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For example, the repeating continued fraction [1;1,1,1,...] is the golden ratio, and the repeating continued fraction [1;2,2,2,...] is the square root of 2. In contrast, the decimal representations of quadratic irrationals are apparently random. The square roots of all (positive) integers that are not perfect squares are quadratic irrationals ...
Symbolic Math Toolbox MathWorks: 1989 2008 2024b: 2024: $3,150 (Commercial), $99 (Student Suite), $700 (Academic), $194 (Home) including price of MATLAB. Proprietary: Provides tools for solving and manipulating symbolic math expressions and performing variable-precision arithmetic. SymPy: Ondřej Čertík 2006 2007 1.13.2: 11 August 2024: Free
[1] [2] [3] The f-number is also known as the focal ratio, f-ratio, or f-stop, and it is key in determining the depth of field, diffraction, and exposure of a photograph. [4] The f-number is dimensionless and is usually expressed using a lower-case hooked f with the format f / N , where N is the f-number.
The field of formal Laurent series over a field k: (()) = [[]] (it is the field of fractions of the formal power series ring [[]]. The function field of an algebraic variety over a field k is lim → k [ U ] {\displaystyle \varinjlim k[U]} where the limit runs over all the coordinate rings k [ U ] of nonempty open subsets U (more ...
Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. [3] Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function.
Notably, Jarkko Kari gave an aperiodic set of Wang tiles based on multiplications by 2 or 2/3 of real numbers encoded by lines of tiles (the encoding is related to Sturmian sequences made as the differences of consecutive elements of Beatty sequences), with the aperiodicity mainly relying on the fact that 2 n /3 m is never equal to 1 for any ...
Based on a proposal by William Kahan and first implemented in the Hewlett-Packard HP-41C calculator in 1979 (referred to under "LN1" in the display, only), some calculators, operating systems (for example Berkeley UNIX 4.3BSD [17]), computer algebra systems and programming languages (for example C99 [18]) provide a special natural logarithm ...
The Fermat numbers satisfy the following recurrence relations: = + = + for n ≥ 1, = + = for n ≥ 2.Each of these relations can be proved by mathematical induction.From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1.