Complex plane

In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal $x$ -axis, called the real axis, is formed by the real numbers, and the vertical $y$ -axis, called the imaginary axis, is formed by the imaginary numbers.

This article is about the geometric representation of complex numbers. For the two-dimensional projective space with complex-number coordinates, see complex projective plane.

The complex plane allows for a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed more 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 or Gauss plane.

Notational conventions[edit]

Complex numbers[edit]

In complex analysis, the complex numbers are customarily represented by the symbol $z$ , which can be separated into its real ( $x$ ) and imaginary ( $y$ ) parts:

$z=x+iy$

for example: $z = 4 + 5 i$ , where $x$ and $y$ are real numbers, and $i$ is the imaginary unit. In this customary notation the complex number $z$ corresponds to the point $(x, y)$ in the Cartesian plane; the point $(x, y)$ can also be represented in polar coordinates with:

${\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta \\r&={\sqrt {x^{2}+y^{2}}}\\\theta &=\arctan {\frac {y}{x}}.\end{aligned}}$

In the Cartesian plane it may be assumed that the range of the arctangent function takes the values $(-π/2, π/2)$ (in radians), and some care must be taken to define the more complete arctangent function for points $(x, y)$ when $x \leq 0$ .^{[note 1]} In the complex plane these polar coordinates take the form

${\begin{aligned}z&=x+iy\\&=|z|\left(\cos \theta +i\sin \theta \right)\\&=|z|e^{i\theta }\end{aligned}}$

where^{[note 2]}

${\begin{aligned}|z|&={\sqrt {x^{2}+y^{2}}}\\\theta &=\arg(z)\\&={\frac {1}{i}}\ln {\frac {z}{|z|}}\\&=-i\ln {\frac {z}{|z|}}.\end{aligned}}$

Here $| z |$ is the absolute value or modulus of the complex number $z$ ; $θ$ , the argument of $z$ , is usually taken on the interval $0 \leq θ < 2 π$ ; and the last equality (to $| z | e iθ$ ) is taken from Euler's formula. Without the constraint on the range of $θ$ , the argument of $z$ is multi-valued, because the complex exponential function is periodic, with period $2 πi$ . Thus, if $θ$ is one value of $arg(z)$ , the other values are given by $arg(z) = θ + 2 nπ$ , where $n$ is any non-zero integer.^[2]

While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers $w$ and $z$ is given by $\Re (w{\overline {z}})$ ; then for a complex number $z$ its absolute value $| z |$ coincides with its Euclidean norm, and its argument $arg(z)$ with the angle turning from 1 to $z$ .

The theory of contour integration comprises a major part of complex analysis. In this context, the direction of travel around a closed curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by $-1$ . By convention the positive direction is counterclockwise. For example, the unit circle is traversed in the positive direction when we start at the point $z = 1$ , then travel up and to the left through the point $z = i$ , then down and to the left through $-1$ , then down and to the right through $- i$ , and finally up and to the right to $z = 1$ , where we started.

Almost all of complex analysis is concerned with complex functions – that is, with functions that map some subset of the complex plane into some other (possibly overlapping, or even identical) subset of the complex plane. Here it is customary to speak of the domain of $f (z)$ as lying in the $z$ -plane, while referring to the range of $f (z)$ as a set of points in the $w$ -plane. In symbols we write

${\begin{aligned}z&=x+iy\\f(z)&=w\\&=u+iv\end{aligned}}$

and often think of the function $f$ as a transformation from the $z$ -plane (with coordinates $(x, y)$ ) into the $w$ -plane (with coordinates $(u, v)$ ).

Complex plane notation[edit]

The complex plane is denoted as $\mathbb {C}$ .

Quadratic spaces[edit]

The complex plane is associated with two distinct quadratic spaces. For a point $z = x + iy$ in the complex plane, the squaring function $z 2$ and the norm-squared $x 2 + y 2$ are both quadratic forms. The former is frequently neglected in the wake of the latter's use in setting a metric on the complex plane. These distinct faces of the complex plane as a quadratic space arise in the construction of algebras over a field with the Cayley–Dickson process. That procedure can be applied to any field, and different results occur for the fields R and C: when R is the take-off field, then C is constructed with the quadratic form $x 2 + y 2$ , but the process can also begin with C and $z 2$ , and that case generates algebras that differ from those derived from R. In any case, the algebras generated are composition algebras; in this case the complex plane is the point set for two distinct composition algebras.

Complex coordinate space

Complex geometry

Complex line

Constellation diagram

which is an extended complex plane

Riemann sphere

s-plane

In-phase and quadrature components

Real line

Flanigan, Francis J. (1983). . Dover. ISBN 0-486-61388-7.

Complex Variables: Harmonic and Analytic Functions

Moretti, Gino (1964). Functions of a Complex Variable. Prentice-Hall.

Wall, H. S. (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company. Reprinted (1973) by Chelsea Publishing Company 0-8284-0207-8.

ISBN

; Watson, G. N. (1927). A Course in Modern Analysis (Fourth ed.). Cambridge University Press.

Whittaker, E. T.

Jean-Robert Argand, "Essai sur une manière de représenter des quantités imaginaires dans les constructions géométriques", 1806, online and analyzed on [for English version, click 'à télécharger']