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 ...
, a transcendental number is a number that is not
algebraic
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
—that is, not the
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of a non-zero
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
of finite degree with
rational coefficients. The best known transcendental numbers are
and .
Though only a few classes of transcendental numbers are known—partly because it can be extremely difficult to show that a given number is transcendental—transcendental numbers are not rare. Indeed,
almost all
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathema ...
real and complex numbers are transcendental, since the algebraic numbers comprise a
countable set, while the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
real numbers and the set of
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 are both
uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal nu ...
s, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are
irrational numbers, since all rational numbers are algebraic.
The
converse is not true: not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping rational, algebraic non-rational and transcendental real numbers.
[ For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation . The golden ratio (denoted or ) is another irrational number that is not transcendental, as it is a root of the polynomial equation . The quality of a number being transcendental is called transcendence.
]
History
The name "transcendental" comes from the Latin ''transcendĕre'' 'to climb over or beyond, surmount', and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that is not an algebraic function of . Euler, in the 18th century, was probably the first person to define transcendental ''numbers'' in the modern sense.
Johann Heinrich Lambert conjectured that and were both transcendental numbers in his 1768 paper proving the number is irrational, and proposed a tentative sketch of a proof of 's transcendence.
Joseph Liouville first proved the existence of transcendental numbers in 1844,[.] and in 1851 gave the first decimal examples such as the Liouville constant
:
in which the th digit after the decimal point is if is equal to ( factorial) for some and otherwise. In other words, the th digit of this number is 1 only if is one of the numbers , etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s than can any irrational algebraic number, and this class of numbers are called Liouville numbers, named in his honour. Liouville showed that all Liouville numbers are transcendental.
The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was , by 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 1873.
In 1874, Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a new method for constructing transcendental numbers. Although this was already implied by his proof of the countability of the algebraic numbers, Cantor also published a construction that proves there are as many transcendental numbers as there are real numbers. Cantor's work established the ubiquity of transcendental numbers.
In 1882, Ferdinand von Lindemann published the first complete proof of the transcendence of . He first proved that is transcendental if is a non-zero algebraic number. Then, since is algebraic (see 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 ...
), must be transcendental. But since is algebraic, therefore must be transcendental. This approach was generalized by Karl Weierstrass to what is now known as the Lindemann–Weierstrass theorem. The transcendence of allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficul ...
.
In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If is an algebraic number that is not zero or one, and is an irrational algebraic number, is necessarily transcendental? The affirmative answer was provided in 1934 by 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 ...
. 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).
Properties
A transcendental number is a (possibly complex) number that is not the root of any integer polynomial. Every real transcendental number must also be irrational, since a rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
is the root of an integer polynomial of degree one. The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, it is impossible for both subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s to be countable. This makes the transcendental numbers uncountable.
No rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
is transcendental and all real transcendental numbers are irrational. The irrational numbers contain all the real transcendental numbers and a subset of the algebraic numbers, including the quadratic irrationals and other forms of algebraic irrationals.
Applying any non-constant single-variable algebraic function to a transcendental argument yields a transcendental value. For example, from knowing that is transcendental, it can be immediately deduced that numbers such as , and 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. For example, and are both transcendental, but is obviously not. It is unknown whether , for example, is transcendental, though at least one of and must be transcendental. More generally, for any two transcendental numbers and , at least one of and must be transcendental. To see this, consider the polynomial . If and were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
, this would imply that the roots of the polynomial, and , must be algebraic. But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental.
The non-computable numbers are a strict subset of the transcendental numbers.
All Liouville numbers are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its continued fraction expansion. 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 , one can show that is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). Kurt Mahler showed in 1953 that 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).
Numbers proven to be transcendental
Numbers proven to be transcendental:
* if is algebraic
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
and nonzero (by the Lindemann–Weierstrass theorem).
* (by the Lindemann–Weierstrass theorem).
* , Gelfond's constant, as well as (by 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 ...
).
* where is algebraic but not 0 or 1, and is irrational algebraic (by the Gelfond–Schneider theorem), in particular:
::, 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 ...
(or Hilbert number)
*, , , , , and , and their hyperbolic counterparts, for any nonzero algebraic number , expressed in radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s (by the Lindemann–Weierstrass theorem).
*The fixed point of the cosine function (also referred to as the Dottie number
The Dottie number is the unique real fixed point of the cosine function.
In mathematics, the Dottie number is a constant that is the unique real root of the equation
: \cos x = x ,
where the argument of \cos is in radians. The decimal expa ...
) – the unique real solution to the equation , where is in radians (by the Lindemann–Weierstrass theorem).
* if is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
* if and are positive integers not both powers of the same integer (by the Gelfond–Schneider theorem).
* The Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
, its first derivative, and the quotient are transcendental when ''ν'' is rational and ''x'' is algebraic and nonzero, and all nonzero roots of and are transcendental when ''ν'' is rational.
* if is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann–Weierstrass theorem), in particular: the omega constant
The omega constant is a mathematical constant defined as the unique real number that satisfies the equation
:\Omega e^\Omega = 1.
It is the value of , where is Lambert's function. The name is derived from the alternate name for Lambert's fu ...
* , the square super-root of any natural number is either an integer or transcendental (by the Gelfond–Schneider theorem)
* , ,[ via Wolfram Mathworld]
Transcendental Number
/ref> and . The numbers