# Complex Plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.

The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates – the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.

The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745–1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.

### Other articles related to "complex plane, complex, plane":

Complex Plane - Terminology
... While the terminology "complex plane" is historically accepted, the object could be more appropriately named "complex line" as it is a 1-dimensional complex vector space ...
Wiener–Hopf Method
... In general, the method works by exploiting the complex-analytical properties of transformed functions ... conditions are transformed and these transforms are used to define a pair of complex functions (typically denoted with '+' and '−' subscripts) which are respectively analytic in the upper and ... These two functions will also coincide on some region of the complex plane, typically, a thin strip containing the real line ...
Mittag-Leffler Star - Examples
... The Mittag-Leffler star of the complex exponential function defined in a neighborhood of a = 0 is the entire complex plane ... The Mittag-Leffler star of the complex logarithm defined in the neighborhood of point a = 1 is the entire complex plane without the origin and the negative real axis ... In general, given the complex logarithm defined in the neighborhood of a point a ≠ 0 in the complex plane, this function can be extended all the way to infinity on any ray starting at a ...
Hilbert's Twenty-first Problem
... For Riemann–Hilbert factorization problems on the complex plane see Riemann–Hilbert ... of n functions of the variable z, regular throughout the complex z-plane except at the given singular points at these points the functions may become infinite of only finite order ... (systems of) differential equations in question are those defined in the complex plane, less a few points, and with a regular singularity at those ...
Specific Types of Mathematical Diagrams - Argand Diagram
... A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram The complex plane is sometimes called the ... to plot the positions of the poles and zeroes of a function in the complex plane ... The concept of the complex plane allows a geometric interpretation of complex numbers ...

### Famous quotes containing the words plane and/or complex:

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Dalton Trumbo (1905–1976)

It’s a complex fate, being an American, and one of the responsibilities it entails is fighting against a superstitious valuation of Europe.
Henry James (1843–1916)