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 and . The quality of a number being transcendental is called transcendence.
Only a few classes of transcendental numbers are known, because it can be difficult to show that a number is transcendental, but transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable, while the real numbers and complex numbers 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 .
History The name "transcendental" comes , s.v. 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 eighteenth 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 proof that 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
in which the th digit after the decimal point is if = ( factorial) for some and otherwise. In other words, the th digit of this number is 1 only if is one of , 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.
The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was , by Charles Hermite in 1873.
In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a new method for constructing transcendental numbers.; Although this was already implied by his proof of the countability of the algebraic numbers, Cantor also published a construction that proves there are as many transcendental numbers as there are real numbers. Cantor's work established the ubiquity of transcendental numbers.
In 1882, Ferdinand von Lindemann published the first complete proof that is transcendental. He first proved that is transcendental if is a non-zero algebraic number. Then, since is algebraic (see Euler's identity), must be transcendental. But since is algebraic, must therefore be transcendental. This approach was generalized by Karl Weierstrass to what is now known as the Lindemann–Weierstrass theorem. The transcendence of implies that geometric constructions involving compass and straightedge only cannot produce certain results, for example squaring the circle.
In 1900, David Hilbert posed a question about transcendental numbers, Hilbert's seventh problem: If is an algebraic number that is not 0 or 1, and is an irrational algebraic number, is necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. 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).
Properties A transcendental number is a (possibly complex) number that is not the root of any integer polynomial. Every real transcendental number must also be irrational, since every rational number is the root of some integer polynomial of degree one. The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, it is impossible for both subsets to be countable. This makes the transcendental numbers uncountable.
No rational number is transcendental and all real transcendental numbers are irrational. The irrational numbers contain all the real transcendental numbers and a subset of the algebraic numbers, including the quadratic irrationals and other forms of algebraic irrationals.
Applying any non-constant single-variable algebraic function to a transcendental argument yields a transcendental value. For example, from knowing that is transcendental, it can be immediately deduced that numbers such as , , , and are transcendental as well.
However, an algebraic function of several variables may yield an algebraic number when applied to transcendental numbers if these numbers are not algebraically independent. For example, and are both transcendental, but is not. It is unknown whether , for example, is transcendental, though at least one of and must be transcendental. More generally, for any two transcendental numbers and , at least one of and must be transcendental. To see this, consider the polynomial . If and were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, and , must be algebraic. But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental.
The non-computable numbers are a strict subset of the transcendental numbers.
All Liouville numbers are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its simple continued fraction expansion. Using a counting argument one can show that there exist transcendental numbers which have bounded partial quotients and hence are not Liouville numbers.
Using the explicit continued fraction expansion of , one can show that is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). Kurt Mahler showed in 1953 that 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. (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).
Numbers proven to be transcendental Numbers proven to be transcendental:
(by the Lindemann–Weierstrass theorem). if is algebraic and nonzero (by the Lindemann–Weierstrass theorem), in particular Euler's number . where is a positive integer; in particular Gelfond's constant (by the Gelfond–Schneider theorem). Algebraic combinations of and such as and (following from their algebraic independence). where is algebraic but not 0 or 1, and is irrational algebraic, in particular the Gelfond–Schneider constant (by the Gelfond–Schneider theorem). The natural logarithm if is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem). if and are positive integers not both powers of the same integer, and is not equal to 1 (by the Gelfond–Schneider theorem). All numbers of the form are transcendental, where are algebraic for all and are non-zero algebraic for all (by Baker's theorem). The trigonometric functions and their hyperbolic counterparts, for any nonzero algebraic number , expressed in radians (by the Lindemann–Weierstrass theorem). Non-zero results of the inverse trigonometric functions and their hyperbolic counterparts, for any algebraic number (by the Lindemann–Weierstrass theorem). , for rational such that . The Dottie number (the fixed point of the cosine function) – the unique real solution to the equation , where is in radians (by the Lindemann–Weierstrass theorem). if is algebraic and nonzero, for any branch of the Lambert W function (by the Lindemann–Weierstrass theorem), in particular the omega constant . if both and the order are algebraic such that , for any branch of the generalized Lambert W function. , the square super-root of any natural number is either an integer or transcendental (by the Gelfond–Schneider theorem). Values of the gamma function of rational numbers that are of the form or . Algebraic combinations of and or of and such as the lemniscate constant (following from their respective algebraic independences). The values of Beta function if and are non-integer rational numbers. The Bessel function of the first kind , its first derivative, and the quotient are transcendental when is rational and is algebraic and nonzero, and all nonzero roots of and are transcendental when is rational. The number , where and are Bessel functions and is the Euler–Mascheroni constant. Values of the Fibonacci zeta function at positive even arguments. Any Liouville number, in particular: Liouville's constant . Numbers with irrationality measure larger than 2, such as the Champernowne constant and Cahen's constant (by Roth's theorem). Numbers artificially constructed not to be algebraic periods. Any non-computable number, in particular: Chaitin's constant. Constructed irrational numbers which are not simply normal in any base. Any number for which the digits with respect to some fixed base form a Sturmian word. Any irrational number for which the nth digit can be generated by a finite-state machine with b states from the base-b expansion of n. Any number for which the simple continued fraction is aperiodic (i.e. a non-quadratic irrational number) and the nth term can be generated by a finite-state machine with b states from the base-b expansion of n. The Prouhet–Thue–Morse constant; and the related rabbit constant. The Komornik–Loreti constant. The paperfolding constant (also named as "Gaussian Liouville number"). The values of the infinite series with fast convergence rate as defined by Y. Gao and J. Gao, such as . Any number of the form (where , are polynomials in variables and , is algebraic and , is any integer greater than 1). Numbers of the form and For where is the floor function. The numbers and with only two different decimal digits whose nonzero digit positions are given by the Moser–de Bruijn sequence and its double. The values of the Rogers-Ramanujan continued fraction where is algebraic and . The lemniscatic values of theta function (under the same conditions for ) are also transcendental. where is algebraic but not imaginary quadratic (i.e, the exceptional set of this function is the number field whose degree of extension over is 2). The constants and in the formula for first index of occurrence of Gijswijt's sequence, where k is any integer greater than 1. Any hypertranscendental number, a number that is not the value at an algebraic point of a function which is the solution of an algebraic differential equation with coefficients in and with algebraic initial conditions.
Conjectured transcendental numbers Numbers which have yet to be proven to be either transcendental or algebraic: