Algebraic numbers are real or complex numbers x that satisfy the equations a_{m}x^{m} + a_{m-1}x^{m-1} + ... + a_{1}x + a_{0} = 0, where a_{i} ∈ ℤ, i.e. are integer, with i ∈ {0, 1, 2, ..., m}, m ∈ ℕ \ {0}, i.e. is natural, a_{m} > 0 and a_{0} ≠ 0. Numbers x ∈ ℝ or x ∈ ℂ that are not algebraic are called transcendental. In the following the behaviour of m and n towards infinity is treated.

In the set theory is shown that the number of the complex algebraic numbers is asymptotically equal to m2^{m}n^{m+1}/ζ(m+1) + O(m2^{m}n^{m}log n) and thus the number of the real algebraic numbers is asymptotically equal to log m 2^{m+1}/π n^{m+1}/ζ(m+1) + O(log m 2^{m}n^{m}log n), where n stands for the maximum absolute values that the a_{i} can take on and m for the maximally permitted degree of the polynomial. In general, one will set m = n. From this the following sufficient criterion for transcendence can be deduced:

A complex number, generated by iteration, is transcendental, if it's main part is generated by addition, multiplication or inversion and the remainder is transcendental or also generated this way and its absolute value is less than 1/(2^{n}n^{n+2}).

Proof of the criterion: If the remainder is less than 1/(2^{n}n^{n+2}), the algebraic number field is left, so the iterated number is transcendental.

Proposition: Every complex number different from zero whose absolute value is less than 1/n or greater than n is already transcendental.

Proof: Define a real series f(1/n) with 0 < f(1/n) < 1. Insertion of f(1/n)/n resp. n/f(1/n) in all polynomials of degree 1 to m yields the proposition (regard orders and rate the a_{i} appropriately).

Main theorem of the theory of the transcendental numbers: Exactly every echt convergent (i.e. especially not becoming constant) infinite iteration is transcendental.

Proof: The remainder falls short of every finite value and the convergent iteration is infinite. Because of the closure of the algebraic number field numbers, generated by addition, multiplication or inversion, become transcendental only by an infinite iteration process.

Note: Infinite iteration processes need an infinite length of time. L can let emerge the result at one stroke by selecting from zis infinite potential, what is not possible in a finite world. Strictly speaking transcendental numbers are no more complex or real, but emerge by adjunction of the infinite.

Definition: Let ∞ be adjoined to the real and complex numbers.

The geometric progression teaches that one cannot just omit zero sequences at the position ∞, if one examines the transcendence of a number. It applies rather the

Proposition: The sum of an algebraic number a and of the value z(∞) ≠ 0 of a zero sequence z(n) is transcendental.

Proof: Inserting of a + z(∞) in a_{m}x^{m} + a_{m-1}x^{m-1} + ... + a_{1}x + a_{0} = 0 yields a polynomial in z(∞) with algebraic coefficients. They cannot all vanish because of a variable substitution. Let i be the smallest index with a_{i} different from 0 and j be the next higher one. Then there exists a representation b_{i} + z(∞) b_{j} (...) = 0. Since z(∞) is arbitrarily small and b_{i} is fixed, follows the proposition.

2) Easier: z(∞) falls short of every finite value. So one leaves the field of algebraic numbers, and it follows the proposition.

Trivial: Sum and product of two transcendental numbers can be both transcendental and also algebraic. Therefore, finite iterations are to examine step by step and number for number.

© 11.01.2009 by Boris Haase

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