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Complex number

The complex numbers are an extension of the real numbers: the real number line is enlarged to get the complex number plane. The complex numbers contain a number i, the imaginary unit, with i2= -1. Every complex number can be represented in the form x+iy, where x and y are real numbers called the real part and the imaginary part of the complex number respectively.

The sum and product of two complex numbers are:

(a+ib) + (c+id) = (a+c) + i(b+d)
(a+ib) · (c+id) = ac-bd + i (bc+ad)

Complex numbers were first introduced in connection with explicit formulas for the roots of cubic[?] polynomials. See the section History below.

Table of contents

Definition

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

  • (a, b) + (c, d) = (a + c, b + d)
  • (a, b) · (c, d) = (ac - bd, bc + ad)
So defined, the complex numbers form a field, the complex number field, denoted by C (or <math>\mathbb{C}</math> 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).

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

z = x + iy = r (cos φ + i sin φ).
The latter expression is sometimes shorthanded as r cis φ, where r is called the absolute value of z and φ is called the complex argument of z. By simple trigonometric identities, we see that
r1 cis φ1 · r2 cis φ2 = r1r2 cis (φ12);
r1 cis φ1 / r2 cis φ2 = r1 / r2 cis (φ12);
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.

Euler's formula states that ei φ = cisφ. The exponential form gives us a better insight then the shorthand rcisφ, which is almost never used in serious mathematical articles.

Absolute value, conjugation and distance

Recall that the absolute value (or modulus or magnitude) of a complex number z = r e 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:

|z + w| ≤ |z| + |w|
|z w| = |z| |w|
|z / w| = |z| / |w|
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 <math>\bar{z}</math> or z*. As seen in the figure, <math>\bar{z}</math> is the "reflection" of z about the real axis. The following can be checked:

<math>\overline{z+w} = \bar{z} + \bar{w}</math>
<math>\overline{zw} = \bar{z}\bar{w}</math>
<math>\overline{(z/w)} = \bar{z}/\bar{w}</math>
<math>\bar{\bar{z}}=z</math>
<math>\bar{z}=z</math> if and only if z is real
<math>|z|=|\bar{z}|</math>
<math>|z|^2 = z\bar{z}</math>
<math>z^{-1} = \bar{z}|z|^{-2}</math> 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.

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

Other properties

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:

C = R[X] / (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.

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.

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 2x2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form

<math>
\begin{pmatrix} a&&-b\\ b&&a \end{pmatrix} </math>

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

<math>
\begin{pmatrix} a&&-b\\

b&&a \end{pmatrix}

a \begin{pmatrix} 1&&0\\ 0&&1 \end{pmatrix} + b \begin{pmatrix} 0&&-1\\ 1&&0 \end{pmatrix} </math> which suggests that we should identify the real number 1 with the matrix
<math>
\begin{pmatrix} 1&&0\\ 0&&1 \end{pmatrix} </math> and the imaginary unit i with
<math>
\begin{pmatrix} 0&&-1\\ 1&&0 \end{pmatrix} </math> 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. Applications

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

f(t)

z eiωt 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 currents. 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.)

The residue theorem of complex analysis is often used in applied fields to compute certain improper integrals.

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.

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

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) = ert.

History

The earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians (see Tartaglia, Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in the 17th century and was meant to be derogatory. The existence of complex numbers was not completely accepted until the above mentioned geometrical interpretation had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss. The formally correct definition using pairs of real numbers was given in the 19th century.

See also

quaternions, complex geometry, local fields, vectors, phasors[?], Leonhard Euler, the most remarkable formula in the world, Hypercomplex 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.



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