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User:Charles R Greathouse IV/Classifying numbers
As there are over 12,000 cons sequences, it can be useful to list some standard methods of identifying transcendental and algebraic numbers.
- Closure: The algebraic numbers are closed under addition and multiplication. Hence if x and y are algebraic, then x+y and xy are algebraic. Example: A135611.
- Closure: The algebraic numbers are closed under conjugation and the taking of real or imaginary parts. Hence if x is algebraic, so are x̅, Re(x), and Im(x). Example: A210462.
- Closure: If x is transcendental and y is algebraic and nonzero, then x+y and xy are both transcendental. Examples: A019669, A091131.
- Exponentiation: By the Lindemann–Weierstrass theorem, if x is algebraic and nonzero, then exp(x) is transcendental. Example: A274540.
- Logarithm: By the Lindemann–Weierstrass theorem, if y > 1 is algebraic, then log(y) is transcendental. (More generally, if exp(x) is algebraic, then either x = 0 or x is transcendental.) Example: A002390.
- Powers: By the Gelfond–Schneider theorem, if x and y are algebraic, x is not 0 or 1, and y is irrational, then xy is transcendental. Example: A078333.
- If are algebraic and not 0 or 1, and are algebraic, irrational, and linearly independent over the rationals, then by Baker's theorem is transcendental. Example: A220782.
Transcendental functions of algebraic numbers
Define a set ℱ of univariate multi-valued real functions by
Then Niven[1] proves that, if α is a nonzero algebraic number and f is in ℱ, then all values of f(α) are transcendental.
Chaphalkar, Hwang, Lee, & Nam[2] give theorems to generalizes Niven's result.
Theorem 1: Let g and h be univariate polynomials and let f in ℱ. If g(f(x)) = h(x) has a solution g(f(α)) ≠ 0 and h(α) ≠ 0, then α is transcendental.
Theorem 2: Let g, h1, and h2 be univariate polynomials and let f in ℱ. For a given equation h1(x)f(g(x)) = h2(x), if the equation has a non-zero solution α such that f(g(α)) ≠ 0 and h1(α) ≠ 0 ≠ h2(α) with h1(x) and h2(x) are relatively prime, then α is transcendental.
They provide further theorems in[3]
References
- ↑ Ivan Niven, Irrational Numbers, The Carus Mathematical Monographs, Number 12, 1967
- ↑ R. M. Chaphalkar, S. G. Hwang, C. H. Lee, Ki-Bong Nam, New Gelfond-type transcendental numbers (arXiv:2106.04055 [math.NT])
- ↑ Suk-Geun Hwang, Choon Ho Lee, Ki-Bong Nam, Rachel M Chaphalkar, Generalized Lindemann-Weierstrass and Gelfond-Schneider-Baker theorems, arXiv:2212.03418 [math.NT].