complex number
    n 1: (mathematics) a number of the form a+bi where a and b are
         real numbers and i is the square root of -1 [syn: complex
         number, complex quantity, imaginary number,
         imaginary]

Complex \Com"plex\ (k[o^]m"pl[e^]ks), a. [L. complexus, p. p. of
   complecti to entwine around, comprise; com- + plectere to
   twist, akin to plicare to fold. See Plait, n.]
   1. Composed of two or more parts; composite; not simple; as,
      a complex being; a complex idea.
      [1913 Webster]

            Ideas thus made up of several simple ones put
            together, I call complex; such as beauty, gratitude,
            a man, an army, the universe.         --Locke.
      [1913 Webster]

   2. Involving many parts; complicated; intricate.
      [1913 Webster]

            When the actual motions of the heavens are
            calculated in the best possible way, the process is
            difficult and complex.                --Whewell.
      [1913 Webster]

   Complex fraction. See Fraction.

   Complex number (Math.), in the theory of numbers, an
      expression of the form a + b[root]-1, when a and b are
      ordinary integers.

   Syn: See Intricate.
        [1913 Webster]

complex number

    A number of the form x+iy where i is the square
   root of -1, and x and y are real numbers, known as the
   "real" and "imaginary" part.  Complex numbers can be plotted
   as points on a two-dimensional plane, known as an Argand
   diagram, where x and y are the Cartesian coordinates.

   An alternative, polar notation, expresses a complex number
   as (r e^it) where e is the base of natural logarithms, and r
   and t are real numbers, known as the magnitude and phase.  The
   two forms are related:

   	r e^it = r cos(t) + i r sin(t)
   	       = x + i y
   where
   	x = r cos(t)
   	y = r sin(t)

   All solutions of any polynomial equation can be expressed as
   complex numbers.  This is the so-called Fundamental Theorem
   of Algebra, first proved by Cauchy.

   Complex numbers are useful in many fields of physics, such as
   electromagnetism because they are a useful way of representing
   a magnitude and phase as a single quantity.

   (1995-04-10)