Search Results for ""

Two lengths are called incommensurate or incommensurable if their ratio cannot be expressed as a ratio of whole numbers. Irrational numbers and transcendental numbers are ...

If algebraic integers alpha_1, ..., alpha_n are linearly independent over Q, then e^(alpha_1), ..., e^(alpha_n) are algebraically independent over Q. The ...

An irreducible algebraic integer which has the property that, if it divides the product of two algebraic integers, then it divides at least one of the factors. 1 and -1 are ...

Gelfond's theorem, also called the Gelfond-Schneider theorem, states that a^b is transcendental if 1. a is algebraic !=0,1 and 2. b is algebraic and irrational. This provides ...

Let K be a field, and A a K-algebra. Elements y_1, ..., y_n are algebraically independent over K if the natural surjection K[Y_1,...,Y_n]->K[y_1,...,y_n] is an isomorphism. ...

The Gelfond-Schneider constant is the number 2^(sqrt(2))=2.66514414... (OEIS A007507) that is known to be transcendental by Gelfond's theorem. Both the Gelfand-Schneider ...

Let lambda_1, ..., lambda_n in C be linearly independent over the rationals Q, then Q(lambda_1,...,lambda_n,e^(lambda_1),...,e^(lambda_n)) has transcendence degree at least n ...

The base-2 transcendental number 0.11011011111011011111..._2 (1) (OEIS A014578), where the nth bit is 1 if n is not divisible by 3 and is the complement of the (n/3)th bit if ...

The transcendence degree of Q(pi), sometimes called the transcendental degree, is one because it is generated by one extra element. In contrast, Q(pi,pi^2) (which is the same ...

Van der Corput's constant is given by m = 2sqrt(2)int_0^(sqrt(pi/2-c))cos(x^2+c)dx (1) = 2pi[coscC(phi)-sincS(phi)] (2) = 3.3643175781... (3) (OEIS A143305), where C(x) and ...