Use Of Complex Numbers Research Articles

Some Elementary Uses of Complex Numbers

A special infinite series of real numbers which appeared in a comic strip is summed. Some of the techniques used are standard in complex calculus, but are now applied in a different context. Two other basic problems where complex numbers are useful are described. One, the solution of cubics is well known. The other, an extension of Ptolemy’s Theorem on cyclic quadrilaterals to general quadrilaterals, is an old result of Bretschneider, but the use of complex numbers to obtain this result seems to be new.

Basic Electricity Courses Revisited

Teaching basic electricity courses to engineering students
 from other fields is a difficult task. Several factors
 compound the problem such as: large student
 groups, lack of interest, the high level of abstract
 thinking required, the varied topics to be covered and
 the use of complex arithmetics. New approaches have
 been implemented to alleviate the problem. Amongst
 them, avoiding the use of complex numbers significantly
 reduces the student apprehension and passive
 resistance to the topic.

A method for digital processing of bitmap images

A method for processing of graphical information is proposed. The method makes it possible to code contour images with the use of complex numbers unambiguously defined by the image shape. Mapping of a noise bitmap image onto the complex plane is studied. The possibility of solving such recognition problems as object identification and determination of the orientation of a figure in a plane is demonstrated.

Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for wigner functions

Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.

The Wien network: an introduction using a graphical calculator

In teaching a basic circuits course for technicians, a proposed geometrical construction, for which a justification is supplied, avoids the use of complex numbers in the sinusoidal signal analysis of the Wien network.

The stability of rational approximations of analytic semigroups

This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex numbers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups inL p , 1<p<∞, while the sufficient condition in the other characterization is meaningful in the case of nonlinear operators.

Remarks on generators of analytic semigroups

This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex numbers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups inLp, 1<p<∞, while the sufficient condition in the other characterization is meaningful in the case of nonlinear operators.

A new compact formulation of the two-dimensional magnetic interpretation problem

It is shown that two-dimensional magnetic interpretation is greatly simplified, if two components of the anomalous field are measured. Making use of complex numbers, it is possible to solve exactly the position and magnetization (two components) of cylinders and infinitely deep thin plates or the thickness and one position parameter of infinitely deep thick plates with known magnetization. The exact solutions are used to construct interpretation operators for more complicated cases, including the interpretation of thick magnetic plates, finitely deep thin plates and two parallel cylinders or infinitely deep plates. Applied examples using thin plate anomalies show that the operators are suitable for rapid sequential interpretation of long magnetic anomaly profiles. The approximate positions of the anomalous bodies are identified from the results of applying the one-plate operator to the whole profile. The iterative operators may then be used to determine more precisely the values of the various plate parameters.

Complex Numbers and the Lagrangian

A method to determine the Lagrangian with the use of complex numbers is described. The kinetic energy is found from the magnitude of the complex velocity, and the potential energy is found from the real part of the complex displacement. Finally this method may be used to determine generalized forces with Lagrange's method of undetermined multipliers.

On the Use of Complex Numbers in Plane Mechanics

First Page

Position Determination from Radio Bearings

A new, simpler derivation is presented for Stansfield's formulas for the most likely position ion and the distribution of its error when a number of bearings are measured from known locations. The simplification results from the use of complex numbers. Simple results are obtained for the case in which bearings are measured at regular, short intervals along a straight reconnaissance flight path, providing asymptotes for Butterly's curves.

Introduction to Complex Numbers

The purpose of this article is to present an elementary introduction to complex numbers with a glance at some applications. The intent is to present the concepts intuitively, more or less as they probably originated, rather than formally by -a rigorous procedure of definitions and postulates. Inevitably this leads to some repetitions in statements but in an elementary presentttion this is not a pedagogical fault. A brief review of the history of complex numbers is interesting and relevant. The first introduction of imaginary numbers occurred in connection with the solution of cubic equations. During the 16th century Italian mathematicians of the then famous university of Bologna discovered that sometimes correct answers could be obtained moreexpeditiously if they assumed 1) a symbol, i, 2) i2 = -1, and 3) in other respects treated i as an ordinary number. One of QCardano's problems was Divide 10 into two parts whose product is 40. Cardano was intrigued by the answer but at the same time he was very doubtful of the method. For several hundred years mathematicians continued to play -with the concept. Euler (1707-1783) achieved brilliant results by the use of complex numbers but the fundamental principles of their logic was either not deemed important or was completely misunderstood. Struik ways that in the 18th century 'There was experimentation with infinite series, with infinite products, ... With the use of symbols such as 0, OD, 0,j?, ... Much of the work of the leading mathematicians impresses us as wildly enthusiastic experimentation. Wessel, a Danish mathematician, probably furnished the first logical foundation for complex numbers, but his paper 'On the Analytical Representation of Direction, presented in 1797, was unknown to mathematicians until republished one hundred years later. Argand, a Paris accountant, also presented in 1806 a logical foundation but his work appears to have received very little attention. Actually Gauss (17771855) first used the phrase complex number, the symbol i forthe imaginary unit, and introduced mathematicians to the true theory of these numbers. Gauss' work was followed by that of Cauchy (1789-1857) and that of Riemann (1826-1866). From their works has arisen the complex function theory which is basic and indispensable for advanced mathematics, and even for a full comprehension of the more elementary theorems of algebraic analysis. The point to be emphasized is this. More than 200 years passed before mathematicians placed complex numbers upon a firm logical basis.

Analysis of an R-C Oscillator

The equations for an R-C oscillator are derived completely from fundamental relationships without the use of alternating current concepts and without the use of complex numbers. Analyzed in this way, the R-C oscillator constitutes a good subject for an experiment dealing with the equation for simple harmonic motion. Theoretical expressions obtained for the frequency and for the damping coefficient can be verified by experiment, and either positive or negative damping may be had. With commonly available apparatus, the period may be made several seconds long. The oscillations and the growth or decay of amplitude with time can thus be observed directly with a milliammeter biased to read zero at its center. Familiarity with a simple two-stage triode amplifier constitutes the needed background in electronics for this experiment.

Mathematics in engineering — A new policy

ENGINEERING, and particularly electrical engineering, has become a science whose future progress depends largely on the extensive use of mathematics and physics. In the writer's early experience it was not unusual to hear prominent engineers state that the place for an integral sign is the violin. The use of complex numbers in circuit analysis, the introduction of operational and transform methods, and the application of tensor analysis in electrical engineering have brought to our profession the realization that not only the integral sign, but also other mathematical hieroglyphics, are indeed just as essential to the engineer as they are to the mathematician.

On the Use of Complex Numbers

On the Use of Complex Numbers