Transcendental number

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, partly because it can be extremely difficult to show that a given number is transcendental, transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers form a countable set, while the set of real numbers ⁠${\displaystyle \mathbb {R} }$⁠ and the set of complex numbers ⁠${\displaystyle \mathbb {C} }$⁠ are both uncountable sets, and therefore larger than any countable set.

All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, 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. The golden ratio (denoted ${\displaystyle \varphi }$ or ${\displaystyle \phi }$) 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', 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

$${\displaystyle {\begin{aligned}L_{b}&=\sum _{n=1}^{\infty }10^{-n!}\\[2pt]&=10^{-1}+10^{-2}+10^{-6}+10^{-24}+10^{-120}+10^{-720}+10^{-5040}+10^{-40320}+\ldots \\[4pt]&=0.{\textbf {1}}{\textbf {1}}000{\textbf {1}}00000000000000000{\textbf {1}}00000000000000000000000000000000000000000000000000000\ \ldots \end{aligned}}}$$

in which the nth digit after the decimal point is 1 if n = k! (k factorial) for some k and 0 otherwise. In other words, the nth digit of this number is 1 only if n is one of 1! = 1, 2! = 2, 3! = 6, 4! = 24, etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by rational numbers than can any irrational algebraic number, and this class of numbers is called the Liouville numbers. Liouville showed that all Liouville numbers are transcendental.