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A typical problem in this area of mathematics is to work out whether a given number is transcendental. Cantor used a cardinality argument to show that there are only countably many algebraic numbers, and hence almost all numbers are transcendental.
A transcendental number is a (possibly complex) number that is not the root of any integer polynomial. Every real transcendental number must also be irrational, since a rational number is the root of an integer polynomial of degree one. [17] The set of transcendental numbers is uncountably infinite.
In computing, a roundoff error, [1] also called rounding error, [2] is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. [3]
Accurate rounding of transcendental mathematical functions is difficult because the number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance. This problem is known as "the table-maker's dilemma".
The theorem is also known variously as the Hermite–Lindemann theorem and the Hermite–Lindemann–Weierstrass theorem.Charles Hermite first proved the simpler theorem where the α i exponents are required to be rational integers and linear independence is only assured over the rational integers, [4] [5] a result sometimes referred to as Hermite's theorem. [6]
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction, multiplication, and division (without the need of taking limits).
Pages in category "Transcendental numbers" The following 16 pages are in this category, out of 16 total. This list may not reflect recent changes. ...
Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers ( Irrationalität und Transzendenz bestimmter Zahlen ).
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