In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients.
Hence, the set of real numbers consists of non-overlapping sets of rational, algebraic irrational, and transcendental real numbers.
) is another irrational number that is not transcendental, as it is a root of the polynomial equation x2 − x − 1 = 0.
The name "transcendental" comes from Latin trānscendere 'to climb over or beyond, surmount',[7] and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that sin x is not an algebraic function of x.
[8] Euler, in the eighteenth century, was probably the first person to define transcendental numbers in the modern sense.
[13] The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was e, by Charles Hermite in 1873.
[a] Cantor's work established the ubiquity of transcendental numbers.
In 1882 Ferdinand von Lindemann published the first complete proof that π is transcendental.
He first proved that ea is transcendental if a is a non-zero algebraic number.
Then, since eiπ = −1 is algebraic (see Euler's identity), iπ must be transcendental.
The transcendence of π implies that geometric constructions involving compass and straightedge only cannot produce certain results, for example squaring the circle.
This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).
For example, from knowing that π is transcendental, it can be immediately deduced that numbers such as
Any Liouville number must have unbounded partial quotients in its simple continued fraction expansion.
Using the explicit continued fraction expansion of e, one can show that e is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded).
Kurt Mahler showed in 1953 that π is also not a Liouville number.
It is conjectured that all infinite continued fractions with bounded terms, that have a "simple" structure, and that are not eventually periodic are transcendental[18] (in other words, algebraic irrational roots of at least third degree polynomials do not have apparent pattern in their continued fraction expansions, since eventually periodic continued fractions correspond to quadratic irrationals, see Hermite's problem).
We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite.
The idea is the following: Assume, for purpose of finding a contradiction, that e is algebraic.
Then there exists a finite set of integer coefficients c0, c1, ..., cn satisfying the equation:
It is difficult to make use of the integer status of these coefficients when multiplied by a power of the irrational e, but we can absorb those powers into an integral which “mostly” will assume integer values.
By splitting respective domains of integration, this equation can be written in the form
Here P will turn out to be an integer, but more importantly it grows quickly with k. There are arbitrarily large k such that
Recall the standard integral (case of the Gamma function)
and those higher degree terms all give rise to factorials
That right hand side is a product of nonzero integer factors less than the prime
Choosing a value of k that satisfies both lemmas leads to a non-zero integer
It follows that the original assumption, that e can satisfy a polynomial equation with integer coefficients, is also impossible; that is, e is transcendental.
A similar strategy, different from Lindemann's original approach, can be used to show that the number π is transcendental.
For detailed information concerning the proofs of the transcendence of π and e, see the references and external links.