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Complex number From Wikipedia, the free encyclopedia Jump to: navigation, search See also: Complex plane A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagramIn mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:[1]

Every complex number can be written in the form a + bi, where a and b are real numbers called the real part and the imaginary part of the complex number, respectively.

Complex numbers are a field, and thus have addition, subtraction, multiplication, and division operations. These operations extend the corresponding operations on real numbers, although with a number of additional elegant and useful properties, e.g., negative real numbers can be obtained by squaring complex (imaginary) numbers.

Complex numbers were first conceived and defined by the Italian mathematician Girolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations.[2] The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, it is always possible to find solutions to polynomial equations of degree one or higher.

The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.[3] A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.

Complex numbers are used in many different fields including applications in engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial and complex Lie algebra.

Contents [hide] 1 Definitions 1.1 Notation 1.2 Formal development 1.3 Equality 1.4 Operations 1.5 Elementary functions 2 The field of complex numbers 2.1 The complex plane 2.2 Absolute value, conjugation and distance 2.3 Geometric interpretation of the operations