A transcendental number is a number that is not a root of any polynomial with integer coefficients. They are the opposite of algebraic numbers, which are numbers that are roots of some integer polynomial. and are the most well-known transcendental numbers.
That is, numbers like and are all algebraic numbers, as they are the roots, or the 0's, of the polynomials
respectively. For instance, for , is a root, because . For these algebraic numbers , there exists some polynomial with integer coefficients where . For transcendental numbers , however, there is no where . For instance is transcendental, so for all possible with integer coefficients..
Transcendental numbers are useful in the study of straightedge-and-compass constructions, particularly in proving the impossibility of squaring the circle (i.e. it proves that it is impossible to construct a square with area equal to the area of any given circle, including , using only a straightedge and a compass, which proves that is transcendental).
Surprisingly, almost all real numbers are transcendental, meaning that a randomly chosen real number will be transcendental with probability 1 (with respect to cardinality). Nonetheless, only a few numbers have been proven transcendental (such as and ), and the vast majority remain unknowns (such as ).
Additionally, an important conjecture in transcendental theory indirectly implies that Euler's identity, the famous equation
is the only nontrivial relation between and . If true, this is quite a remarkable result.
Leibniz was the first to use the term "transcendental" in a 1682 paper, where he proved the function is not an algebraic function of , which roughly means that there are no integer polynomials such that
for all .
However, it was not until 1748 that the term was applied to transcendental numbers when Euler conjectured that if is irrational, it is also transcendental. However, this conjecture went unproved for over a century.
Nearly 100 years later, Liouville demonstrated that transcendental numbers existed, using a constructive proof involving continued fractions. However, his proof was only strong enough to demonstrate specifically crafted numbers (known as Liouville numbers) were transcendental, and in particular was not strong enough to detect the transcendentality of .
Some time later, Cantor demonstrated a deeper result: not only do transcendental numbers exist, but they also comprise almost all real numbers. This was a consequence of a deeper result regarding the uncountability of the real numbers, which also has deep applications to the study of infinities.
In 1955, the Thue-Siegel-Roth theorem (often shortened to Roth's theorem) improved significantly upon Liouville's work, allowing for the discovery of many other transcendental numbers, most notably the Champernowne constant . This work earned Roth the 1958 Fields Medal.
The field received a new boost with the discovery of the Lindemann-Weierstraß theorem, which finally put to rest the question of squaring the circle, showing it was impossible to construct a square with area equal to a given circle using only a straightedge and compass. In particular, the theorem proved that is transcendental, and it was previously established that only algebraic lengths can be made using a straightedge and compass, so the problem was answered in the negative.
The latest major result was the Gelfand-Schneider theorem, which showed that is transcendental for any algebraic where is not and is irrational This answers Hilbert's seventh problem affirmatively, from the famous 23 problems Hilbert proposed at the turn of the century.
Liouville's approach, intuitively speaking, was to argue that algebraic numbers cannot be approximated particularly well by rational numbers, so any number that can be approximated well by rational numbers must be transcendental. More formally, if is an algebraic number with degree meaning the polynomial with the smallest degree that has as a root has degree then the inequality
has only finitely many solutions , where is any positive number.
However, this result is difficult to use in showing a number is transcendental, as it is necessary to demonstrate infinite solutions for every value of . As a result, Liouville's proof was only sufficient to show specially crafted numbers were transcendental, most notably the number
known as Liouville's constant. More generally, i.e. is transcendental for , and this class of numbers is known as Liouville numbers.
These numbers are chosen specifically to satisfy the following property:
For any integer and Liouville number , there exist an infinite number of pairs satisfying the inequality
If proved, this implies is transcendental, as if it were algebraic with degree then there would only be finitely many solutions to , contradicting the fact that there are infinite solutions for any .
The proof makes use of the following lemma:
If is an algebraic number with degree , then there exists a real number such that for all and
Now, if is algebraic, then there exists some such that
for all . But from the above property, there exist integers such that
for any . In particular, setting gives
so choosing a big enough such that ,
for any , which contradicts the original assertion that is algebraic. Hence, must be transcendental.
Cantor demonstrated that transcendental numbers exist in his now-famous diagonal argument, which demonstrated that the real numbers are uncountable. In other words, there is no bijection between the real numbers and the natural numbers, meaning that there are "more" real numbers than there are natural numbers (despite there being an infinite number of both).
Cantor also showed that there is a bijection between the natural numbers and the algebraic numbers, meaning that there are "more" real numbers than there are algebraic numbers. This directly implies that there must be real numbers that are not algebraic, which means that transcendental numbers exist.
In fact, Cantor's argument is stronger than this, since it demonstrates an important result:
Almost all real numbers are transcendental.
In this sense, the phrase "almost all" has a specific meaning: all numbers except a countable set. In particular, if a real number were chosen randomly (the term randomly is used loosely here, as it is impossible to choose a real number uniformly at random, but the intuition suffices), the probability it will be algebraic is 0.
Liouville's theorem was sufficient to show transcendental numbers existed, but only specially crafted ones. Thue was the first to realize the broader application of Liouville's theorem to Diophantine equations if the exponent in the inequality
could be reduced. Thue's theorem, in 1909, improved the bound to
This was later improved again by Siegel, and later Dyson, who both demonstrated a bound of
where big O notation is used. Finally, Roth effectively settled Liouville's work and earned a Fields Medal in the process, by improving the bound to
which is quite a remarkable improvement. This theorem is interchangeably known as the Thue-Siegel-Roth theorem, honoring the progression of improved bounds, and as Roth's theorem after its ultimate discoverer.
It is worth noting the purpose of the in the previous section: the statement fails to remain true when the exponent is reduced to 2. In particular,
there are infinitely many satisfying for any real , whether algebraic or transcendental.
It is clearer to prove a stronger result:
For any integer , there exist such that and
Consider the interval
where there are sub-intervals. Now consider the fractional parts . By the pigeonhole principle, some two of these fractional parts belong to the same interval, so there exist for which and belong to the same interval. But then for some integer , since is within of an integer . As a result,
precisely as desired.
The greatest triumph of transcendental number theory was the Lindemann-Weierstrauss theorem:
If are algebraic numbers that are linearly independent over , then are algebraically independent over , meaning that the set is not the solution to any nontrivial polynomial with rational coefficients.
This helps to discover several more transcendental numbers, most notably and :
and are both transcendental.
Note that is a linearly independent set, so is an algebraically independent set. This implies that is not the solution to any nontrivial polynomial, so is transcendental by definition. This same logic shows that is transcendental for any algebraic .
Now suppose were algebraic. Since is also algebraic, this implies that is algebraic. Hence is transcendental as is transcendental for any algebraic . However, , which is clearly not transcendental, so must be transcendental as desired.
Surprisingly, relatively little is known about transcendental numbers in general. In fact, there is little general strategy available for determining whether a specific number is transcendental, especially when the number is unrelated to the exponential function and the logarithmic function . For example, while some numbers are known to be transcendental:
- Champernowne's constant
- Gelfond's constant
- Gelfond-Schneider constant
- which is different from
the majority are not. For instance, it is unknown whether
- (though it is known that at least one of and is transcendental)
- the Euler-Mascheroni constant
- Apéry's constant (Apéry showed in 1978 that it is irrational)
are transcendental. More specifically, while the Gelfond-Schneider theorem showed that is transcendental for any algebraic (other than the trivial cases and is rational), this is still a countable set of numbers. In fact, it is not known whether many of the constants on the above list are even rational.
Several conjectures have been made regarding these types of numbers, such as Schanuel's conjecture which states that has transcendence degree at least , which generalizes the Lindemann-Weierstrauss theorem above. If proved, it would establish the nature of numbers such as and . In some sense, this also implies Euler's theorem is the only relation between and .
Additionally, as an improvement to Roth's theorem, Lang conjectured that
which is as yet unproved.