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.

Read more about Complex Plane: Notational Conventions, Stereographic Projections, Cutting The Plane, Gluing The Cut Plane Back Together, Use of The Complex Plane in Control Theory, Other Meanings of "complex Plane", Terminology

### 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 ...

... 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 ...

... 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 ...

... 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 ...

... 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:

“We’ve got to figure these things a little bit different than most people. Y’know, there’s something about going out in a *plane* that beats any other way.... A guy that washes out at the controls of his own ship, well, he goes down doing the thing that he loved the best. It seems to me that that’s a very special way to die.”

—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)