Complex numbers

Type of physical science: Mathematical methods

Field of study: Algebra

The complex numbers are useful extensions of the real numbers. A complex number is usually written as z = x + iy, where √-1, and x and y are real numbers. It is convenient to represent a complex number as an ordered pair (x,y), a p point in a two-dimensional plane called the complex plane.

Overview

It is universally believed by mathematicians and the users of complex numbers that the words "complex" or "imaginary" should never have been used for these numbers. These numbers are quite simple and also very useful in many real life situations. Nevertheless, now their use is too well established, and the dictionary meanings of the words "complex" and "imaginary" do not apply to the context of numbers they signify.

It is customary and convenient to begin with the root-extracting problem. Extracting square roots of numbers has been a matter of interest in classical arithmetic. In geometry, such problems arise in finding the lengths of diagonals of simple figures such as squares, rectangles, and cubes. Now, the number 4 has two square roots, namely 2 and -2, because when squared, they both give the number 4. A natural question that comes to mind is: What are the square roots of the real number -4? If one looks for a real number for the answer, there is none. In fact, since the square of a real number, positive or negative, is always a positive number, there will be no real number whose square is a negative real number. Thus, square roots of negative numbers cannot be real numbers. Obviously, numbers other than real numbers have to be considered. This was, perhaps, considered a rather complex situation, and therefore the numbers that resolved this difficulty came to be called complex numbers.

The singularly most important number in the study of complex numbers is i = √ - 1. All integral powers of this number take one of the values 1, -1, i, or -i. The set of complex numbers of the form z = x + iy, where x and y are real numbers, includes all the real numbers.

The real numbers are complex numbers when coefficient of i, that is, imaginary part, is zero. Interestingly, the rules of addition and multiplication of real numbers carry over to complex numbers. This reduces calculations with complex numbers to calculations with real numbers.

A complex number is zero if and only if its real and imaginary parts are separately zero.

Also, two complex numbers are equal if and only if the real part of one is equal to the real part of the other and the imaginary part of one is equal to the imaginary part of the other. If two complex numbers are not equal to each other, however, it cannot be said that one is lesser or greater than the other. The order property of real numbers does not hold in complex numbers. The sum, difference, product, and quotient (except when divisor is zero) of two complex numbers is a complex number. Under addition and multiplication laws, the complex numbers satisfy the properties of being a field, an unordered field.

In root-finding problems, complex numbers resolve the problem of finding square roots of all negative numbers, and, in fact, all roots (square, cubic, or any), of all numbers including complex numbers (therefore of positive or negative real numbers) can be found.

Notation i for √-1 has an advantage. It eliminates the type of paradox (√-1)2 = -1 as well as (√-1)( √-1) = (-1)(-1) = √1 = 1.

Classical algebra deals with equations in powers of one or more variables. The simplest of these are algebraic equations of positive integral degrees in a single variable with integral coefficients. The central problem, then, is of finding the roots of algebraic equations. Some of the simplest such as x² + 1 = 0 have no solution in real numbers. Complex numbers help one find all the roots of such an equation of any degree. The fundamental theorem of algebra guarantees that an algebraic equation of positive integral degree has a solution (in the domain of complex numbers).

The complex roots occur in conjugate pairs, and some roots may occur more than once.

In fact, the result of the fundamental theorem of algebra can be strengthened to say that if a root is counted as many times as it occurs, then an algebraic equation of degree n (positive integer) has exactly n roots in the domain of complex numbers. The result of the fundamental theorem of algebra hold for polynomials or algebraic equations with complex coefficients.

The passing from real to the complex number system simplifies the analysis of polynomials and clarifies many of their properties. This, in a sense, is considered to be the original motivation for introducing the complex number system.

Ordered pair representation, such as (x, y) for complex number x + iy, brings in their elegant geometrical characteristics. The point (x, y) is plotted on a two-dimensional plane called complex plane. The two rectangular x- and y-axes are called real and imaginary axes. With these conventions, the arithmetic laws of addition and multiplication of two complex numbers are geometrically explained through Argand diagrams. The addition follows the law of parallelogram for combining vectors. The geometrical multiplication law can be viewed as a method of defining the product of two ordered pairs, something that has no parallel in Cartesian geometry. Multiplication by i amounts to rotation by right angles in the counter-clockwise direction, without a change in magnitude.

Geometric representation in polar coordinates [z = r(cosθ +isinθ)], brings in a function θ called argument of z. The argument, as a function of complex numbers, follows the logarithmic laws of addition and difference [arg(z₁z₂) = arg(z₁) + arg(z₂)]. This is interesting because the argument of a complex number is the only other practical function with these properties. A very elegant result that follows from this representation is Euler's identity. This, in turn, gives a breath-holding relation in mathematical constants e, i, π, and 1, namely, e^iπ = -1.

Geometrical considerations further help represent a complex number in terms of trigonometric functions, sine and cosine of its argument. Such a representation is unique and is very convenient for certain purposes. A useful result called De Moivre's theorem is proved for complex numbers in this form. With the help of this theorem, all the roots of a number (complex in general) can be systematically found. There are many interesting infinite series that can be summed up with De Moivre's theorem. In addition, series expansions of some trigonometric functions can be derived with this theorem.

These definitions and results, particularly Euler's formula, involving trigonometric functions, paved the way for extending the definitions of sine, cosine, and other trigonometric functions as functions of a complex variable. It may be pointed out that it is found interesting, and in many cases easier, in advanced mathematics to analyze and study all important functions in the domain of complex number systems. The complex function theory provides mathematicians a clearer analysis and abundance of new results, which have wider applications in the physical world.

The ease with which algebra of complex numbers is built with the help of operations of real numbers carried over to them makes Euler's notation x + iy convenient. This notation is therefore largely used. Nevertheless, it raises some logical problems. The question that is raised is about the meaning of +. Is it the same as the + that occurs elsewhere in the algebraic development? The ordered-pair approach (x, y), formulated by Sir William Rowan Hamilton, is free of this difficulty and shows that no distinction needs to be made in the two uses of +.

Another representation of complex numbers is in the form of 2 x 2 matrices. The complex number x + iy in the matrix form is written as In this representation, the operations of addition and multiplication are carried out according to the usual rules for matrices.

From complex numbers, a mathematician goes to complex analysis, wherein many mathematical properties of functions of complex variables are studied. Three main avenues in complex analysis are the Cauchy-Riemann equations, theory of complex integration, and Weierstrass' theory of power series. The theory of residues provides a powerful and simple technique of evaluating many useful definite integrals. In the study of complex analysis on conformal mappings, one finds that a nonconstant analytic function maps a domain from one complex plane onto another such that any two smooth intersecting curves from the first plane are mapped into curves, which intersect at the same angle. By means of conformal mapping, problems of fluid flow, electrostatics, and many other areas can be transformed into simpler problems of the same general type.

Applications

Kinematics is the study of possible motions of mechanical systems. The concept of complex numbers is very convenient for describing the motion of a particle in a plane. The position of a moving particle is expressed by a complex variable, in Cartesian or polar coordinates. On differentiation with respect to time, complex velocity with components along the axes in Cartesian form, and radial and transversal, are easily determined. On further differentiation, the complex acceleration or its components along the axes or radial and tangential directions can be directly determined. In polar coordinate representation, Coriolis acceleration, tangential acceleration, centripetal acceleration, and radial acceleration follow in a rather straightforward manner.

Another application in kinematics is the proof of the law of conservation of angular momentum for a particle moving under the influence of a central force. Denoting the complex location of the particle by z, its mass by a real positive number m, and the (complex) force acting on the particle by F, Newton's second law of motion yields that iF is equal to the product of mass and second derivative of z with respect to time.

Now, the angular momentum--that is, one directed orthogonally to the z plane has magnitude that is the product of mass and imaginary part of conjugate complex of z and the derivative of z with respect to time. From that point, it follows that the derivative of the expression for the angular momentum with respect to time is zero. This established that the angular momentum is conserved.

The theory of analytic functions of a complex variable is a mine of effective tools for handling important problems in physics, namely, in heat conduction, diffusion, elasticity, electrostatics, and the flow of electric currents. The chief reason for such wide applications may be seen in the fact that real and imaginary parts of an analytic function satisfy Pierre-Simon Laplace's equation. It is connected to the study of problems in these areas, and all potential functions satisfy it. For the two-dimensional flow of an ideal fluid, the study not only leads to Laplace's equation but also provides a physical interpretation of the two Cauchy-Riemann equations that arise. If the velocity at a point depends only on the position of that point, not on time, then velocity can be represented by a two-dimensional vector. The equation of continuity and the equation of motion together can be formulated. The velocity potential function and the stream function among themselves characterize the flow.

Problems of sources and sinks are also best handled through complex analysis. When the distribution of electrical charge, instead of mass, does not change with time, but instead redistributes itself, the problem has identical formulation. More complicated flows are studied by superposing individual flows resulting from point sources and/or vortex point sources.

Transform calculus is an important tool for many mathematical applications in physical sciences. Fourier transform(s) with kernel as the complex exponential function and Fourier analysis have very wide applications. Signal processing depends directly on Fourier transform considerations.

In general a signal is a function, f(t), of time. Its Fourier transform F(w), a function of frequency variable w, is the frequency spectrum. The duration of signal and the bandwidth duration principle (2σ_t, σ_w ≥ 1, where σ_t and σ_w, are, respectively, the "statistical duration" or standard deviations of times of signal occurrence of the signal, and "statistical [angular] bandwidth") is established by this technique. In terms of the statistical bandwidth, σ_v = σ_w/2π, and the Planck hypothesis E = hv, the uncertainty principle for time t and energy E(σ_tσ_E ≥ h/2), is also similarly derived.

The filtering problem of signaling is also worked out by application of Fourier transforms. The input and output signal of a channel are functions of time, say x(t), and y(t) with their Fourier transforms X(s) and Y(s), respectively. A signal is strictly band-limited if its spectrum (Fourier transform) is zero outside a limited range of frequencies. The transform function, G(s), which characterizes the relation between X(s) and Y(s), and the system function or frequency response of the filter are then studied. The Sampling theorem follows from Fourier analysis of the signals.

Context

Historically, the concept of complex numbers, initially used in a vague way and referred to as imaginary numbers was known to workers in the field of algebra. British mathematician John Wallis (1616-1703) discussed them in his ALGEBRA (1673), written in Latin. In 1797, Norwegian surveyor Caspar Wessel (1745-1818) published a paper on graphic representation of complex numbers. Nevertheless, German mathematician Carl Friedrich Gauss (1777-1855) was the first to give a coherent account of complex numbers and to interpret them as points in a plane. Gauss was also the first to provide a rigorous proof of the fundamental theorem of algebra. The notation i for ≥ -1 was first introduced in 1779 by Swiss mathematician Leonhard Euler.

The development of complex numbers was greatly facilitated by geometric representations. The complex plane provides so much motivation and intuitive insight that it seems direct to introduce the functions of complex variables. Many applications in physics are prompted by such a representation.

Throughout the nineteenth century, the attention of the mathematical world, to a large extent, concentrated on complex function theory. Some of the greatest mathematicians of this century, namely Gauss, Augustin-Louis Cauchy, Carl Gustav Jacob Jacobi, Georg Friedrich Bernhard Riemann, Karl Theodor Wilhelm Weierstrass, Albert Einstein, Henri Poincare, and Niels Hendrik Abel made substantial contributions to this theory. One of the major twentieth century open problems of mathematics, from the twenty-three famous problems of David Hilbert (1900), related to the distribution of zeros of the Riemann's ζ function, is in the field of complex analysis.

Complex analysis occupies central position in pure and applied mathematics. Though it is considered a classical area in mathematics, it has assumed great importance in several different areas of mathematics in modern times. For example, the work of W. Thurston (1982) shows vital importance of Mobius transformations. The subject is directly linked to development of abstract theories of categories and homologies in mathematics of the twentieth century.

Principal terms

ARGUMENT OR PHASE: the angle that the line joining a complex number to the origin of coordinates in the complex plane makes with the x-axis (also called real axis)

COMPLEX CONJUGATE: a number x - iy that is closely related to the complex number z = x + iy (obtained by replacing i by -i)

IMAGINARY PART: the real number y in z = x + iy is called the imaginary part of the complex number z; if the imaginary part is zero, then the number z is a purely real number

IMAGINARY UNIT: the square root of number -1; ≥ -1 is called the imaginary unit; in mathematics and physics, it is usually denoted by the letter i, and in engineering sometimes by j

MODULUS OR ABSOLUTE VALUE: the distance of a complex number from the origin; alternatively, its square is equal to the product of a complex number and its complex conjugate

REAL PART: the real number x in z = x + iy is called the real part of complex number z; if the real part is zero, the number is called imaginary

Bibliography

Ahlfors, Lars V. COMPLEX ANALYSIS. 3d ed. New York: McGraw-Hill, 1979. Geared for mathematicians, this book deals with the subject in a mathematically sound manner using modern terminology and approach.

Churchill, Ruel V., and James W. Brown. COMPLEX VARIABLES AND APPLICATIONS. 4th ed. New York: McGraw-Hill, 1984. This technical book is oriented to applications in physical sciences. In particular, the book considers applications of conformal mapping to steady temperatures, steady temperatures in a half plane, temperatures in a quadrant with part of one boundary insulated, electrostatic potential, potential in a cylindrical space, two-dimensional fluid flow, fluid flow in a channel through a slit, and electrostatic potential about an edge of a conducting plate.

Greenleaf, Frederick P. INTRODUCTION TO COMPLEX VARIABLES. Philadelphia: W. B. Saunders, 1972. Contains a number of good applications. The approach is technical and mathematically sophisticated.

Guillemin, Ernst A. THE MATHEMATICS OF CIRCUIT ANALYSIS. New York: John Wiley & Sons, 1949. Deals with the concepts of complex numbers and functions. Considers many situations where ideas of complex numbers are applied in circuit analysis.

Hille, Einar. ANALYTIC FUNCTION THEORY. 2 vols. 2d ed. New York: Chelsea, 1973. A well-written reference for mathematicians.

Thompson, Milne. THEORETICAL HYDRODYNAMICS. London: Macmillan, 1955. A very extensive book on applications of complex function theory in the field of fluid mechanics.

Complex number x + iy in matrix form

Centrifugal/Centripetal and Coriolis Accelerations