In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. It follows that all transcendental numbers are irrational. However, not all irrational numbers are transcendental; √2 (the square root of 2) is irrational, but is a solution of the polynomial x² − 2 = 0.

The set of all transcendental numbers is uncountable. The proof is simple: Since the polynomials with integer coefficients are countable, and since each such polynomial has a finite number of zeroes, the set of algebraic numbers is countable. But the reals are uncountable; so the set of all transcendental numbers must also be uncountable. In a very real sense, then, there are many more transcendental numbers than algebraic ones. However, only a few classes of transcendental numbers are known and proving that a given number is transcendental can be extremely difficult.

The existence of transcendental numbers was first proved in 1844 by Joseph Liouville, who exhibited examples, including the Liouville constant:

\sum_k=1^\infty 10^-k! = 0.110001000000000000000001000..

in which the nth digit after the decimal point is 1 if n is a factorial (i.e., 1, 2, 6, 24, 120, 720, .., etc.) and 0 otherwise. The first number to be proved transcendental without having been specifically constructed to achieve this was e, by Charles Hermite in 1873. In 1882, Ferdinand von

See also Lindemann-Weierstrass theorem.

Here is a list of some numbers known to be transcendental:

• e^a if a is algebraic and nonzero. In particular, e itself is transcendental.

• π

• e^π Gelfond's constant

• 2√2, the Gelfond-Schneider constant, or more generally a^b where a ≠ 0,1 is algebraic and b is algebraic but not rational (Gelfond-Schneider theorem and Hilbert's seventh problem).

• sin(1)

• ln(a) if a is positive, rational and ≠ 1

• Γ(1/3), Γ(1/4), and Γ(1/6) (see gamma function).

• Ω, Chaitin's constant.

• 2.536027081689339.., Herkommer number

• $\backslash sum\_k=0^\backslash infty\; 10^-\backslash lfloor\; \backslash beta^k\; \backslash rfloor;\backslash qquad\; \backslash beta\; >\; 1\backslash ;\; ,$

where $\backslash beta\backslash mapsto\backslash lfloor\; \backslash beta\; \backslash rfloor$ is the floor function. For example if β = 2 then this number is 0.11010001000000010000000000000001000…

Any non-constant algebraic function of a single transcendental number is also transcendental. However, an algebraic function of several transcendental numbers may be algebraic if they are not algebraically independent: π and 1−π are both transcendental, but π+(1−π)=1 is obviously not. It is unknown whether π+e, for example, is transcendental, though at least one of π+e and π e must be transcendental. More generally, for any two transcendental numbers a and b, at least one of a+b and a b must be transcendental. To see this, consider the polynomial (x−a) (x−b) = x² − (a+b) x + a b. If (a+b) and a b were both

The discovery of transcendental numbers allowed the proof of the impossibility of several ancient geometric problems involving ruler-and-compass construction; the most famous one, squaring the circle, is impossible because π is transcendental.

Proof that $e$ is transcendental

The first proof that $e$ is transcendental dates from 1873. We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. The idea is the following:

Assume, for purpose of finding a contradiction, that $e$ is algebraic. Then there exists a finite set of integer coefficients $c\_0,c\_1,\backslash ldots,c\_n,$ not all equal to zero, satisfying the equation:

$c\_0+c\_1e+c\_2e^2+\backslash ldots+c\_ne^n=0$

Depending on the value of n, we specify a sufficently large positive integer k (to meet our needs later), and multiply both sides of the above equation by $\backslash int^\backslash infty\_0$, where the notation $\backslash int^b\_a$ will be used in this proof as shorthand for the integral:

$\backslash int^b\_a:=\backslash int^b\_az^k^k+1e^-zdz$.

Note that

$z^k^k+1e^-z$ is the product of the functions $^k$ and $(z-1)(z-2)\backslash ldots(z-n)e^-z.$

We have arrived at the equation:

$c\_0\backslash int^\backslash infty\_0+c\_1e\backslash int^\backslash infty\_0+\backslash ldots+c\_ne^n\backslash int^\backslash infty\_0$ = 0

which can now be written in the form $P\_1+P\_2=0$ where

$P\_1=c\_0\backslash int^\backslash infty\_0+c\_1e\backslash int^\backslash infty\_1+c\_2e^2\backslash int^\backslash infty\_2+\backslash ldots+c\_ne^n\backslash int^\backslash infty\_n$

$P\_2=c\_1e\backslash int^1\_0+c\_2e^2\backslash int^2\_0+\backslash ldots+c\_ne^n\backslash int^n\_0$

The plan of attack now is to show that for k sufficiently large, the above relations are impossible to satisfy because

$\backslash fracP\_1k!$ is a non-zero integer and $\backslash fracP\_2k!$ is not.

The fact that $\backslash fracP\_1k!$ is a nonzero integer results from the relation:

$\backslash int^\backslash infty\_0x^ke^-x=k!$

Showing that

for sufficiently large k can be done with, amongst other things, some straightforward estimates.

A similar strategy can be used to show that the number $\backslash pi$ is transcendental. Besides the gamma-function and some estimates as in the proof for $e$, facts about symmetric polynomials play a vital role in the proof.

The original proof of the transcedence of $\backslash pi$ first showed that $e$ to any algebraic power is transcedental, and since $e^i\backslash pi\; =\; -1$ is algebraic, $i\backslash pi$ and therefore $\backslash pi$ must be transcendental.

For detailed information concerning the proofs of the transcendence of $\backslash pi$ and $e$ see the references and external links.

References

• David Hilbert, "Über die Transcendenz der Zahlen $e$ und $\backslash pi$", Mathematische Annalen 43:216–219 (1893).
• Michael Spivak, Calculus. New York, Amsterdam: W. A. Benjamin, Inc. (1967).

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