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 x^y is transcendental. Example: A078333.
• If $x_{1},\ldots ,x_{n}$ are algebraic and not 0 or 1, and $y_{1},\ldots ,y_{n}$ are algebraic, irrational, and linearly independent over the rationals, then by Baker's theorem $x_{1}^{y_{1}}x_{2}^{y_{2}}\cdots x_{n}^{y_{n}}$ is transcendental. Example: A220782.

Transcendental functions of algebraic numbers

Define a set ℱ of univariate multi-valued real functions by

${\mathcal {F}}=\{\exp(x),\log(x),\sin(x),\cos(x),\tan(x),\csc(x),\sec(x),\cot(x),\sinh(x),\cosh(x),\tanh(x),\coth(x)\}.$

Then Niven 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 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