In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . 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 comprise a countable set, while the set of real numbers and the set of complex numbers 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 rational, algebraic non-rational 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 . The golden ratio (denoted or ) is another irrational number that is not transcendental, as it is a root of the polynomial equation . The quality of a number being transcendental is called transcendence. History The name "transcendental" comes from the Latin transcendĕre 'to climb over or beyond, surmount', and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that is not an algebraic function of . Euler, in the 18th century, was probably the first person to define transcendental numbers in the modern sense. Johann Heinrich Lambert conjectured that and were both transcendental numbers in his 1768 paper proving the number is irrational, and proposed a tentative sketch of a proof of 's transcendence. Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant in which the th digit after the decimal point is if is equal to ( factorial) for some and otherwise. In other words, the th digit of this number is 1 only if is one of the numbers , 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 are called Liouville