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
rationalcoefficients. The best known transcendental numbers are
π and e.^{
[1]}^{
[2]}
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 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.^{
[3]}^{
[4]}^{
[5]}^{
[6]} 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.^{
[3]} 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 x^{2} − 2 = 0. The
golden ratio (denoted $\varphi$ or $\phi$) is another irrational number that is not transcendental, as it is a root of the polynomial equation x^{2} − x − 1 = 0. 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',^{
[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]}^{
[9]}Euler, in the 18th century, was probably the first person to define transcendental numbers in the modern sense.^{
[10]}
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 of a proof of π's transcendence.^{
[11]}
Joseph Liouville first proved the existence of transcendental numbers in 1844,^{
[12]} and in 1851 gave the first decimal examples such as the
Liouville constant
in which the nth digit after the decimal point is 1 if n is equal to k! (kfactorial) for some k and 0 otherwise.^{
[13]} In other words, the nth digit of this number is 1 only if n is one of the numbers 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 are called
Liouville numbers, named in his honour. Liouville showed that all Liouville numbers are transcendental.^{
[14]}
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.
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.^{
[15]}^{
[16]} 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.^{
[17]} Cantor's work established the ubiquity of transcendental numbers.
In 1882,
Ferdinand von Lindemann published the first complete proof of the transcendence of π. He first proved that e^{a} is transcendental if a is a non-zero algebraic number. Then, since e^{iπ} = −1 is algebraic (see
Euler's identity), iπ must be transcendental. But since i is algebraic, π therefore must be transcendental. This approach was generalized by
Karl Weierstrass to what is now known as the
Lindemann–Weierstrass theorem. The transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving
compass and straightedge, including the most famous one,
squaring the circle.
In 1900,
David Hilbert posed a question about transcendental numbers,
Hilbert's seventh problem: If a is an algebraic number that is not zero or one, and b is an irrational
algebraic number, is a^{b} 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).^{
[18]}
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 a
rational number is the root of an integer polynomial of
degree one.^{
[19]} 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 π is transcendental, it can be immediately deduced that numbers such as 5π, π-3/√2, (√π-√3)^{8}, and ^{4}√π^{5}+7 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, π and (1 − π) are both transcendental, but π + (1 − π) = 1 is obviously not. It is unknown whether e + π, for example, is transcendental, though at least one of e + π and eπ must be transcendental. More generally, for any two transcendental numbers a and b, at least one of a + b and ab must be transcendental. To see this, consider the polynomial (x − a)(x − b) = x^{2} − (a + b)x + ab. If (a + b) and ab 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, a and b, must be algebraic. But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental.
All
Liouville numbers are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its
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 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 are not eventually periodic are transcendental (eventually periodic continued fractions correspond to quadratic irrationals).^{
[20]}
The
fixed point of the cosine function (also referred to as the
Dottie numberd) – the unique real solution to the equation cos x = x, where x is in radians (by the Lindemann–Weierstrass theorem).^{
[21]}
lna if a is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
log_{b}a if a and b are positive integers not both powers of the same integer, and a is not equal to 1 (by the Gelfond–Schneider theorem).
The
Bessel functionJ_{ν}(x), its first derivative, and the quotient J'_{ν}(x)/J_{ν}(x) are transcendental when ν is rational and x is algebraic and nonzero,^{
[22]} and all nonzero roots of J_{ν}(x) and J'_{ν}(x) are transcendental when ν is rational.^{
[23]}
W(a) if a is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann–Weierstrass theorem), in particular: Ω the
omega constant
√x_{s}, the
square super-root of any natural number is either an integer or transcendental (by the Gelfond–Schneider theorem)
Γ(1/3),^{
[24]}Γ(1/4),^{
[25]} and Γ(1/6).^{
[25]} The numbers $\Gamma (2/3)$, $\Gamma (3/4)$ and $\Gamma (5/6)$ are also known to be transcendental. The numbers $\Gamma (1/4)^{4}/\pi$ and $\Gamma (1/3)^{2}/\pi$ are also transcendental.^{
[26]}
where $\beta \mapsto \lfloor \beta \rfloor$ is the
floor function.
3.300330000000000330033... and its reciprocal 0.30300000303..., two numbers with only two different decimal digits whose nonzero digit positions are given by the
Moser–de Bruijn sequence and its double.^{
[39]}
The number π/2Y_{0}(2)/J_{0}(2)-γ, where Y_{α}(x) and J_{α}(x) are Bessel functions and γ is the
Euler–Mascheroni constant.^{
[40]}^{
[41]}
Nesterenko proved in 1996 that $\pi ,e^{\pi }$ and $\Gamma (1/4)$ are algebraically independent.^{
[26]}
Possible transcendental numbers
Numbers which have yet to be proven to be either transcendental or algebraic:
Most sums, products, powers, etc. of the number π and the
number e, e.g. eπ, e + π, π − e, π/e, π^{π}, e^{e}, π^{e}, π^{√2}, e^{π2} are not known to be rational, algebraic, irrational or transcendental. A notable exception is e^{π√n} (for any positive integer n) which has been proven transcendental.^{
[42]}
The
Euler–Mascheroni constantγ: In 2010 M. Ram Murty and N. Saradha found an infinite list of numbers containing γ/4 such that all but at most one of them are transcendental.^{
[43]}^{
[44]} In 2012 it was shown that at least one of γ and the
Euler–Gompertz constantδ is transcendental.^{
[45]}
Assume, for purpose of finding a contradiction, that e is algebraic. Then there exists a finite set of integer coefficients c_{0}, c_{1}, ..., c_{n} satisfying the equation:
Lemma 1. For an appropriate choice of k, ${\tfrac {P}{k!}}$ is a non-zero integer.
Proof. Each term in P is an integer times a sum of factorials, which results from the relation
$\int _{0}^{\infty }x^{j}e^{-x}\,dx=j!$
which is valid for any positive integer j (consider the
Gamma function).
It is non-zero because for every a satisfying 0< a ≤ n, the integrand in
$c_{a}e^{a}\int _{a}^{\infty }f_{k}e^{-x}\,dx$
is e^{−x} times a sum of terms whose lowest power of x is k+1 after substituting x for x+a in the integral. Then this becomes a sum of integrals of the form
$A_{j-k}\int _{0}^{\infty }x^{j}e^{-x}\,dx$ Where A_{j-k} is integer.
with k+1 ≤ j, and it is therefore an integer divisible by (k+1)!. After dividing by k!, we get zero
modulo (k+1). However, we can write:
So when dividing each integral in P by k!, the initial one is not divisible by k+1, but all the others are, as long as k+1 is prime and larger than n and |c_{0}|. It follows that ${\tfrac {P}{k!}}$ itself is not divisible by the prime k+1 and therefore cannot be zero.
Lemma 2.$\left|{\tfrac {Q}{k!}}\right|<1$ for sufficiently large $k$.
where $u(x)$ and $v(x)$ are continuous functions of $x$ for all $x$, so are bounded on the interval $[0,n]$. That is, there are constants $G,H>0$ such that
$\left|f_{k}e^{-x}\right|\leq |u(x)|^{k}\cdot |v(x)|<G^{k}H\quad {\text{ for }}0\leq x\leq n.$
So each of those integrals composing $Q$ is bounded, the worst case being
where $M$ is a constant not depending on $k$. It follows that
$\left|{\frac {Q}{k!}}\right|<M\cdot {\frac {G^{k}}{k!}}\to 0\quad {\text{ as }}k\to \infty ,$
finishing the proof of this lemma.
Choosing a value of $k$ satisfying both lemmas leads to a non-zero integer ($P/k!$) added to a vanishingly small quantity ($Q/k!$) being equal to zero, is an impossibility. It follows that the original assumption, that e can satisfy a polynomial equation with integer coefficients, is also impossible; that is, e is transcendental.
The transcendence of π
A similar strategy, different from
Lindemann's original approach, can be used to show that the
number π is transcendental. Besides the
gamma-function and some estimates as in the proof for e, facts about
symmetric polynomials play a vital role in the proof.
For detailed information concerning the proofs of the transcendence of π and e, see the references and external links.
^Cantor 1878, p. 254. Cantor's construction builds a
one-to-one correspondence between the set of transcendental numbers and the set of real numbers. In this article, Cantor only applies his construction to the set of irrational numbers.
^J J O'Connor and E F Robertson:
Alan Baker. The MacTutor History of Mathematics archive 1998.
^Hardy 2005 harvnb error: no target: CITEREFHardy2005 (
help).
^Allouche & Shallit 2003, pp. 385, 403. The name 'Fredholm number' is misplaced: Kempner first proved this number is transcendental, and the note on page 403 states that Fredholm never studied this number.
^Allouche, Jean-Paul; Cosnard, Michel (2000), "The Komornik–Loreti constant is transcendental", American Mathematical Monthly, 107 (5): 448–449,
doi:
10.2307/2695302,
JSTOR2695302,
MR1763399
Burger, Edward B.; Tubbs, Robert (2004). Making transcendence transparent. An intuitive approach to classical transcendental number theory.
Springer.
ISBN978-0-387-21444-3.
Zbl1092.11031.
Calude, Cristian S. (2002). Information and Randomness: An Algorithmic Perspective. Texts in Theoretical Computer Science (2nd rev. and ext. ed.).
Springer.
ISBN978-3-540-43466-5.
Zbl1055.68058.
Lambert, Johann Heinrich (1768). "Mémoire sur quelques propriétés remarquables des quantités transcendantes, circulaires et logarithmiques". Mémoires de l'Académie Royale des Sciences de Berlin: 265–322.
Leibniz, Gottfried Wilhelm; Gerhardt, Karl Immanuel; Pertz, Georg Heinrich (1858).
Leibnizens mathematische Schriften. Vol. 5. A. Asher & Co. pp. 97–98.