Natural numbers are called the numbers 0, 1, 2, 3, ... (often also without the 0). Integer numbers are the numbers ..., -3, -2, -1, 0, 1, 2, 3, ... Rational numbers are represented by fractions with integer numerator and natural denominator. Real numbers are constructed from the rational numbers, complex from the real ones through extension by the imaginary unit. The set of natural numbers is denoted with ℕ, the set of integer numbers with ℤ, the set of rational numbers with ℚ, the set of real numbers with ℝ and the set of complex numbers with ℂ.

The sum a_{m}x^{m} + a_{m-1}x^{m-1} + ... + a_{1}x + a_{0} is called polynomial. Here m ∈ ℕ is the degree of the polynomial if a_{m} ≠ 0, and the a_{i} ∈ ℤ with i ∈ {0, 1, 2, ..., m} are called coefficients (of the polynomial). The complex numbers x that let the sum become zero (so-called zeros of the polynomial) are called algebraic. For m = 1 this is true exactly for all x ∈ ℚ. In the special case a_{m} = 1 one speaks of algebraic integers. For m = 1 this is exactly true for all x ∈ ℤ.

Real numbers that are not rational are called irrational. Complex numbers x that let no polynomial become zero are called transcendental. Concerning their evidence, one finds a sufficient criterion under "Transcendental Numbers". In the following, the behaviour for n tending to infinity is considered. The number of elements of a set M will be denoted with |M|.

The assertion that |ℝ| should equal 2^{|ℕ|} states that all real numbers can be developed into dual numbers. Against this on the one hand speaks that many rational numbers can be represented only approximately as dual numbers and much more that also real numbers may be sums of greater powers than 2^{n}. For example, the number ⅓ is only approximately included as 0.01010101... and must be distinguished from it.

In the following, the degree of the polynomial is m > 0. The number of algebraic numbers corresponds to the number of zeros of normalised irreducible polynomials. These are not reducible, i.e. they cannot be divided into products of polynomials, the greatest common divisor gcd of its coefficients is 1 and it is a_{m} > 0. In the following, n stands for the maximal absolute value that the a_{i} can take on and m for the maximally permitted degree of the polynomial.

The number of the algebraic numbers (of polynomial degree m and thus generally) is asymptotically equal to z(m)2^{m}n^{m+1}/ζ(m+1) + O(z(m)2^{m}n^{m}log n), where ζ stands for the Riemann zeta function and z(m) is the (average) number of the zeros of the polynomial. In the complex case, z(m) = m applies after the fundamental theorem of algebra.

This results with the aid of a polynomial long division. To determine the number of reducible polynomials multiply a left polynomial factor with coefficients whose absolute value is maximally one successively with two and check how large the coefficients of a right polynomial factor may maximally be when building the products. Sum up the corresponding frequencies of the built products and show the proposition.

Furthermore, z(m) is in the real case for m towards infinity equal to 2/π log m + O(1) after a theorem by Mark Kac (On the average number of real roots of a random algebraic equation, Bull Amer. Soc. 49 (1943) 314-20). The factor 1/ζ(m+1) ensures the elimination of polynomials with gcd(a_{0}, a_{1}, ... , a_{m}) ≠ 1. Generally, one will set m = n. It is not worth to calculate the values of the Riemann zeta function for odd natural arguments exactly, since because of ζ(m) = 1 + 2^{-m} + O(3^{-m}) it does not matter whether one looks up the Bernoulli numbers for small arguments, or the function values themselves, if one assumes that these require similar computational effort for odd as for even natural numbers, and because the accuracy of the approximate formula grows rapidly for greater m.

For m = 1 one obtains 12n^{2}/π^{2} + O(n log n) rational solutions. For m = 2 one obtains 4n^{3}/ζ(3) + O(n^{2}log n) real solutions, since a real polynomial of degree 2 has two zeros with probability ½. For a_{m} = 1 < m one obtains z(m)2^{m}n^{m}/ζ(m) + O(z(m)2^{m}n^{m-1}log n) algebraic integer solutions. The said above about z(m) here also applies.

The algebraic numbers are dense in ℂ, the rational ones are dense in the real algebraic numbers. I.e. between two different complex (real algebraic) numbers is at least one algebraic (rational) number. The set of algebraic numbers is a field and thus closed regarding addition, multiplication and inversion. I.e. the set of algebraic numbers is not left by these operations. In this set are all algebraic numbers whose absolute value is greater than 1/(2^{n}n^{n+2}) or less than 2^{n}n^{n+2}. The number of the set of the complex (real) numbers includes so many elements as one permits. These can be created, among other techniques, by continued exponentiation.

The determination of the number of elements of any constructed infinite set must consider exactly its construction before it can be related to the number of natural numbers. This should be, due to its simple construction, used as basis. Without knowing the construction of a set, their number cannot be (uniquely) determined.

If there are several construction options, the most plausible should be used, i.e. it should represent infinity in the best possible way, in the sense of differentiation. Since this requires a value judgement, it has not to be uniquely determined. If one does not agree on a construction option, despite rational reasoning, so the calculated number of elements of the set is to state with its construction.

Mathematics is not value-free. There can be always stated (rational) reasons why one procedure is to choose over the other one.

© 24.05.2009 by Boris Haase

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