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  2. nth root - Wikipedia

    en.wikipedia.org/wiki/Nth_root

    The four 4th roots of −1, none of which are real The three 3rd roots of −1, one of which is a negative real. An n th root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth power is x:

  3. Root of unity - Wikipedia

    en.wikipedia.org/wiki/Root_of_unity

    Geometric representation of the 2nd to 6th root of a general complex number in polar form. For the nth root of unity, set r = 1 and φ = 0. The principal root is in black. An n th root of unity, where n is a positive integer, is a number z satisfying the equation [1] [2] =

  4. Tetration - Wikipedia

    en.wikipedia.org/wiki/Tetration

    Analogously, the inverses of tetration are often called the super-root, and the super-logarithm (In fact, all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function =, the two inverses are the cube super-root of y and the super-logarithm base y of x.

  5. Quotient group - Wikipedia

    en.wikipedia.org/wiki/Quotient_group

    The twelfth roots of unity, which are points on the complex unit circle, form a multiplicative abelian group ⁠ ⁠, shown on the picture on the right as colored balls with the number at each point giving its complex argument. Consider its subgroup made of the fourth roots of unity, shown as red balls. This normal subgroup splits the group ...

  6. Nested radical - Wikipedia

    en.wikipedia.org/wiki/Nested_radical

    In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to be any specific complex cube root of the complex radicand, and by defining the second cube root to be the complex conjugate of the first one.

  7. Lill's method - Wikipedia

    en.wikipedia.org/wiki/Lill's_method

    A quadratic with two real roots, for example, will have exactly two angles that satisfy the above conditions. For complex roots, one must also find a series of similar triangles, but with the vertices of the root path displaced from the polynomial path by a distance equal to the imaginary part of the root. In this case, the root path will not ...

  8. Laguerre's method - Wikipedia

    en.wikipedia.org/wiki/Laguerre's_method

    Laguerre's method may even converge to a complex root of the polynomial, because the radicand of the square root may be of a negative number, in the formula for the correction, , given above – manageable so long as complex numbers can be conveniently accommodated for the calculation. This may be considered an advantage or a liability ...

  9. Descartes' rule of signs - Wikipedia

    en.wikipedia.org/wiki/Descartes'_rule_of_signs

    Any nth degree polynomial has exactly n roots in the complex plane, if counted according to multiplicity. So if f(x) is a polynomial with real coefficients which does not have a root at 0 (that is a polynomial with a nonzero constant term) then the minimum number of nonreal roots is equal to (+),

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