What is "complex number"

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.

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

2. Involving many parts; complicated; intricate.

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

   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.

n. (context complex analysis English) A number of the form ''a + bi'', where ''a'' and ''b'' are real numbers and ''i'' denotes the imaginary unit

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

A complex number is a number that can be expressed in the form , where and are real numbers and is the imaginary unit, that satisfies the equation . In this expression, is the and is the of the complex number.

Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.

As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century.