In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Gelfond's constant, named after
Aleksandr Gelfond
Alexander Osipovich Gelfond (russian: Алекса́ндр О́сипович Ге́льфонд; 24 October 1906 – 7 November 1968) was a Soviet mathematician. Gelfond's theorem, also known as the Gelfond-Schneider theorem is named after hi ...
, is , that is, raised to the
power . Like both and , this constant is a
transcendental number
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 rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
. This was first established by Gelfond and may now be considered as an application of the
Gelfond–Schneider theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.
History
It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
Statement
: If ''a'' and ''b'' a ...
, noting that
where is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. Since is algebraic but not rational, is transcendental. The constant was mentioned in
Hilbert's seventh problem
Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers (''Irrationalität und Transzendenz bestimmter Zahlen'').
Statement of the p ...
.
A related constant is , known as the
Gelfond–Schneider constant
The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two:
:2 = ...
which was proved to be a transcendental number by Rodion Kuzmin in 1930.
In 1934, Aleksandr Gelfond and Theodor Schneider independently pr ...
. The related value + is also irrational.
Numerical value
The decimal expansion of Gelfond's constant begins
:
...
Construction
If one defines and
for , then the sequence
converges rapidly to .
Continued fraction expansion
This is based on the digits for the
simple continued fraction:
As given by the integer sequence oeis:A058287, A058287.
Geometric property
The Volume of an n-ball, volume of the ''n''-dimensional ball (or n-ball, ''n''-ball), is given by
where is its radius, and is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
. Any even-dimensional ball has volume
and, summing up all the unit-ball () volumes of even-dimension gives
[Connolly, Francis. University of Notre Dame]
Similar or related constants
Ramanujan's constant
This is known as Ramanujan's constant. It is an application of
Heegner number In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factori ...
s, where 163 is the Heegner number in question.
Similar to , is very close to an integer:
:
...
As it was the Indian mathematician
Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, ...
who first predicted this almost-integer number, it has been named after him, though the number was first discovered by the French mathematician
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
...
in 1859.
The coincidental closeness, to within 0.000 000 000 000 75 of the number is explained by
complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
and the
''q''-expansion of the
j-invariant, specifically:
and,
where is the error term,
which explains why is 0.000 000 000 000 75 below .
(For more detail on this proof, consult the article on
Heegner number In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factori ...
s.)
The number
The decimal expansion of is given by
A018938:
:
...
Despite this being nearly the integer 20, no explanation has been given for this fact and it is believed to be a
mathematical coincidence.
The number
The decimal expansion of is given by
A059850:
:
...
It is not known whether or not this number is transcendental. Note that, by
Gelfond-Schneider theorem, we can only infer definitively that is transcendental if is algebraic and is not rational ( and are both considered
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, also , ).
In the case of , we are only able to prove this number transcendental due to properties of complex exponential forms, where is considered the modulus of the complex number , and the above equivalency given to transform it into , allowing the application of Gelfond-Schneider theorem.
has no such equivalence, and hence, as both and are transcendental, we can make no conclusion about the transcendence of .
The number
As with , it is not known whether is transcendental. Further, no proof exists to show whether or not it is irrational.
The decimal expansion for is given by
A063504:
:
...
The number
Using the
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a posit ...
of the
complex logarithm
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm of a nonzero complex number z, defined to b ...
,
The decimal expansion of is given by
A049006:
:
...
Because of the equivalence, we can use the Gelfond-Schneider theorem to prove that the reciprocal square root of Gelfond's constant is also transcendental:
is both algebraic (a solution to the polynomial ), and not rational, hence is transcendental.
See also
*
Transcendental number
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 rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
*
Transcendental number theory, the study of questions related to transcendental numbers
*
Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality
e^ + 1 = 0
where
: is Euler's number, the base of natural logarithms,
: is the imaginary unit, which by definition satisfies , and
: is pi, the ratio of the circ ...
*
Gelfond–Schneider constant
The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two:
:2 = ...
which was proved to be a transcendental number by Rodion Kuzmin in 1930.
In 1934, Aleksandr Gelfond and Theodor Schneider independently pr ...
References
Further reading
*
Alan Baker and
Gisbert Wüstholz, ''Logarithmic Forms and Diophantine Geometry'', New Mathematical Monographs 9, Cambridge University Press, 2007, {{ISBN, 978-0-521-88268-2
External links
Gelfond's constant at ''MathWorld''
Mathematical constants
Real transcendental numbers