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Powers of Complex Numbers (DeMoivre's Theorem) definitions

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  • Polar Form
    A representation of complex numbers using a modulus and an angle, often written as r cis θ or r(cos θ + i sin θ).
  • Modulus
    The distance from the origin to the point representing a complex number in the complex plane, denoted as r.
  • Argument
    The angle measured from the positive real axis to the line representing a complex number in polar form, denoted as θ.
  • De Moivre's Theorem
    A shortcut for raising complex numbers in polar form to integer powers by exponentiating the modulus and multiplying the angle.
  • cis Notation
    A shorthand for cos θ + i sin θ, used to simplify expressions of complex numbers in polar form.
  • Root of Unity
    A solution to the equation z^n = 1 in the complex plane, resulting in multiple equally spaced points on the unit circle.
  • Principal Root
    The primary or first solution when extracting roots of a complex number, typically corresponding to k = 0.
  • Multiple Roots
    The phenomenon where extracting roots of a complex number yields several distinct solutions due to periodicity.
  • Periodicity
    The property of complex numbers where adding full rotations (360° or 2π) to the angle results in the same point.
  • Exponent
    The integer or fractional power to which a complex number is raised, affecting both modulus and argument.
  • Radians
    A unit for measuring angles, where a full circle is 2π radians, often used in complex number calculations.
  • Degrees
    A unit for measuring angles, where a full circle is 360°, commonly used in polar form representations.
  • k Value
    An integer parameter used to enumerate all possible roots of a complex number, ranging from 0 to n-1.
  • Complex Plane
    A two-dimensional plane where complex numbers are represented, with the real part on the x-axis and imaginary part on the y-axis.
  • Full Rotation
    An addition of 360° or 2π to an angle, resulting in the same position in the complex plane due to periodicity.