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  2. Square root of 5 - Wikipedia

    en.wikipedia.org/wiki/Square_root_of_5

    The / diagonal of a half square forms the basis for the geometrical construction of a golden rectangle.. The golden ratio φ is the arithmetic mean of 1 and . [4] The algebraic relationship between , the golden ratio and the conjugate of the golden ratio (Φ = − ⁠ 1 / φ ⁠ = 1 − φ) is expressed in the following formulae:

  3. Casus irreducibilis - Wikipedia

    en.wikipedia.org/wiki/Casus_irreducibilis

    Casus irreducibilis occurs when none of the roots are rational and when all three roots are distinct and real; the case of three distinct real roots occurs if and only if ⁠ q 2 / 4 ⁠ + ⁠ p 3 / 27 ⁠ < 0, in which case Cardano's formula involves first taking the square root of a negative number, which is imaginary, and then taking the ...

  4. Cube rule - Wikipedia

    en.wikipedia.org/wiki/Cube_rule

    Expected fraction of seats won, s vs fraction of votes received, v (solid black) according to the cube rule, with a plot of the seat:vote ratio (dashed red) The cube rule or cube law is an empirical observation regarding elections under the first-past-the-post system. The rule suggests that the party getting the most votes is over-represented ...

  5. Natural logarithm of 2 - Wikipedia

    en.wikipedia.org/wiki/Natural_logarithm_of_2

    In a third layer, the logarithms of rational numbers r = ⁠ a / b ⁠ are computed with ln(r) = ln(a) − ln(b), and logarithms of roots via ln n √ c = ⁠ 1 / n ⁠ ln(c).. The logarithm of 2 is useful in the sense that the powers of 2 are rather densely distributed; finding powers 2 i close to powers b j of other numbers b is comparatively easy, and series representations of ln(b) are ...

  6. Square root of 6 - Wikipedia

    en.wikipedia.org/wiki/Square_root_of_6

    A regular octahedron with an inscribed sphere, illustrating the square root of 6 ratio between edge length and radius Root rectangles illustrate a construction of the square root of 6 An equilateral triangle with circumscribed rectangle and square; the side of the square is (+) /, and the diagonal of the rectangle is the square root of 7.

  7. Generalized continued fraction - Wikipedia

    en.wikipedia.org/wiki/Generalized_continued_fraction

    The roots of this equation are the fixed points of f(z). If the discriminant (c − b) 2 + 4ad is zero the LFT fixes a single point; otherwise it has two fixed points. If ad ≠ bc the LFT is an invertible conformal mapping of the extended complex plane onto itself. In other words, this LFT has an inverse function

  8. Rabdology - Wikipedia

    en.wikipedia.org/wiki/Rabdology

    The rods were used to multiply, divide and even find the square roots and cube roots of numbers. The second device was a promptuary (Latin promptuarium meaning storehouse) and consisted of a large set of strips that could multiply multidigit numbers more easily than the bones. In combination with a table of reciprocals, it could also divide ...

  9. Rational root theorem - Wikipedia

    en.wikipedia.org/wiki/Rational_root_theorem

    If the rational root test finds no rational solutions, then the only way to express the solutions algebraically uses cube roots. But if the test finds a rational solution r, then factoring out (x – r) leaves a quadratic polynomial whose two roots, found with the quadratic formula, are the remaining two roots of the cubic, avoiding cube roots.

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