Transcendental Numbers

Transcendental numbers are never is irrational, but is a solution of the polynomial ''x''² − 2 = 0.

The set of all transcendental numbers is , and since each such polynomial has a finite number of Zeroes , the set of algebraic numbers is countable. But Cantor's Diagonal Argument establishes that the reals (and therefore also the complex numbers) 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.

HISTORY

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\ldots

in which the ''n''th digit after the decimal point is 1 if ''n'' is a Factorial (i.e., 1, 2, 6, 24, 120, 720, ...., etc.) and 0 otherwise. Liouville showed that this number is what we now call a Liouville Number ; this essentially means that it can be particularly well approximated by Rational Number s. Liouville showed that all Liouville numbers are transcendental.

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 argument described 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 algebraic power is transcendental, and since

e^{i\pi} = -1

is algebraic (see Euler's Identity ),

i\pi

and therefore

\pi

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 problems involving Compass And Straightedge construction, including the most famous one, Squaring The Circle .

In 1900, . This work was extended by Alan Baker in the 1960s.

KNOWN TRANSCENDENTAL NUMBERS AND OPEN PROBLEMS

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

'' E ^a'' if a is algebraic and nonzero (a consequence of the Lindemann-Weierstrass Theorem ). In particular, ''e'' itself is transcendental.

π .

''e''^π Gelfond's Constant . (A consequence of the Gelfond-Schneider Theorem .)

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 (a), Cos (a) and Tan (a) for any nonzero rational number a.

Ln (a) if a is positive, rational and ≠ 1

Γ(1/3), Γ(1/4), and Γ(1/6) (see Gamma Function ).

the Champernowne Constant is an example of a transcendental number that is Normal in base 10.

Ω, is transcendental (since all algebraic numbers are computable).

2.536027081689339..., Herkommer Number

Prouhet-Thue-Morse Constant

$\sum_{k=0}^\infty 10^{-\lfloor \beta^{k}
floor};\qquad \beta > 1\; ,$

\beta\mapsto\lfloor \beta 
floor

Floor Function

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 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 value when applied to transcendental numbers if these numbers are not , this would imply that the roots of the polynomial, which happen to be ''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 was transcendental.

Numbers for which it is unknown whether they are transcendental or not include

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

ζ(3), Apéry's Constant

All Liouville numbers are transcendental, however not all transcendental numbers are Liouville numbers. Any Liouville number must have unbounded terms in its Continued Fraction expression, and so using a counting argument one can show that there exist transcendental numbers which are not Liouville. Using the explicit continued fraction expansion of ''e'', one can show that ''e'' is not a Liouville number. 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).

PROOF SKETCH THAT <MATH>E</MATH> 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},\ldots,c_{n},

satisfying the equation:

:

c_{0}+c_{1}e+c_{2}e^{2}+\ldots+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 sufficently 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}

{Link without Title} ^{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

:

rac{P_{1}}{k!}

is a non-zero integer and

rac{P_{2}}{k!}

is not.

The fact that

rac{P_{1}}{k!}

is a nonzero integer results from the relation

:

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

which is valid for any positive integer ''j'' and can be proved using Integration By Parts and Mathematical Induction .

To show that

Using Upper Bounds For <math>x(x-1)(x-2)\cdots(x-n)</math> And <math>(x-1)(x-2)\cdots(x-n)e^{-x}</math> On The

"http://wwwinformationdelightinfo/encyclopedia/entry/interval_(mathematics)" class="copylinks">Interval {Link without Title} and employing the fact

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