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Hilbert's seventh problem concerns the irrationality and transcendence of certain numbers (Irrationalität und Transzendenz bestimmter Zahlen). In its geometric formulation, it asks whether the following statement is provably true: In mathematics, an irrational number is any real number that is not a rational number, i. ...
In mathematics, a transcendental number is any irrational number that is not an algebraic number, i. ...
Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ...
- In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, then the ratio between base and side is always transcendental.
A special case of this problem asks: For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...
In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree...
In mathematics, an irrational number is any real number that is not a rational number, i. ...
In mathematics, a transcendental number is any irrational number that is not an algebraic number, i. ...
- Is ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b?
When b is rational, ab will be algebraic. In mathematics, a transcendental number is any irrational number that is not an algebraic number, i. ...
In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree...
In mathematics, an irrational number is any real number that is not a rational number, i. ...
The special problem was solved by Aleksandr Gelfond in 1934, and refined by Theodor Schneider (1911 - ) in 1935. They proved that ab is transcendental when b is both algebraic and irrational. This result is known as Gelfond's theorem or the Gelfond-Schneider theorem. 1934 was a common year starting on Monday (link will take you to calendar). ...
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1935 was a common year starting on Tuesday (link will take you to calendar). ...
In mathematics, the Gelfond-Schneider theorem is the following statement, originally proved by Aleksandr Gelfond: If is an algebraic number (with and ), and is an irrational algebraic number, then is a transcendental number. ...
From the point of view of generalisations, this is the case
- blog (α) + log(β) = 0
of the general linear form in logarithms.
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