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Encyclopedia > Transcendental number

In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. The most prominent examples of transcendental numbers are π and e. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... e is the unique number such that the value of the derivative (slope) of f(x)=ex for any value of x is equal to the value of f(x). ...


Although transcendental numbers are never rational, some irrational numbers are not transcendental: the square root of 2 is irrational, but it is a solution of the polynomial x2 − 2 = 0, so it is algebraic. In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ...


The transcendental numbers are uncountable. 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. But Cantor's diagonal argument proves that the reals (and therefore also the complex numbers) are uncountable; so the set of all transcendental numbers must also be uncountable. In a very real sense, then, there are many more transcendental numbers than algebraic ones. However, only a few classes of transcendental numbers are known and proving that a given number is transcendental can be extremely difficult. In mathematics, an uncountable set is a set which is not countable. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics the term countable set is used to describe the size of a set, e. ... Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ...

Contents

History

Leibniz was probably the first person to believe in the existence of numbers which do not satisfy polynomials with rational coefficients. The name "transcendentals" comes from Leibniz in his 1682 paper where he proved sin(x) is not an algebraic function of x. The existence of transcendental numbers was first proved in 1844 by Joseph Liouville, who exhibited examples, including the Liouville constant: Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... 1844 was a leap year starting on Monday (see link for calendar). ... Joseph Liouville (born March 24, 1809, died 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 p and q with q > 1 and such that 0 < |x − p/q| < 1/qn. ...


sum_{k=1}^infty 10^{-k!} = 0.110001000000000000000001000ldots in which the nth digit after the decimal point is 1 if n is a factorial (i.e., 1, 2, 6, 24, 120, 720, ...., etc.) and 0 otherwise. 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. Liouville showed that all Liouville numbers are transcendental. The beginning of the sequence of factorials (sequence A000142 in OEIS) In mathematics, the factorial of a natural number n is the product of all positive integers less than or equal to n. ... In number theory, a Liouville number is a real number x with the property that, for any positive integer n, there exist integers p and q with q > 1 and such that 0 < |x − p/q| < 1/qn. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...


Johann Heinrich Lambert conjectured that e and π were both transcendental numbers, in his paper proving the number π is irrational. The first number to be proven transcendental without having been specifically constructed for the purpose was e, by Charles Hermite in 1873. In 1874, Georg Cantor found the argument mentioned above establishing the ubiquity of transcendental numbers. Johann Heinrich Lambert Johann Heinrich Lambert (August 26, 1728 &#8211; September 25, 1777), was a mathematician, physicist and astronomer. ... When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... e is the unique number such that the value of the derivative (slope) of f(x)=ex for any value of x is equal to the value of f(x). ... Charles Hermite (pronounced in IPA, , or phonetically air-meet) (December 24, 1822 - January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... 1873 (MDCCCLXXIII) was a common year starting on Wednesday (see link for calendar). ... 1874 (MDCCCLXXIV) was a common year starting on Thursday (see link for calendar). ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ...


In 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. He first showed that e to any algebraic power is transcendental, and since eiπ = − 1 is algebraic (see Euler's identity), iπ and therefore π must be transcendental. This approach was generalized by Karl Weierstrass to 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. 1882 (MDCCCLXXXII) was a common year starting on Sunday (see link for calendar) of the Gregorian calendar or a common year starting on Tuesday of the 12-day slower Julian calendar. ... Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that &#960; is a transcendental number, i. ... When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one... Karl Weierstraß Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Biography Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... In mathematics, the Lindemann–Weierstrass theorem states that if α1,...,αn are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... This square and circle have the same area. ...


In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If the number a is algebraic, but is neither 0 nor 1, and the number b is irrational and algebraic, is ab necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond-Schneider theorem. This work was extended by Alan Baker in the 1960s. David Hilbert (January 23, 1862, Wehlau, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Hilberts seventh problem concerns the irrationality and transcendence of certain numbers (Irrationalität und Transzendenz bestimmter Zahlen). ... In mathematics, an irrational number is any real number that is not a rational number, i. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, the Gelfond-Schneider theorem is the following statement, originally proved by Aleksandr Gelfond: If is an algebraic number (with and ), and is an irrational algebraic number, then is a transcendental number. ... Alan Baker (born on August 19, 1939) is an English mathematician. ...


Known transcendental numbers and open problems

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

  • ea if a is algebraic and nonzero (a consequence of the Lindemann-Weierstrass theorem). In particular, e itself is transcendental.
  • π.
  • eπ Gelfond's constant. (A consequence of the Gelfond-Schneider theorem.)
  • 22, the Gelfond-Schneider constant, or more generally ab where a ≠ 0, 1 is algebraic and b is algebraic but not rational (Gelfond-Schneider theorem and Hilbert's seventh problem).
  • sin(a), cos(a) and tan(a) for any nonzero rational number a.
  • ln(a) if a is positive, rational and ≠ 1
  • Γ(1/3), Γ(1/4), and Γ(1/6) (see gamma function).
  • the Champernowne constant is an example of a transcendental number that is normal in base 10.
  • Ω, Chaitin's constant, and more generally: every non-computable number is transcendental (since all algebraic numbers are computable).
  • Prouhet-Thue-Morse constant
  • sum_{k=0}^infty 10^{-lfloor beta^{k} rfloor}; where β > 1 and betamapstolfloor beta rfloor is the floor function.

Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument. So for example, from knowing that π is transcendental, we can immediately deduce that 5π, (π − 3)/√2, (√π − √3)8 and (π5 + 7)1/7 are transcendental as well. e is the unique number such that the value of the derivative (slope) of f(x)=ex for any value of x is equal to the value of f(x). ... In mathematics, the Lindemann-Weierstrass theorem states that if &#945;1,...,&#945;n are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . ... When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... In mathematics, Gelfonds constant, named after Aleksandr Gelfond, is that is, e to the power of Ï€. Like both e and Ï€, this constant is a transcendental number. ... In mathematics, the Gelfond-Schneider theorem is the following statement, originally proved by Aleksandr Gelfond: If is an algebraic number (with and ), and is an irrational algebraic number, then is a transcendental number. ... The Gelfond-Schneider constant is which Aleksandr Gelfond proved to be a transcendental number using the Gelfond-Schneider theorem , answering one of the questions raised in Hilberts seventh problem. ... In mathematics, the Gelfond-Schneider theorem is the following statement, originally proved by Aleksandr Gelfond: If is an algebraic number (with and ), and is an irrational algebraic number, then is a transcendental number. ... Hilberts seventh problem concerns the irrationality and transcendence of certain numbers (Irrationalität und Transzendenz bestimmter Zahlen). ... All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. ... All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. ... All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ... In mathematics, the Champernowne constant C10 is a certain real number, named after mathematician D. G. Champernowne. ... In mathematics, a normal number is, roughly speaking, a real number whose digits show a random distribution with all digits being equally likely. ... In the computer science subfield of algorithmic information theory the Chaitin constant or halting probability is a construction by Gregory Chaitin which describes the probability that a randomly generated program for a given model of computation or programming language will halt. ... In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the subset of the real numbers consisting of the numbers which can be computed by a finite, terminating algorithm. ... In mathematics and its applications, the Prouhet-Thue-Morse constant is the number whose binary expansion is the Prouhet-Thue-Morse sequence. ... The floor and fractional part functions In mathematics, the floor function of a real number x, denoted or floor(x), is the largest integer less than or equal to x (formally, ). For example, floor(2. ... This article or section does not cite its references or sources. ...


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 π + e, for example, is transcendental, though at least one of π + e and π e must be transcendental. More generally, for any two transcendental numbers a and b, at least one of a + b and a b must be transcendental. To see this, consider the polynomial (xa) (xb) = x2 − (a + b) x + a b. 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. In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. This means that for every finite sequence α1, ..., αn of elements of S, no two the... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ...


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

All Liouville numbers are transcendental, however not all transcendental numbers are Liouville numbers. Any Liouville number must have unbounded terms in its continued fraction expression, and so using a counting argument one can show that there exist transcendental numbers which are not Liouville. Using the explicit continued fraction expansion of e, one can show that e is not a Liouville number. 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). When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... e is the unique number such that the value of the derivative (slope) of f(x)=ex for any value of x is equal to the value of f(x). ... The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is γ ≈ 0. ... Catalans constant K, which occasionally appears in estimates in combinatorics, is defined by or equivalently along with where K(x) is a complete elliptic integral of the first kind, and has nothing to do with the constant itself. ... In mathematics, Apérys constant is a curious number that occurs in a variety of situations. ... In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... Kurt Mahler is a British mathematician and a Fellow of the Royal Society. ...


Proof sketch that e is transcendental

The first proof that the base of the natural logarithms, e, is transcendental dates from 1873. We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. The idea is the following: e is the unique number such that the value of the derivative (slope) of f(x)=ex for any value of x is equal to the value of f(x). ... David Hilbert (January 23, 1862, Wehlau, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Charles Hermite (pronounced in IPA, , or phonetically air-meet) (December 24, 1822 - January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ...


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 c0 and cn 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}eint^{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}eint^{infty}_{1}+c_{2}e^{2}int^{infty}_{2}+cdots+c_{n}e^{n}int^{infty}_{n}
P_{2}=c_{1}eint^{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, possibly simpler, integrals. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...


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 In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

lim_{ktoinfty}frac{G^k}{k!}=0 for every real number G

is then sufficient to finish the proof.


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. When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ... e is the unique number such that the value of the derivative (slope) of f(x)=ex for any value of x is equal to the value of f(x). ... In mathematics, a symmetric polynomial is a polynomial in n variables , such that if some of the variables are interchanged, the polynomial stays the same. ...


For detailed information concerning the proofs of the transcendence of π and e see the references and external links. When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... e is the unique number such that the value of the derivative (slope) of f(x)=ex for any value of x is equal to the value of f(x). ...


See also

In mathematics, transcendence theory investigates transcendental numbers, in a qualitative and quantitative way. ...

References

  • David Hilbert, "Über die Transcendenz der Zahlen e und π", Mathematische Annalen 43:216–219 (1893).
  • Alan Baker, Transcendental Number Theory, Cambridge University Press, 1975, ISBN 0-521-39791-X.

David Hilbert (January 23, 1862, Wehlau, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... 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)

  Results from FactBites:
 
PlanetMath: transcendental number (94 words)
A transcendental number is a complex number that is not an algebraic number.
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The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.
In the base-ten number system, they are written as a string of digits, with a period (decimal point) (in, for example, the US and UK) or a comma (in, for example, continental Europe) to the right of the ones place; negative real numbers are written with a preceding minus sign.
The existence of complex numbers was not completely accepted until the geometrical interpretation had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion.
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