(mod , a prime).

The late writers (from 1884) on the general theory include Weierstrass, Schwarz, Richard Dedekind, Otto Hölder, Berloty, Henri Poincaré, Eduard Study, and Macfarlane.

The formally correct definition using pairs of real numbers was given in the 19th century.

Definition

Formally we may define complex numbers as ordered pairs of real numbers (a, b) together with the operations:

So defined, the complex numbers form a field, the complex number field, denoted by C (or in blackboard bold).

We identify the real number a with the complex number (a, 0), and in this way the field of real numbers R becomes a subfield of C. The imaginary unit i is the complex number (0,1).

In C, we have:

• additive identity ("zero"): (0,0)
• multiplicative identity ("one"): (1,0)
• additive inverse of (a,b): (−a,−b)
• multiplicative inverse of non-zero (a,b):

C could also be defined as the topological closure of algebraic numbers and the algebraic closure of R.

Geometry

A complex number can also be viewed as a point or a position vector on the two dimensional Cartesian coordinate system. This representation is sometimes called an Argand diagram. In the figure, we have

The latter expression is sometimes shorthanded as r cis φ, where r = |z| is called the absolute value of z and φ = arg(z) is called the complex argument of z. However, Euler's formula states that e^{i φ} = cisφ. The exponential form gives us a better insight than the shorthand rcisφ, which is almost never used in serious mathematical articles. By simple trigonometric identities, we see that

and that

Now the addition of two complex numbers is just the vector addition of two vectors, and the multiplication with a fixed complex number can be seen as a simultaneous rotation and stretching.

Multiplication with i corresponds to a counter clockwise rotation by 90 degrees. The geometric content of the equation i² = -1 is that a sequence of two 90 degree rotation results in a 180 degree rotation. Even the fact (-1) · (-1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.

Absolute value, conjugation and distance

Recall that the absolute value (or modulus or magnitude) of a complex number z = r e^iφ is defined as |z| = r. Algebraically, if z = a + ib, then |z| = √(a² + b² ).

One can check readily that the absolute value has three important properties:

for all complex numbers z and w. By defining the distance function d(z, w) = |z - w| we turn the complex numbers into a metric space and we can therefore talk about limits and continuity. The addition, subtraction, multiplication and division of complex numbers are then continuous operations. Unless anything else is said, this is always the metric being used on the complex numbers.

The complex conjugate of the complex number z = a + ib is defined to be a - ib, written as or z^*. As seen in the figure, is the "reflection" of z about the real axis. The following can be checked:

  if and only if z is real

  if z is non-zero

The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.

That conjugate commutes with all the algebraic operations (and many functions; e.g. ) is rooted in the ambiguity in choice of i (-1 has two square roots); note, however, that conjugate is not differentiable (see holomorphic).

The complex argument of z=re^iφ is φ. Note that the complex argument is unique modulo 2π.

Matrix representation of complex numbers

While usually not useful, alternative representations of complex field can give some insight into their nature. One particularly elegant representation interprets every complex number as 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form

with real numbers a and b. The sum and product of two such matrices is again of this form. Every non-zero such matrix is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as

which suggests that we should identify the real number 1 with the matrix

and the imaginary unit i with

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to -1.

The absolute value of a complex number expressed as a matrix is equal to the square root of the determinant of that matrix. If the matrix is viewed as a transformation of a plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z.

If the matrix elements are themselves complex numbers, than the resulting algebra is that of the quaternions. In this way, the matrix representation can be seen as a way of expressing the Cayley-Dickson construction of algebras.

Some properties

Real vector space

C is a two-dimensional real vector space. Unlike the reals, complex numbers cannot be ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field.

Solutions of polynomial equations

A root of the polynomial p is a complex number z such that p(z) = 0. A most striking result is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the Fundamental Theorem of Algebra, and shows that the complex numbers are an algebraically closed field.

Indeed, the complex number field is the algebraic closure of the real number field. It can be identified as the quotient ring of the polynomial ring R[X] by the ideal generated by the polynomial X² + 1:

This is indeed a field because X² + 1 is irreducible. The image of X in this quotient ring becomes the imaginary unit i.

Algebraic characterization

The field C is (up to field isomorphism) characterized by the following three facts:

• its characteristic is 0
• its transcendence degree over the prime field is the cardinality of the continuum
• it is algebraically closed

Consequently, C contains many proper subfields which are isomorphic to C.

Complex analysis

The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

Applications

Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.

In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are

• in the right half plane, it will be unstable,
• all in the left half plane, it will be stable,
• on the imaginary axis, it will be marginally stable.

If a system has zeros in the right half plane, it is a nonminimum phase system.

Signal analysis

Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value |z| is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency.

If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as the real part of complex valued functions of the form

where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.

In electrical engineering, this is done for varying voltages and currentss. The treatment of resistors, capacitors and inductors can then be unified by introducing imaginary frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents.)

Improper integrals

In applied fields, the use of complex analysis is often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this, see methods of contour integration.

Quantum mechanics

The complex number field is also of utmost importance in quantum mechanics since the underlying theory is built on (infinite dimensional) Hilbert spaces over C.

Relativity

In Special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary.

Applied mathematics

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = e^rt.

Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in 2d.

Fractals

Certain fractals are plotted in the complex plane e.g. Mandelbrot set and Julia set.

See also

• Complex geometry
• De Moivre's formula
• Euler's identity
• Hypercomplex number
• Leonhard Euler
• Local field
• Phasor
• Quaternions
• Split-complex number

Further reading

• An Imaginary Tale, by Paul J. Nahin; Princeton University Press; ISBN 0691027951 (hardcover, 1998). A gentle introduction to the history of complex numbers and the beginnings of complex analysis.

External links

• John and Betty's Journey Through Complex Numbers
• Complex Number from MathWorld
• SOS Math - Complex Variables
• Windows calculator that supports complex numbers

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