Transcendental number

In mathematics, a transcendental number is an irrational number that is not algebraic, that is, not a solution of a non-zero polynomial equation with rational coefficients. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, a coefficient is a Constant multiplicative factor of a certain object

The most prominent examples of transcendental numbers are π and e. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line Only a few classes of transcendental numbers are known, indicating that it can be extremely difficult to show that a given number is transcendental.

However, transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable, but the sets of real and complex numbers are uncountable. See also Generic property In Mathematics, the phrase almost all has a number of specialised uses

1 History
2 Properties
3 Known transcendental numbers and open problems
4 Proof sketch that e is transcendental
5 See also
6 References
7 External links

History

Euler was probably not the first person to define transcendental numbers in the modern sense. ^[1] The name "transcendentals" comes from Leibniz in his 1682 paper where he proved sin x is not an algebraic function of x. In Mathematics, an algebraic function is informally a function which satisfies a Polynomial equation whose coefficients are themselves polynomials ^[2] Joseph Liouville first proved the existence of transcendental numbers in 1844,^[3] and in 1851 gave the first decimal examples such as the Liouville constant

$\sum_{k=1}^\infty 10^{-k!} = 0.110001000000000000000001000\ldots$

in which the nth digit after the decimal point is 1 if n is a factorial (i. Joseph Liouville ( March 24 1809 &ndash September 8 1882) was a French Mathematician. In Number theory, a Liouville number is a Real number x with the property that for any positive Integer n, there exist integers Definition The factorial function is formally defined by n!=\prod_{k=1}^n k e. , 1, 2, 6, 24, 120, 720, . . . . , etc. ) and 0 otherwise. ^[4] Liouville showed that this number is what we now call a Liouville number; this essentially means that it can be particularly well approximated by rational numbers. In Number theory, a Liouville number is a Real number x with the property that for any positive Integer n, there exist integers In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions Liouville showed that all Liouville numbers are transcendental.

Johann Heinrich Lambert conjectured that e and π were both transcendental numbers, in his 1761 paper proving the number π is irrational. Johann Heinrich Lambert ( August 26, 1728 &ndash September 25 1777) was a Swiss Mathematician, Physicist and IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction The first number to be proven transcendental without having been specifically constructed for the purpose was e, by Charles Hermite in 1873. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line Charles Hermite (ʃaʁl ɛʁˈmit ( December 24, 1822 &ndash January 14, 1901) was a French Mathematician who did Year 1873 ( MDCCCLXXIII) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common In 1874, Georg Cantor found the argument mentioned above establishing the ubiquity of transcendental numbers. Year 1874 ( MDCCCLXXIV) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar (or a Common Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia.

In 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. Year 1882 ( MDCCCLXXXII) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common Carl Louis Ferdinand von Lindemann ( April 12, 1852 &ndash March 6 1939) was a German Mathematician, noted for his proof IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems He first showed that e to any nonzero algebraic power is transcendental, and since e^iπ = −1 is algebraic (see Euler's identity), iπ and therefore π must be transcendental. In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where This approach was generalized by Karl Weierstrass to the Lindemann–Weierstrass theorem. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician In Mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers 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. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles Squaring the circle is a problem proposed by ancient Geometers.

In 1900, David Hilbert posed an influential 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. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Hilbert's seventh problem is one of David Hilbert 's list of open mathematical problems posed in 1900 In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or In Mathematics, the Gelfond–Schneider theorem is a result which establishes the transcendence of a large class of numbers 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). Alan Baker (born on August 19 1939) is an English Mathematician.

Properties

The set of transcendental numbers is uncountably infinite. The proof is simple: Since the polynomials with integer coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true This article is about the zeros of a function which should not be confused with the value at zero. In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or But Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable; so the set of all transcendental numbers must also be uncountable. Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891

Transcendental numbers are never rational, but some irrational numbers are not transcendental. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction For example, the square root of 2 is irrational, but it is a solution of the polynomial x² − 2 = 0, so it is algebraic, not transcendental. The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2} or √2 In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or

Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument. In Mathematics, an algebraic function is informally a function which satisfies a Polynomial equation whose coefficients are themselves polynomials So, for example, from knowing that π is transcendental, we can immediately deduce that numbers such as 5π, (π − 3)/√2, (√π − √3)⁸ and (π⁵ + 7)^1/7 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. In Abstract algebra, a Subset S of a field L is algebraically independent over a subfield K if the elements For example, π and 1 − π are both transcendental, but π + (1 − π) = 1 is obviously not. It is unknown whether π + e, for example, is transcendental, though at least one of π + e and πe 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) = x² − (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. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental.

Every non–computable number is also transcendental since all algebraic numbers are computable. In Mathematics, Theoretical computer science and Mathematical logic, the computable numbers, also known as the recursive numbers or the

All Liouville numbers are transcendental; however, not all transcendental numbers are Liouville numbers. In Number theory, a Liouville number is a Real number x with the property that for any positive Integer n, there exist integers Any Liouville number must have unbounded partial quotients in its continued fraction expansion. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} 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. Kurt Mahler ( 26 July 1903, Krefeld, Germany – 25 February 1988, Canberra, Australia) was a mathematician 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). ^[5]

Known transcendental numbers and open problems

Here is a list of some numbers known to be transcendental:

e^a if a is algebraic and nonzero (by the Lindemann–Weierstrass theorem), and in particular, e itself. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line In Mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers
π (by the Lindemann–Weierstrass theorem). IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems In Mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers
e^π, Gelfond's constant, as well as e^-π/2=iⁱ (by the Gelfond–Schneider theorem). In Mathematics, Gelfond's constant, named after Aleksandr Gelfond, is e^\pi \ that is e to the In Mathematics, the Gelfond–Schneider theorem is a result which establishes the transcendence of a large class of numbers
a^b where a is non-zero algebraic and b is irrational algebraic (by the Gelfond–Schneider theorem), in particular:
- $2^\sqrt{2}$ , the Gelfond–Schneider constant (Hilbert number),
sin(a), cos(a) and tan(a), and their multiplicative inverses csc(a), sec(a) and cot(a), for any nonzero algebraic number a (by the Lindemann–Weierstrass theorem). In Mathematics, the Gelfond–Schneider theorem is a result which establishes the transcendence of a large class of numbers The Gelfond–Schneider constant is 2^{\sqrt{2}}=26651441 which was proved by Rodion Kuzmin to be a Transcendental number. In Mathematics, Hilbert number, named after David Hilbert, has different meanings In Mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers
ln(a) if a is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem). The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational In Mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers
Γ(1/3),^[6] Γ(1/4),^[7] and Γ(1/6). In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function ^[7]
0. 12345678910111213141516. . . , the Champernowne constant. In Mathematics, the Champernowne Constant C10 is a transcendental real constant whose decimal expansion has important properties ^[8]
Ω, Chaitin's constant (since it is a non-computable number). In the Computer science subfield of Algorithmic information theory a Chaitin Constant or halting probability is a Real number that
Prouhet–Thue–Morse constant
$\sum_{k=0}^\infty 10^{-\lfloor \beta^{k} \rfloor};$ where $β > 1$ and $\beta\mapsto\lfloor \beta \rfloor$ is the floor function. In Mathematics and its applications the Prouhet-Thue-Morse constant is the number \tau whose binary expansion. In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The

Numbers for which it is unknown whether they are transcendental or not:

Sums, products, powers, etc. (except for Gelfond's constant) of the number π and the number e: π + e, π − e, π·e, π/e, π^π, e^e, π^e
the Euler–Mascheroni constant γ (which has not even been proven to be irrational)
Catalan's constant, also not known to be irrational
Apéry's constant, ζ(3), and in fact, ζ(2n + 1) for any positive integer n (see Riemann zeta function). In Mathematics, Gelfond's constant, named after Aleksandr Gelfond, is e^\pi \ that is e to the IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line The Euler–Mascheroni constant (also called the Euler constant) is a Mathematical constant recurring in analysis and Number theory, usually In Mathematics, Catalan's constant G, which occasionally appears in estimates in Combinatorics, is defined by G = \beta(2 = \sum_{n=0}^{\infty} In Mathematics, Apéry's Constant is a curious number that occurs in a variety of situations In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in

Conjectures:

Schanuel's conjecture

Proof sketch that e is transcendental

The first proof that the base of the natural logarithms, e, is transcendental dates from 1873. In Mathematics, specifically Transcendence theory, Schanuel's conjecture is the following statement Given any n Complex numbers The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Charles Hermite (ʃaʁl ɛʁˈmit ( December 24, 1822 &ndash January 14, 1901) was a French Mathematician who did The idea is the following:

Assume, for purpose of finding a contradiction, that e is algebraic. Then there exists a finite set of integer coefficients $c_{0},c_{1},\ldots,c_{n},$ satisfying the equation:

$c_{0}+c_{1}e+c_{2}e^{2}+\cdots+c_{n}e^{n}=0$

and such that $c 0$ and $c n$ are both non-zero.

Depending on the value of n, we specify a sufficiently large positive integer k (to meet our needs later), and multiply both sides of the above equation by $\int^{\infty}_{0}$ , where the notation $\int^{b}_{a}$ will be used in this proof as shorthand for the integral:

$\int^{b}_{a}:=\int^{b}_{a}x^{k}[(x-1)(x-2)\cdots(x-n)]^{k+1}e^{-x}\,dx.$

We have arrived at the equation:

$c_{0}\int^{\infty}_{0}+c_{1}e\int^{\infty}_{0}+\cdots+c_{n}e^{n}\int^{\infty}_{0} = 0$

which can now be written in the form

$P_{1}+P_{2}=0\;$

where

$P_{1}=c_{0}\int^{\infty}_{0}+c_{1}e\int^{\infty}_{1}+c_{2}e^{2}\int^{\infty}_{2}+\cdots+c_{n}e^{n}\int^{\infty}_{n}$

$P_{2}=c_{1}e\int^{1}_{0}+c_{2}e^{2}\int^{2}_{0}+\cdots+c_{n}e^{n}\int^{n}_{0}$

The plan of attack now is to show that for k sufficiently large, the above relations are impossible to satisfy because

$\frac{P_{1}}{k!}$ is a non-zero integer and $\frac{P_{2}}{k!}$ is not.

The fact that $\frac{P_{1}}{k!}$ is a nonzero integer results from the relation

$\int^{\infty}_{0}x^{j}e^{-x}\,dx=j!$

which is valid for any positive integer j and can be proved using integration by parts and mathematical induction. In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that

To show that

$\left|\frac{P_{2}}{k!}\right|<1$ for sufficiently large k

we first note that $x^{k}[(x-1)(x-2)\cdots(x-n)]^{k+1}e^{-x}$ is the product of the functions $[x(x-1)(x-2)\cdots(x-n)]^{k}$ and $(x-1)(x-2)\cdots(x-n)e^{-x}$ . Using upper bounds for $|x(x-1)(x-2)\cdots(x-n)|$ and $|(x-1)(x-2)\cdots(x-n)e^{-x}|$ on the interval [0,n] and employing the fact

$\lim_{k\to\infty}\frac{G^k}{k!}=0$ for every real number G

is then sufficient to finish the proof. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set

A similar strategy, different from Lindemann's original approach, can be used to show that the number π is transcendental. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems Besides the gamma-function and some estimates as in the proof for e, facts about symmetric polynomials play a vital role in the proof. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line In Mathematics, a symmetric polynomial is a polynomial P ( X 1 X 2 &hellip X n)

For detailed information concerning the proofs of the transcendence of π and e see the references and external links. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line

References

^ Paul Erdős, Underwood Dudley (November 1983). In Mathematics, transcendence theory is a branch of Number theory that investigates Transcendental numbers in both qualitative and quantitative ways Paul Erdős ( Hungarian: Erdős Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 &ndash "Some Remarks and Problems in Number Theory Related to the Work of Euler". Mathematics Magazine 56 (5): 292-298.
^ Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer.
^ Aubrey J. Kempner (October 1916). "On Transcendental Numbers". Transactions of the American Mathematical Society 17 (4): 476-482.
^ Weisstein, Eric W. "Liouville's Constant", MathWorld [1]
^ Boris Adamczewski and Yann Bugeaud (March 2005). "On the complexity of algebraic numbers, II. Continued fractions". Acta Mathematica 195 (1): 1-20.
^ Le Lionnais, F. Les nombres remarquables (ISBN 2705614079). Paris: Hermann, p. 46, 1979. via Wolfram Mathworld, Transcendental Number
^ ^a ^b Chudnovsky, G. V. Contributions to the Theory of Transcendental Numbers (ISBN 0821815008). Providence, RI: Amer. Math. Soc. , 1984. via Wolfram Mathworld, Transcendental Number
^ K. Mahler (1937). "Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen". Proc. Konin. Neder. Akad. Wet. Ser. A. (40): 421–428.

David Hilbert, "Über die Transcendenz der Zahlen $e$ und $π$ ", Mathematische Annalen 43:216–219 (1893). David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most
Alan Baker, Transcendental Number Theory, Cambridge University Press, 1975, ISBN 0-521-39791-X. Alan Baker (born on August 19 1939) is an English Mathematician.

External links

(English) Proof that $e$ is transcendental
(German) Proof that $e$ is transcendental (PDF)
(German) Proof that $π$ is transcendental (PDF)

Dictionary