Transcendental number

In mathematics, a transcendental number is a number (possibly a complex number) which is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients.

The most prominent examples of transcendental numbers are π and e. Only a few classes of transcendental numbers are known. This is partly because it can be extremely difficult to show that a given number is transcendental.

However, transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable, but the sets of real and complex numbers are uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental, eg. the square root of 2.

1 History
2 Properties
3 Known transcendental numbers and open problems
4 Sketch of a proof that e is transcendental
5 See also
6 Notes
7 References
8 External links

History

Euler was probably the first person to define transcendental numbers in the modern sense.^[1] The name "transcendentals" comes from Leibniz in his 1682 paper where he proved sin x is not an algebraic function of x.^[2]^[3]
Joseph Liouville first proved the existence of transcendental numbers in 1844,^[4] and in 1851 gave the first decimal examples such as the Liouville constant

$\sum_{k=1}^\infty 10^{-k!} = 0.110001000000000000000001000\ldots$

in which the nth digit after the decimal point is 1 if n is equal to k factorial (i.e., 1, 2, 6, 24, 120, 720, ...., etc.) and 0 otherwise.^[5] Liouville showed that this number is what we now call a Liouville number; this essentially means that it can be too well approximated by rational numbers, better than any algebraic number can. Liouville showed that all Liouville numbers are transcendental.^[6]

Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1761 paper proving the number π is irrational. The first number to be proven transcendental without having been specifically constructed for the purpose was e, by Charles Hermite in 1873. In 1874, Georg Cantor found the countability argument mentioned above establishing the ubiquity of transcendental numbers.

In 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. He first showed that e to any nonzero algebraic power is transcendental, and since e^iπ = −1 is algebraic (see Euler's identity), iπ and therefore π must be transcendental. This approach was generalized by Karl Weierstrass to the Lindemann–Weierstrass theorem. The transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle.

In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number, that is not zero or one, and b is an irrational algebraic number, is a^b necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. 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).^[7]

Properties

The set of transcendental numbers is uncountably infinite. Since the polynomials with integer coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. But Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable; so the set of all transcendental numbers must also be uncountable.

No rational number is transcendental and all real transcendental numbers are irrational. However, some irrational numbers are not transcendental. For example, the square root of 2 is irrational and not transcendental (because it is a solution of the polynomial x² − 2 = 0).

Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument. So, for example, from knowing that π is transcendental, we can immediately deduce that numbers such as 5π, (π − 3)/√2, (√π − √3)⁸ and (π⁵ + 7)^1/7 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, 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 πe must be transcendental. More generally, for any two transcendental numbers a and b, at least one of a + b and ab must be transcendental. To see this, consider the polynomial (x − a) (x − b) = x² − (a + b)x + ab. If (a + b) and ab were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, a and b, 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; however, not all transcendental numbers are Liouville numbers. 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 e, one can show that e is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). Kurt Mahler showed in 1953 that π 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).^[8]

A related class of numbers are closed-form numbers, which may be defined in various ways, including rational numbers (and in some definitions all algebraic numbers), but also allow exponentiation and logarithm.

Known transcendental numbers and open problems

Numbers known to be transcendental:

e^a if a is algebraic and nonzero (by the Lindemann–Weierstrass theorem), and in particular, e itself.
π (by the Lindemann–Weierstrass theorem).
e^π, Gelfond's constant, as well as e^-π/2=i ⁱ (by the Gelfond–Schneider theorem).
a^b where a ≠ 0, 1 is algebraic and b is irrational algebraic (by the Gelfond–Schneider theorem), in particular:
- $2^\sqrt{2}$ , the Gelfond–Schneider constant (Hilbert number),
- iⁱ - raising i (the imaginary square root of -1) to the i power.[3]
sin(a), cos(a) and tan(a), and their multiplicative inverses csc(a), sec(a) and cot(a), for any nonzero algebraic number a (by the Lindemann–Weierstrass theorem).
ln(a) if a is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
Γ(1/3),^[9] Γ(1/4),^[10] and Γ(1/6).^[10]
0.12345678910111213141516..., the Champernowne constant.^[11]
Ω, Chaitin's constant (since it is a non-computable number).
Prouhet–Thue–Morse constant
$\sum_{k=0}^\infty 10^{-\left\lfloor \beta^{k} \right\rfloor};$ where $\beta > 1$ and $\beta\mapsto\lfloor \beta \rfloor$ is the floor function.

Numbers which may or may not be transcendental:

Sums, products, powers, etc. (except for Gelfond's constant) of the number π and the number e: π + e, π − e, π·e, π/e, π^π, e^e, π^e
the Euler–Mascheroni constant γ (which has not even been proven to be irrational)
Catalan's constant, also not known to be irrational
Apéry's constant, ζ(3), and in fact, ζ(2n + 1) for any positive integer n (see Riemann zeta function).
The Feigenbaum constants, $\delta$ and $\alpha$

Conjectures:

Schanuel's conjecture

Sketch of a proof that e is transcendental

The first proof that the base of the natural logarithms, 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 $c_{0},c_{1},\ldots,c_{n},$ satisfying the equation:

$c_{0}+c_{1}e+c_{2}e^{2}+\cdots+c_{n}e^{n}=0$

and such that $c_0$ and $c_n$ are both non-zero.

Depending on the value of n, we specify a sufficiently large positive integer k (to meet our needs later), and multiply both sides of the above equation by $\int^{\infty}_{0}$ , where the notation $\int^{b}_{a}$ will be used in this proof as shorthand for the integral:

$\int^{b}_{a}:=\int^{b}_{a}x^{k}[(x-1)(x-2)\cdots(x-n)]^{k+1}e^{-x}\,dx.$

We have arrived at the equation:

$c_{0}\int^{\infty}_{0}+c_{1}e\int^{\infty}_{0}+\cdots+c_{n}e^{n}\int^{\infty}_{0} = 0$

which can now be written in the form

$P_{1}+P_{2}=0\;$

where

$P_{1}=c_{0}\int^{\infty}_{0}+c_{1}e\int^{\infty}_{1}+c_{2}e^{2}\int^{\infty}_{2}+\cdots+c_{n}e^{n}\int^{\infty}_{n}$

$P_{2}=c_{1}e\int^{1}_{0}+c_{2}e^{2}\int^{2}_{0}+\cdots+c_{n}e^{n}\int^{n}_{0}$

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

$\frac{P_{1}}{k!}$ is a non-zero integer and $\frac{P_{2}}{k!}$ is not.

The fact that $\frac{P_{1}}{k!}$ is a nonzero integer results from the relation

$\int^{\infty}_{0}x^{j}e^{-x}\,dx=j!$

which is valid for any positive integer j and can be proved using integration by parts and mathematical induction.

It is non-zero because for every $i$ satisfying $0<i\le n$ , the integrand in $c_{i}e^{i}\int^{\infty}_{i}$ is $e^{-x}$ times a sum of terms whose lowest power of x is k+1, and it is therefore a product of $(k+1)!$ . Thus, after division by $k!$ , we get zero modulo (k+1) (i.e. a product of (k+1)). However, the integrand in $\int^{\infty}_{0}$ has a term of the form $n! (-1)^{(k+1)}e^{-x}x^k$ and thus ${\frac{1}{k!}}c_{0}\int^{\infty}_{0} = c_{0}n! (-1)^{(k+1)} (mod ~k+1)$ . By choosing $k+1$ which is prime and larger than $n$ and $c_0$ , we get that $\frac{P_{1}}{k!}$ is non-zero modulo (k+1) and is thus non-zero.

To show that

$\left|\frac{P_{2}}{k!}\right|<1$ for sufficiently large k

we construct an auxiliary function $x^{k}[(x-1)(x-2)\cdots(x-n)]^{k+1}e^{-x}$ , noting that it is the product of the functions $[x(x-1)(x-2)\cdots(x-n)]^{k}$ and $(x-1)(x-2)\cdots(x-n)e^{-x}$ . Using upper bounds for $|x(x-1)(x-2)\cdots(x-n)|$ and $|(x-1)(x-2)\cdots(x-n)e^{-x}|$ on the interval [0,n] and employing the fact

$\lim_{k\to\infty}\frac{G^k}{k!}=0$ for every real number G

is then sufficient to finish the proof.

A similar strategy, different from Lindemann's original approach, can be used to show that the number π 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.

For detailed information concerning the proofs of the transcendence of π and e see the references and external links.

Notes

↑ Paul Erdős, Underwood Dudley (November 1983). "Some Remarks and Problems in Number Theory Related to the Work of Euler". Mathematics Magazine 56 (5): 292–298. doi:10.2307/2690369. http://jstor.org/stable/2690369.
↑ Gottfried Wilhelm Leibniz, Karl Immanuel Gerhardt, Georg Heinrich Pertz (1858). Leibnizens mathematische Schriften. 5. A. Asher & Co.. pp. 97–98. [1]
↑ Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer. pp. 74.
↑ Aubrey J. Kempner (October 1916). "On Transcendental Numbers". Transactions of the American Mathematical Society (American Mathematical Society) 17 (4): 476–482. doi:10.2307/1988833. http://jstor.org/stable/1988833.
↑ Weisstein, Eric W. "Liouville's Constant", MathWorld [2]
↑ J. Liouville, "Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques," J. Math. Pures et Appl. 18, 883-885, and 910-911, (1844).
↑ J J O'Connor and E F Robertson: Alan Baker. The MacTutor History of Mathematics archive 1998.
↑ Boris Adamczewski and Yann Bugeaud (March 2005). "On the complexity of algebraic numbers, II. Continued fractions". Acta Mathematica 195 (1): 1–20. doi:10.1007/BF02588048.
↑ Le Lionnais, F. Les nombres remarquables (ISBN 2-7056-1407-9). Paris: Hermann, p. 46, 1979. via Wolfram Mathworld, Transcendental Number
↑ ^10.0 ^10.1 Chudnovsky, G. V. Contributions to the Theory of Transcendental Numbers (ISBN 0-8218-1500-8). Providence, RI: Amer. Math. Soc., 1984. via Wolfram Mathworld, Transcendental Number
↑ K. Mahler (1937). "Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen". Proc. Konin. Neder. Akad. Wet. Ser. A. (40): 421–428.

References

David Hilbert, "Über die Transcendenz der Zahlen $e$ und $\pi$ ", Mathematische Annalen 43:216–219 (1893).
Alan Baker, Transcendental Number Theory, Cambridge University Press, 1975, ISBN 0-521-39791-X.
Peter M Higgins, "Number Story" Copernicus Books, 2008, ISBN 978-84800-000-1.

External links

(English) Proof that $e$ is transcendental
(German) Proof that $e$ is transcendental (PDF)
(German) Proof that $\pi$ is transcendental (PDF)

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