Transcendental numbers

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Constant problem

Given an expression involving known constants, integration in finite terms, computation of limits, etc., determine if the expression is equal to zero. The constant problem, sometimes also called the identity problem (Richardson 1968) is a very difficult unsolved problem in transcendental number theory. However, it is known that the problem is undecidable if the expression involves oscillatory functions such as sine. However, the Ferguson-Forcade algorithm is a practical algorithm for determining if there exist integers for given real numbers such thator else establishing bounds within which no relation can exist (Bailey 1988).

Six exponentials theorem

Let and be two sets of complex numbers linearly independent over the rationals. Then at least one ofis transcendental (Waldschmidt 1979, p. 3.5). This theorem is due to Siegel, Schneider, Lang, and Ramachandra. The corresponding statement obtained by replacing with is called the four exponentials conjecture and remains unproven.

Shidlovskii theorem

Let , ..., for be a set of E-functions that (1) form a solution of the system of differential equationsfor and , ..., and (2) are algebraically independent over . Then for all , where denotes the set of algebraic numbers with and distinct from singularities of the differential equations, the numbers , ..., are algebraically independent (Nesterenko 1999).

Gelfond's theorem

Gelfond's theorem, also called the Gelfond-Schneider theorem, states that is transcendental if 1. is algebraic and 2. is algebraic and irrational. This provides a partial solution to the seventh of Hilbert'sproblems. It was proved independently by Gelfond (1934ab) and Schneider (1934ab).This establishes the transcendence of Gelfond's constant (since ) and the Gelfond-Schneider constant .Gelfond's theorem is implied by Schanuel's conjecture(Chow 1999).

Schanuel's conjecture

Let , ..., be linearly independent over the rationals , thenhas transcendence degree at least over . Schanuel's conjecture implies the Lindemann-Weierstrass theorem and Gelfond's theorem. If the conjecture is true, then it follows that and are algebraically independent. Macintyre (1991) proved that the truth of Schanuel's conjecture also guarantees that there are no unexpected exponential-algebraic relations on the integers (Marker 1996).At present, a proof of Schanuel's conjecture seems out of reach (Chow 1999).

Liouville's constant

Liouville's constant, sometimes also called Liouville's number, is the real number defined by(OEIS A012245). Liouville's constant is a decimal fraction with a 1 in each decimal place corresponding to a factorial , and zeros everywhere else. Liouville (1844) constructed an infinite class of transcendental numbers using continued fractions, but the above number was the first decimal constant to be proven transcendental (Liouville 1850). However, Cantor subsequently proved that "almost all" real numbers are in fact transcendental.A recurrence plot of the binary digits is illustratedabove.Liouville's constant nearly satisfieswhich has solution 0.1100009999... (OEIS A093409), but plugging into this equation gives instead of 0.Liouville's constant has continued fraction [0, 9, 11, 99, 1, 10, 9, 999999999999, 1, 8, 10, 1, 99, 11, 9, 999999999999999999999999999999999999999999999999999999999999999999999999,..

Radical integer

A radical integer is a number obtained by closing the integers under addition, multiplication, subtraction, and root extraction. An example of such a number is . The radical integers are a subring of the algebraic integers.There exist cubic algebraic integers which are not radical integers, namely those which can't be expressed in terms of radicals. R. Schroeppel (pers. comm., May 11, 1997) proved that these are the only ones; i.e., if an algebraic integer can be expressed in terms of radicals, then it can be done so without using division.

Gelfond's constant

The constant that Gelfond's theorem established to be transcendental seems to lack a generally accepted name. As a result, in this work, it will be dubbed Gelfond's constant. Both the Gelfond-Schneider constant and Gelfond's constant were singled out in the 7th of Hilbert's problems as examples of numbers whose transcendence was an open problem (Wells 1986, p. 45).Gelfond's constant has the numerical value(1)(OEIS A039661) and simplecontinued fraction(2)(OEIS A058287).Its digits can be computed efficiently using the iteration(3)with , and then plugging in to(4)(Borwein and Bailey 2003, p. 137).

Liouville number

A Liouville number is a transcendental number which has very close rational number approximations. An irrational number is called a Liouville number if, for each , there exist integers and such thatNote that the first inequality is true by definition, since it follows immediately from the fact that is irrational and hence cannot be equal to for any values of and .Liouville's constant is an example of a Liouville number and is sometimes called "the" Liouville number or "Liouville's number" (Wells 1986, p. 26). Mahler (1953) proved that is not a Liouville number.

Four exponentials conjecture

Let and be two sets of complex numbers linearly independent over the rationals. Then the four exponential conjecture posits that at least one ofis transcendental (Waldschmidt 1979, p. 3.5). The corresponding statement obtained by replacing with has been proven and is known as the six exponentials theorem.

Transcendental number

A transcendental number is a (possibly complex) number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree. Every real transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one.A complex number can be tested to see if it is transcendental using the Wolfram Language command Not[Element[x, Algebraics]].Transcendental numbers are important in the history of mathematics because their investigation provided the first proof that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was, in fact, insoluble. Specifically, in order for a number to be produced by a geometric construction using the ancient Greek rules, it must be either rational or a very special kind of algebraic number known as a Euclidean number. Because the number is transcendental, the construction..

Algebraically independent

Let be a field, and a -algebra. Elements , ..., are algebraically independent over if the natural surjection is an isomorphism. In other words, there are no polynomial relations with coefficients in .

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