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 (called polynomial equations), where a_{i} ∈ ℤ, i.e. integer, with i ∈ {0, 1, 2, ..., m}, m ∈ ℕ (in the following defined without 0), i.e. natural, a_{m} > 0 and a_{0} ≠ 0. Numbers x ∈ ℝ (real) or x ∈ ℂ (complex) that are not algebraic are called transcendental. In the following, the behaviour of m and n towards infinity is treated and let be i, j and k ∈ ℕ.

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), where n resp. n_{0} stands for the maximum absolute value that the a_{i} can take on and m for the maximally permitted degree of the polynomial. From this the following sufficient criterion for transcendence can be deduced: The set of algebraic numbers does not contain any numbers whose absolute value is less than 1/(2^{|ℕ|}|ℕ|^{|ℕ|+2}) or greater than 2^{|ℕ|}|ℕ|^{|ℕ|+2}.

Definition: A complex number is called *finitely algebraic* if the set of the algebraic numbers is defined finite. It is called *finitely transcendental* if it is not finitely algebraic. It is called *k-transcendental (k-algebraic)* if it is transcendental (algebraic) or (and) the k-th dual place (and all succeeding) of its real or (and) imaginary part in binary representation 1 (0).

Thus all finitely algebraic numbers are algebraic and all transcendental numbers finitely transcendental. The inverse does not apply. It may correspond k+1 (because of the sign bit), for example, to the maximally representable binary digits of a number or its mantissa in the computer. Then the k-transcendental numbers are the ones just not (exactly) representable there.

From algebra is known that the sum, difference, product, quotient of two finitely algebraic numbers of degree j and k are finitely algebraic of maximally degree jk, and the 1/j-th power of a finitely algebraic number of degree k is finitely algebraic of maximally degree jk.

(Finitely) transcendental numbers consist therefore especially of an (finitely) algebraic main part and a (finitely) transcendental remainder. One cannot thus simply omit the sequence values a_{i} for great i and for i → ∞ if one examines the (finite) transcendence of a number: They are decisive. (Finitely) transcendental numbers are all numbers that lie between the (finitely) algebraic ones and beyond them. Between two different (finitely) transcendental numbers ((finitely) algebraic numbers of degree k), lying closely enough together, is no (finitely) algebraic number (of degree < k).

Bound proposition for transcendental numbers: Every complex number different from zero, whose absolute value of the real and/or imaginary part is less or equal 1/(|ℕ|+1) or greater or equal |ℕ|+1, is already transcendental.

Proof: If one sets a_{m} = 1 and a_{i} = -|ℕ| for i < m, then the assertion follows from the geometric series, if one still builds the reciprocal. One receives the exact limit values, if one replaces |ℕ|+1 in both cases by |ℕ|+1-µ with µ = ℕ/(|ℕ|+1-µ)^{m}. In the complex case, the statements of the form x = (|ℕ|+1)(1+bi) for b and x ∈ ℝ yield the desired.

Coefficient proposition for (finitely) transcendental numbers: Let be n_{0} ∈ ℕ finite. All normalised irreducible polynomials in that is |a_{i}| > |ℕ| (n_{0}) for at least one a_{i} have only (finitely) transcendental zeros.

Proof: The zeros of normalised irreducible polynomials are pairwise different and uniquely determined. Since they are not (finitely) algebraic, they must be (finitely) transcendental.

Definition: A number a + bi with a, b ∈ ℚ is called *complex-rational*. A number ±b_{1}/b_{2} ± ib_{3}/b_{4} is called *trans-complex-rational*, if it is not complex-rational and b_{2}b_{4} ≠ 0 applies, with at least one b_{k} that emerged from continued addition of one to |ℕ|, and integer b_{k} else with k ∈ {1, 2, 3, 4}. A complex number that satisfies a polynomial equation of degree > |ℕ| with (trans-) integer coefficients is called *trans-algebraic*.

Product and sum proposition for transcendental numbers: All products that cannot be further simplified of a trans-complex-rational number t and a complex algebraic or trans-algebraic number z (her also the sum) as well as sums and products of a complex trans-algebraic and a complex algebraic (complex trans-algebraic) number with coprime polynomial degree are transcendental.

Proof: The simplification is unique, if it also includes minimising the absolute value of the coefficients of the minimal polynomial of z and of each numerator and denominator of the trans-complex-rational number t. If z is complex-rational, tz only satisfies an equation of a minimal polynomial of degree ≤ 2, with at least one coefficient with absolute value > |ℕ|. If z is irrational, tz cannot be zero of the equation of the minimal polynomial of z, since this equation is, after inserting z, not identical with the one in z, what it would have to be. Would be tz zero of another equation of a minimal polynomial, so this polynomial equation in tx could be extended by multiplication with the minimal polynomial in t'x, with t' conjugated to t, to a polynomial equation in x, whose zero would be also z and where the absolute value of at least one coefficient would be > |ℕ|. This is impossible, since z can be only zero of one minimal polynomial. The involvement of a complex trans-algebraic number leads always to a equation of a minimal polynomial with at least the same polynomial degree, if the number is coprime.

Proposition: The irrational quotients of two finite sums of finite powers, multiplied by finite complex-rational numbers, with finite natural exponent of the same (finitely) transcendental number, are all finitely transcendental.

Proof: Inserting into the polynomial equations and suitable multiplying by the conjugated complex yields directly the assertion.

Together with the rational numbers, the trans-(complex-)rational numbers build numerically already complete ℝ (ℂ). Beyond the numerical, the algebraic numbers are characterised only by the fact that they satisfy the equations defined in the first paragraph. This is the reason why they are so difficult to distinguish from the transcendental numbers.

Who wants to examine the transcendence of a number with Liouville's approximation theorem should ensure that the approximating rational numbers are actually still rational and not trans-rational, such as the number 10^{|ℕ|}. Also, who considers prime numbers greater than |ℕ| in transcendence proofs should safely move in the uncountable.

Proposition: The sum of all natural powers of a complex-rational number x that is different from zero and no root of unity, (geometric series) is already transcendental.

Proof: The absolute value of the numerator or denominator of x^{|ℕ|+1} is for |x| < 1 (|x| > 1) less than 2^{-|ℕ|+1} (greater than 2^{|ℕ|-1}). The subtraction of 1 and the division by 1 - x do not change anything of the transcendence.

Proposition: Euler's number e is transcendental.

Proof: From the definition of e via the exponential series follows e = (k|ℕ|+1)/|ℕ|! with k > |ℕ|. Therewith, numerator and denominator of the stated fraction must be > |ℕ|+1, since in the numerator neither |ℕ| nor a prime divisor of k can be cancelled against |ℕ|!. If one defines e as (1+1/|ℕ|)^{|ℕ|}, so the assertion is trivial. Notice that here two different numbers are defined.

Approximation proposition for algebraic numbers: Every irrational algebraic number of degree k can be approximated by an algebraic number of degree j < k with an average error that is in the real case asymptotically equal to π ζ(j+1)/(2 log j |2ℕ|^{j}).

Proof: On the real axis the number of the algebraic numbers increases per degree more circa by the factor 2|ℕ|, without that their area increases also. The error corresponds to the distance of the algebraic numbers among each other. The non-real algebraic numbers lie less dense.

Conclusion: Two different real algebraic numbers have at least the average distance π/(log |ℕ| |2ℕ|^{|ℕ|}). The precise determination of the minimal distance requires solving an infinite non-linear non-convex optimisation problem.

Proposition: The maximal distance of two real algebraic numbers ≠ 0 amounts 1/|ℕ|^{2} - |O(1/|ℕ|^{3})|.

Proof: The real algebraic numbers ≠ 0 are farthest apart beyond ±1. A rational number r > 1 (0 < r < 1) can be better approximated by a real algebraic x that satisfies the polynomial equation |ℕ|/r x^{m} - |ℕ|x^{m-1} = 1 (x^{m} - |ℕ|x = -|ℕ|r). For negative r applies similar. If one wants to approximate 1 by a greater real algebraic x, it must coercively satisfy a polynomial equation with a_{m} = |ℕ|-1 and a_{0} = -|ℕ|. If one sets still a_{1} = -|ℕ| and a_{2} = |ℕ|, follows the assertion, since the maximal distance cannot be further diminished.

By algebraic approximation cannot be determined whether a complex number is algebraic of degree k, but only after criteria from the inserting of this number into the polynomial equations of degree k; for the difference of the complex number and an appropriate algebraic approximation is always transcendental.

The conventional differentiation and integration obliterates, by building the limit, the precise distinction of transcendence and algebraicity. This is problematic, e.g. for the exact determination of zeros. Therefore, the nonstandardanalysis on this homepage goes another (more precise) way.

© 01.10.2009 by Boris Haase

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