Complex Number

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This Demonstration shows the basic elements of representations of a complex number in a two-dimensional Cartesian coordinate system or in a polar coordinate system.

Contributed by: Štefan Porubský (October 2008)
Open content licensed under CC BY-NC-SA



A complex number can be visually represented using a two-dimensional Cartesian coordinate system as an ordered pair of real numbers on the complex plane. The representation of a complex number in terms of its Cartesian coordinates in the form , where is the imaginary unit, is called the algebraic form of that complex number. The coordinate is called the real part and the imaginary part of the complex number, respectively. The absolute value (or the modulus) of a complex number is defined by .

Alternatively to the Cartesian system, the polar coordinate system may used. In polar coordinates , the radial coordinate and the angular coordinate , where , is called the argument (or angle) of the complex number . A complex number is then represented in the trigonometric form , or in the exponential form . In technical applications the argument is often chosen from the interval and it is called phase. The corresponding exponential form is then called phasor form.

(Author was supported by project 1ET200300529 of the program Information Society of the National Research Program of the Czech Republic and by the Institutional Research Plan AV0Z10300504; the Demonstration was submitted 2008-10-01.)

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