. Later, in the 1850s, he gave a necessary condition for a number to be algebraic, and thus a sufficient condition for a number to be transcendental. This transcendence criterion was not strong enough to be necessary too, and indeed it fails to detect that the number e is transcendental. But his work did provide a larger class of transcendental numbers, now known as Liouville numbers in his honour.

Liouville's criterion essentially said that algebraic numbers cannot be very well approximated by rational numbers. So if a number can be very well approximated by rational numbers then it must be transcendental. The exact meaning of "very well approximated" in Liouville's work relates to a certain exponent. He showed that if α is an algebraic number of degree d ≥ 2 and ε is any number greater than zero, then the expression

can be satisfied by only finitely many rational numbers p/q. Using this as a criterion for transcendence is not trivial, as one must check whether there are infinitely many solutions p/q for every d ≥ 2.