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. The best-known transcendental numbers are π and e. The quality of a number being transcendental is called transcendence. Though only a few classes of transcendental numbers are known, because it can be difficult to show that a number is transcendental, transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable, while the real numbers R {\displaystyle \mathbb {R} } and complex numbers C {\displaystyle \mathbb {C} } are both uncountable, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational, since all rational numbers are algebraic. The converse is not true: Not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping sets of rational, algebraic irrational, and transcendental real numbers. For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x2 − 2 = 0.
History The name "transcendental" comes from Latin trānscendere 'to climb over or beyond, surmount', 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. Euler, in the eighteenth century, was probably the first person to define transcendental numbers in the modern sense. Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch proof that π is transcendental. Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant
L b = ∑ n = 1 ∞ 10 − n ! = 10 − 1 + 10 − 2 + 10 − 6 + 10 − 24 + 10 − 120 + 10 − 720 + 10 − 5040 + 10 − 40320 + … = 0. 1 1 000 1 00000000000000000 1 00000000000000000000000000000000000000000000000000000 … {\displaystyle {\begin{aligned}L_{b}&=\sum _{n=1}^{\infty }10^{-n!}\\[2pt]&=10^{-1}+10^{-2}+10^{-6}+10^{-24}+10^{-120}+10^{-720}+10^{-5040}+10^{-40320}+\ldots \\[4pt]&=0.{\textbf {1}}{\textbf {1}}000{\textbf {1}}00000000000000000{\textbf {1}}00000000000000000000000000000000000000000000000000000\ \ldots \end{aligned}}}
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