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Using tangent half-angle formulas, it follows immediately that α = sin A = 2r / (1 + r2) and β = cos A = (1 − r2) / (1 + r2) are both rational and that α2 + β2 = 1. Multiplying up by the smallest integer that clears the denominators of α and β recovers the original primitive Pythagorean triple.
If x is rational, it will have two continued fraction representations that are finite, x 1 and x 2, and similarly a rational y will have two representations, y 1 and y 2. The coefficients beyond the last in any of these representations should be interpreted as +∞; and the best rational will be one of z(x 1, y 1), z(x 1, y 2), z(x 2, y 1), or ...
Excel maintains 15 figures in its numbers, but they are not always accurate; mathematically, the bottom line should be the same as the top line, in 'fp-math' the step '1 + 1/9000' leads to a rounding up as the first bit of the 14 bit tail '10111000110010' of the mantissa falling off the table when adding 1 is a '1', this up-rounding is not undone when subtracting the 1 again, since there is no ...
There are three methods for displaying formulas in Wikipedia: raw HTML, HTML with math templates (abbreviated here as { {math}}), and a subset of LaTeX implemented with the HTML markup <math></math> (referred to as LaTeX in this article).
2 + 8x 2 − 1 = 0. Since P 2 (x) < 0 for x = 1 / 9 , and P 2 (x) > 0 for all x > 1 / 8 , the next term in the greedy expansion is 1 / 9 . If x 3 is the remaining fraction after this step of the greedy expansion, it satisfies the equation P 2 (x 3 + 1 / 9 ) = 0, which can again be expanded as a polynomial equation ...
Legendre polynomials occur in the solution of Laplace's equation of the static potential, ∇ 2 Φ(x) = 0, in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle).
Algebraic fraction. In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are and . Algebraic fractions are subject to the same laws as arithmetic fractions. A rational fraction is an algebraic fraction whose numerator and denominator are both polynomials.
Microsoft Excel is a spreadsheet editor developed by Microsoft for Windows, macOS, Android, iOS and iPadOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro programming language called Visual Basic for Applications (VBA). Excel forms part of the Microsoft 365 suite of software.