|
In mathematics, the complex plane is a way of visualising the space of the complex numbers. It can be thought of as a modified cartesian plane, with the real part represented in the x-axis and the imaginary part represented in the y-axis. The x-axis is also called the real axis and the y-axis is called the imaginary axis. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ...
In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i. ...
The complex plane is sometimes called the Argand plane for its use in Argand diagrams. Its creation is generally credited to Jean-Robert Argand, although it was first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel. Jean-Robert Argand (July 18, 1768 - August 13, 1822) was a non-professional mathematician. ...
Caspar Wessel (June 8, 1745 - March 25, 1818) was a Norwegian-Danish mathematician. ...
The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors, and the multiplication of complex numbers can be expressed simply using polar coordinates, where the magnitude of the product is the product of those of the terms, and the angle from the real axis of the product is the sum of those of the terms. Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ...
Addition (or summation) is one of the basic operations of arithmetic. ...
The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ...
In its simplest form, multiplication is the sum of a list of identical numbers. ...
This article describes some of the common coordinate systems that appear in elementary mathematics. ...
Argand diagrams are frequently used to plot the positions of poles and zeros of a function in the complex plane. In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...
In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
Use of the complex plane in control theory In control theory, one use of a complex plane is that known as the 's-plane'. It is used to visualise the roots of the equation describing a system's behaviour (the characteristic equation) graphically. The equation is normally expressed as a polynomial in the Laplace operator 's', hence the name 's' plane. Jump to: navigation, search In engineering and mathematics, control theory deals with the behavior of dynamical systems over time. ...
Jump to: navigation, search In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator (or a hyperbolic operator, when defined on pseudo-Riemannian manifolds), with many applications in mathematics and physics. ...
In addition, a complex plane (not the "s" plane) is used with the Nyquist stability criterion. This is a geometric principle which allows the stability of a control system to be determined from inspection of a Nyquist plot of its frequency-phase response in the complex plane. The Nyquist stability criterion is a unique and powerful method for determining the stability of a closed-loop control system. ...
A Nyquist plot is a graph used in signal processing in which the magnitude and phase of a frequency response are plotted on orthogonal axes. ...
The 'z-plane' is a discrete-time version of the s-plane, where z-transforms are used instead of the Laplace transformation. In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...
See also The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
A constellation diagram is a representation of a digital modulation scheme in the complex plane. ...
Jump to: navigation, search In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by: The lower limit of is short notation to mean and assures the inclusion of the entire...
In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...
External links |
Feeds |
Contact