# Transcendent number

In mathematics is a real number (or more generally a complex number ) transcendent, if not zero one (different from zero polynomial) the polynomial with integer coefficients. Otherwise it is an algebraic number . Every real transcendent number is also irrational .

## Classification without prior mathematical knowledge

In number theory (the science that deals with whole numbers and their properties) the question of how naturally a number can be characterized is of importance . Since the whole numbers appear in a particularly natural way, because they are directly connected to the process of counting , it is natural to ask how any number is related to them. For example , it is simply a quotient of two whole numbers (a “fraction”) - purely algebraically, it can be used as a solution to the whole number equation ${\ displaystyle \ dotsc, -3, -2, -1,0,1,2,3, \ dotsc}$${\ displaystyle x}$${\ displaystyle x = - {\ tfrac {4} {5}}}$${\ displaystyle x}$${\ displaystyle x}$

${\ displaystyle 5x + 4 = 0}$

be characterized. Since only a simple one occurs in such an equation , the rational numbers are also referred to as “numbers of degree 1”. If one takes the naturalness of adding and multiplying as a basis, it is logical to see the powers as natural (algebraic) relatives of the original number , as well as sums and differences and products of these numbers. If we succeed in combining the number zero from a finite number of such numbers, a close relationship to the whole numbers is established. For example, we can take the square root of 3, namely , a finite number of elementary algebraic steps to zero we multiply it by yourself, get with it , and pull on the outcome of three, so . Since the number has been multiplied twice, it has at most "degree 2" (in the case of , the degree is indeed equal to 2, since this is not a rational number ). ${\ displaystyle x}$${\ displaystyle x ^ {2}, x ^ {3}, \ dotsc}$${\ displaystyle x}$${\ displaystyle x = {\ sqrt {3}}}$${\ displaystyle x ^ {2} = 3}$${\ displaystyle x ^ {2} -3 = 0}$${\ displaystyle x}$${\ displaystyle x = {\ sqrt {3}}}$

Transcendent numbers are numbers that, after any number of elementary algebraic manipulations, can never be made the number zero. Therefore, from an algebra point of view, they are "invisible" in some ways. An important example of a transcendent number is the circle number . It plays an elementary role geometrically, since it indicates the ratio of the circumference of a circle to its diameter, but from an algebraic point of view it is extremely mysterious. An illustration of this is provided by the idea that a circle has “infinitely many corners” and when the border crosses from very fine corners to the circle (all with algebraically “visible” perimeter) the perimeter increases in degree to ultimately “completely out of the” Algebra to disappear ”. ${\ displaystyle \ pi = 3 {,} 1415926 \ dots}$${\ displaystyle n}$

Although transcendent numbers are so intangible, they are far more numerous than algebraic numbers. This is because the quality of being algebraic is a very exquisite one and comes with far-reaching consequences and structural properties. The idea that the other way round algebraic numbers are “particularly rare” is therefore more obvious. A subjectively observed particularly frequent occurrence of algebraic numbers can be explained by the fact that many phenomena in everyday life and science are based on very elementary and natural processes. In addition, real numbers are greatly simplified in everyday use by, for example, rounding , whereby algebraic questions only have to be answered approximately, if at all. Since even algebraic number theory always works on the basis of strong structures, transcendent numbers play only a limited role in this discipline, despite their “natural frequency”. Questions about transcendent numbers, such as whether a particular number is transcendent and methods of finding it, are extremely difficult and the subject of extensive mathematical research.

## definition

A complex number is called transcendent if it is not an algebraic number, i.e. if it is not a polynomial ${\ displaystyle b}$

${\ displaystyle p (x) = a_ {n} x ^ {n} + a_ {n-1} x ^ {n-1} + a_ {n-2} x ^ {n-2} + \ dotsb + a_ {1} x + a_ {0} \ qquad (n \ geq 1, \, a_ {0}, \ dotsc, a_ {n} \ in \ mathbb {Z}, a_ {n} \ neq 0)}$

exists with . ${\ displaystyle p (b) = 0}$

## Historical development of the concept of transcendence

### Discovery of the concept

The idea of ​​mathematical transcendence emerged very gradually in the course of the 18th century in the considerations of great mathematicians such as Gottfried Wilhelm Leibniz (omnem rationem transcendunt, Latin: You are beyond all reason) and Leonhard Euler , who did not have a strict definition of this term , but were nevertheless certain that there had to be such mathematically “elusive” numbers, of which Euler wrote that they “exceed [...] the effectiveness of algebraic methods”. In 1748 Euler even claimed in his textbook Introductio in Analysin Infinitorum that with positive, rational and natural , which is not a square number, the number is not rational , but is also “no longer irrational” (whereby under “irrational numbers” he refers to today's algebraic numbers understood the mentioned range of numbers). In fact, this assumption of transcendence was confirmed in 1934 as a special case of a result by the Russian mathematician Alexander Ossipowitsch Gelfond and the German mathematician Theodor Schneider . Their evidence differs on essential points. ${\ displaystyle a \ neq 1}$${\ displaystyle b}$${\ displaystyle a ^ {\ sqrt {b}}}$

### First constructions of transcendent numbers

Joseph Liouville was the first to prove the existence of transcendent numbers in 1844 and to provide explicit examples using his constructive proof method. In his work he was able to show that there is a constant for every algebraic number of degree , so that for every rational approximation : ${\ displaystyle x}$${\ displaystyle n \ geq 2}$${\ displaystyle c> 0}$ ${\ displaystyle p / q}$

${\ displaystyle \ left | x - {\ frac {p} {q}} \ right |> {\ frac {c} {q ^ {n}}}}$

applies. It follows that the Liouville constant

${\ displaystyle L = \ sum _ {k = 1} ^ {\ infty} 10 ^ {- k!} = 0 {,} 110001000000000000000001000 \ dots}$

is transcendent.

### Proof of uncountability by Georg Cantor

In 1874 Georg Cantor was not only able to prove the existence of transcendent numbers again, but even show that there are “more” transcendent numbers than algebraic numbers. In contrast to Liouville , Cantor's proof of existence for transcendent numbers did not use any number-theoretic properties of algebraic numbers, but is (from today's perspective) of a purely set- theoretical nature. The mathematically exact formulation of the term ' more ' was certainly the most important result of Cantor's work, because it revolutionized the knowledge of the real number system. However, for a long time his new ideas could not prevail against influential conservative critics such as Leopold Kronecker . Cantor proved that the set of algebraic real numbers (in modern parlance) is countable , while the set of all real numbers is uncountable (infinite, but not countable). It follows easily from this that the set of all transcendent numbers is equal to the set of all real numbers (in particular: also uncountable).

## Uncountability

The amount is uncountable. This means that it is not possible to make a complete list by “counting” transcendent numbers, for example in the form of a list, even if it is infinitely long. A proof can be given indirectly via the countability of the algebraic numbers (for which such a list exists) and the uncountability of the set of all complex numbers. ${\ displaystyle \ mathbb {T} \ subset \ mathbb {C}}$${\ displaystyle z_ {1}, z_ {2}, z_ {3}, \ dotsc}$

For the countability of the algebraic numbers, the idea that a countable listing of lists turns out to be a countable list helps. If you mentally combine the lists and etc., the resulting list will again be a count. This explains why there is a counting of all polynomials with whole coefficients, since these are in the form ${\ displaystyle a_ {1}, a_ {2}, a_ {3}, \ dotsc}$${\ displaystyle b_ {1}, b_ {2}, b_ {3}, \ dotsc}$

${\ displaystyle \ bigoplus _ {n = 0} ^ {\ infty} \ mathbb {Z} x ^ {n}}$

given are. However, if the list of polynomials is countable, so too is the list of their (always at most finitely many) solutions.

This fact can be formulated as follows:

If denotes the set of transcendent numbers and the set of real numbers, then: ${\ displaystyle \ mathbb {T}}$${\ displaystyle \ mathbb {R}}$

${\ displaystyle {\ hbox {card}} \, \ mathbb {T} = {\ hbox {card}} \, \ mathbb {R} = 2 ^ {\ aleph _ {0}}}$

Here is the set theoretical symbol for the power of ; (pronounced “ Aleph zero”) is the set-theoretical symbol for the power of a countably infinite set, especially the set of natural numbers . ${\ displaystyle 2 ^ {\ aleph _ {0}}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle {\ aleph _ {0}}}$${\ displaystyle \ mathbb {N}}$

## Six Exponentials Theorem

The Six Exponentials Theorem makes the following statement: If two are linearly independent complex numbers and three are linearly independent complex numbers, then at least one of the six numbers with and is transcendent. For example, it can be used to show that at least one of the numbers is transcendent. ${\ displaystyle x_ {1}, x_ {2}}$${\ displaystyle \ mathbb {Q}}$${\ displaystyle y_ {1}, y_ {2}, y_ {3}}$${\ displaystyle \ mathbb {Q}}$${\ displaystyle \ mathrm {e} ^ {x_ {m} y_ {n}}}$${\ displaystyle 1 \ leq m \ leq 2}$${\ displaystyle 1 \ leq n \ leq 3}$${\ displaystyle 2 ^ {\ pi}, 3 ^ {\ pi}, 5 ^ {\ pi}}$

The sentence comes from Serge Lang and Kanakanahalli Ramachandra with preparatory work by Carl Ludwig Siegel and Theodor Schneider.

## Schanuel's guess

One of the most far-reaching conjectures in the theory of transcendent numbers is the so-called Schanuel conjecture. This means: If complex numbers are linearly independent over , then is the degree of transcendence of the body ${\ displaystyle \ alpha _ {1}, \ alpha _ {2}, \ dotsc, \ alpha _ {n}}$${\ displaystyle \ mathbb {Q}}$

${\ displaystyle K = \ mathbb {Q} (\ alpha _ {1}, \ alpha _ {2}, \ dotsc, \ alpha _ {n}, \ mathrm {e} ^ {\ alpha _ {1}}, \ dotsc, \ mathrm {e} ^ {\ alpha _ {n}})}$

at least . This means that there must be at least numbers in , so that for a polynomial with variables and rational coefficients: From already follows that the constant must be a null function. ${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle a_ {1}, \ dotsc, a_ {n}}$${\ displaystyle K}$${\ displaystyle f}$${\ displaystyle n}$${\ displaystyle f (a_ {1}, \ dotsc, a_ {n}) = 0}$${\ displaystyle f}$

## Elliptical functions and modular forms

### The Weierstrasse function${\ displaystyle \ wp}$

The function of Weierstrass over a grating is an elliptical (i. E. Double periodic meromorphic ) function, that for each , the differential equation of ${\ displaystyle \ wp}$ ${\ displaystyle L}$ ${\ displaystyle \ wp (z + \ lambda) = \ wp (z)}$${\ displaystyle \ lambda \ in L}$

${\ displaystyle (\ wp ') ^ {2} = 4 \ wp ^ {3} -g_ {2} (L) \ wp -g_ {3} (L)}$

enough. If the Eisenstein series (two complex numbers) belonging to the grid are both algebraic, then the value is transcendent for every algebraic number . This has important consequences for the non-trivial periods to elliptic curves: non-vanishing periods of any elliptic curve with algebraic and are necessarily transcendent. ${\ displaystyle g_ {2} (L), g_ {3} (L)}$${\ displaystyle \ alpha \ notin L}$${\ displaystyle \ wp (\ alpha)}$ ${\ displaystyle y ^ {2} = 4x ^ {3} -g_ {2} x-g_ {3}}$${\ displaystyle g_ {2}}$${\ displaystyle g_ {3}}$

Further it can be shown that if and are algebraic and is some complex number that is not a pole of , at least one of the two numbers and is transcendent. ${\ displaystyle g_ {2} (L)}$${\ displaystyle g_ {3} (L)}$${\ displaystyle z_ {0}}$ ${\ displaystyle \ wp}$${\ displaystyle \ mathrm {e} ^ {z_ {0}}}$${\ displaystyle \ wp (z_ {0})}$

### The -invariant${\ displaystyle j}$

In the case of the -invariants, it is known from Schneider's theorem that for algebraic numbers the function value is algebraic if and only if there is a so-called CM point (where CM stands for complex multiplication ). First of all, this just means that solves a quadratic equation . For example is ${\ displaystyle j}$${\ displaystyle z}$${\ displaystyle j (z)}$ ${\ displaystyle z}$${\ displaystyle z}$ ${\ displaystyle Aw ^ {2} + Bw + C = 0}$

${\ displaystyle j \ left ({\ frac {1+ \ mathrm {i} {\ sqrt {67}}} {2}} \ right) = - 147 \ 197 \ 952 \ 000}$

even a whole number .

## Proofs of transcendence of e and π

The original evidence of the transcendence of and comes from Charles Hermite and Ferdinand von Lindemann, respectively . The evidence is very difficult to understand, however. Over time, however, this evidence has been simplified again and again. The famous mathematician David Hilbert (1862–1943) published a very “elegant” proof in his essay “On the Transcendence of Numbers and ” in 1893 . ${\ displaystyle e}$${\ displaystyle \ pi}$${\ displaystyle e}$${\ displaystyle \ pi}$

## Examples of transcendent numbers

• ${\ displaystyle \ pi = 3 {,} 1415926535897932384626433832795 \ ldots}$
From the transcendence of , which was proven by Carl Louis Ferdinand von Lindemann , follows the insolubility of squaring the circle with a compass and ruler.${\ displaystyle \ pi}$
• ${\ displaystyle e = 2 {,} 7182818284590452353602874713526 \ ldots}$,
the Euler number whose transcendence in 1873 by Charles Hermite could be proven.
• ${\ displaystyle e ^ {a}}$for algebraic . See also Lindemann-Weierstrass theorem .${\ displaystyle a \ neq 0}$
• ${\ displaystyle 2 ^ {\ sqrt {2}}}$. More generally, Gelfond 1934 and Theodor Schneider 1934 were able to show independently of one another using different methods: Is , algebraic, algebraic and irrational, then is a transcendent number. This is a partial solution to Hilbert's seventh problem . This sentence obviously does not apply to the transcendent , since z. B. (see also theorem by Gelfond-Schneider ).${\ displaystyle 0 \ neq a \ neq 1}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle a ^ {b}}$${\ displaystyle b}$${\ displaystyle 3 ^ {\ log _ {3} 2} = 2}$
• Liouville numbers . They result from constructions with better rational approximations to unreal numbers than given by Liouville's theorem . Likewise, examples result from the tightening of Liouville's theorem in Thue-Siegel-Roth's theorem .
• The sine of an algebraic number (see again Lindemann-Weierstrass theorem ).${\ displaystyle \ sin (a)}$${\ displaystyle a \ neq 0}$
• The logarithm of a rational positive number .${\ displaystyle \ ln (a)}$${\ displaystyle a \ neq 1}$
• ${\ displaystyle \ Gamma ({\ tfrac {1} {3}})}$and (see gamma function )${\ displaystyle \ Gamma ({\ tfrac {1} {4}})}$
• ${\ displaystyle \ textstyle \ sum _ {k = 0} ^ {\ infty} 10 ^ {- \ lfloor \ beta ^ {k} \ rfloor}}$, . The bracket here is the Gaussian bracket .${\ displaystyle \ beta> 1}$${\ displaystyle \ lfloor \ cdot \ rfloor}$
• The number , formed by joining the numbers in the decimal system, and similarly formed numbers in place value systems on a basis other than 10 ( Kurt Mahler 1946).${\ displaystyle \ tau = 123456789101112131415 \ cdots}$

## generalization

In the context of general body extensions one also considers elements in which are algebraically or transcendently over . See also algebraic element . ${\ displaystyle L / K}$${\ displaystyle L}$${\ displaystyle K}$

## literature

• Alan Baker : Transcendental number theory. Reprinted edition. Cambridge University Press, London a. a. 1990, ISBN 0-521-39791-X (A demanding standard work that develops profound theorems, but requires profound prior knowledge).
• Peter Bundschuh: Introduction to Number Theory. 4th, revised and updated edition. Springer, Berlin a. a. 1998, ISBN 3-540-64630-2 (Offers an introductory overview on the subject of “transcendent numbers”).
• Naum Iljitsch Feldman , Juri Walentinowitsch Nesterenko : Number Theory IV: Transcendental numbers, (Encyclopaedia of Mathematical Sciences 44), Springer 1997, ISBN 978-3540614678
• David Hilbert : About the transcendence of numbers and . ${\ displaystyle e}$${\ displaystyle \ pi}$In: Mathematical Annals . Vol. 43, No. 2/3, 1893, pp. 216-219, doi : 10.1007 / BF01443645 .
• Arthur Jones, Sidney A. Morris, Kenneth R. Pearson: Abstract Algebra and Famous Impossibilities. Corrected 2nd printing. Springer, New York a. a. 1994, ISBN 0-387-97661-2 (Contains a detailed step-by-step explanation of Lindemann's proof of transcendence for .)${\ displaystyle \ pi).}$
• Kurt Mahler : Lectures on transcendental numbers , Lecture notes in mathematics 566, Springer 1976
• M. Ram Murty , Purusottam Rath: Transcendental Numbers. Springer, New York 2014, ISBN 978-1-4939-0831-8 .
• Oskar Perron : Irrational Numbers (= Göschen's teaching library . Group 1: Pure Mathematics. Vol. 1, ZDB -ID 503797-9 ). de Gruyter, Berlin a. a. 1921.
• Theodor Schneider : Introduction to the transcendent numbers (= the basic teachings of the mathematical sciences in individual representations. Vol. 81, ). Springer, Berlin a. a. 1957.
• Carl Ludwig Siegel : Transcendental numbers , Annals of Mathematical Studies, Princeton UP 1949, German translation Transzendente Numbers , BI university pocket books 1967
• Andrei Borissowitsch Schidlowski : Transcendental numbers (= De Gruyter Studies in Mathematics. Vol. 12). de Gruyter, Berlin a. a. 1989, ISBN 3-11-011568-9 (easier to read than Baker's book, but similarly founded).
• Fridtjof Tönniessen: The secret of the transcendent numbers , Spektrum Akademischer Verlag 2010
• Michel Waldschmidt : Transcendence Methods , Queens Papers in Pure and Applied Mathematics 52, Queens University, Kingston 1979