Complex Numbers and their Applications

✅ Paper Type: Free Essay	✅ Subject: Mathematics
✅ Wordcount: 2931 words	✅ Published: 29th Jul 2021

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INTRODUCTION

A complex number is a number comprising area land imaginary part. It can be written in the form a+ib, where a and b are real numbers, and i is the standard imaginary unit with the property i2=-1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication.

HISTORY OF COMPLEX NUMBERS:

Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them “fictitious”, during his attempts to find solutions to cubic equations. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

The rules for addition, subtraction and multiplication of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton.

COMPLEX NUMBER INTERPRETATION:

A number in the form of x+iy where x and y are real numbers and i = -1 is called a complex number.

Let z = x+iy

X is called real part of z and is denoted by R (z)

Y is called imaginary part of z and is denoted by I (z)

CONJUGATE OF A COMPLEX NUMBER:

A pair of complex numbers x+iy and x-iy are said to be conjugate of each other.

PROPERTIES OF COMPLEX NUMBERS ARE:

If x1+ iy1 = x2 + iy2 then x1- iy1 = x2 – iy2
Two complex numbers x1+ iy1 and x2 + iy2 are said to be equal
If R (x1 + iy1) = R (x2 + iy2)
I (x1 + iy1) = I (x2 + iy2)
Sum of the two complex numbers is
(x1 + iy1) + (x2 + iy2) = (x1+ x2) + i(y1+ y2)
Difference of two complex numbers is
(x1 + iy1) – (x2 + iy2) = (x1-x2) + i(y1 – y2)
Product of two complex numbers is
(x1+ iy1) ( x2 + iy2) = x1x2 – y1y2 + i(y1x2 + y2 x1)
Division of two complex numbers is
(x1 + iy1) (x2 + iy2) = x1x2 + y1 y2)x22+y22 + iy1x2 y2 x1x22+y22
Every complex number can be expressed in terms of r (cosθ + i sinθ)
               R (x+ iy) = r cosθ
               I (x+ iy) = r sinθ
               r = x2+y2 and θ = tan-1yx

REPRESENTATION OF COMPLEX NUMBERS IN A PLANE

The set of complex numbers is two-dimensional, and a coordinate plane is required to illustrate them graphically. This is in contrast to the real numbers, which are one-dimensional, and can be illustrated by a simple number line. The rectangular complex number plane is constructed by arranging the real numbers along the horizontal axis, and the imaginary numbers along the vertical axis. Each point in this plane can be assigned to a unique complex number, and each complex number can be assigned to a unique point in the plane.

Modulus and Argument of a complex number:

The number r = x2+y2 is called modulus of x+ iy and is written by mod (x+ iy) or x+iy

θ = tan-1yx is called amplitude or argument of x + iy and is written by amp (x + iy) or arg (x + iy)

Application of imaginary numbers:

For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. Fractions such as 2/3 and 1/8 are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Negative numbers such as -3 and -5 are meaningless when measuring the mass of an object, but essential when keeping track of monetary debits and credits. Similarly, imaginary numbers have essential concrete applications in a variety of sciences and related areas such as signal processing, control theory, electromagnetism, quantum mechanics, cartography, vibration analysis, and many others.

APPLICATION OF COMPLEX NUMBER IN ENGINEERING:

Control Theory

Incontrol theory, systems are often transformed from thetime domainto thefrequency domainusing theLaplace transform. The system’spolesandzerosare then analyzed in the complex plane. Theroot locus,Nyquist plot, andNichols plottechniques all make use of the complex plane.

In the root locus method, it is especially important whether thepolesandzerosare 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 beunstable,
all in the left half plane, it will bestable,
on the imaginary axis, it will havemarginal stability.

If a system has zeros in the right half plane, it is anonminimum phasesystem.

Signal analysis

Complex numbers are used insignal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value |z| of the corresponding z is the amplitude and the argument arg (z) the phase.

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

ω f (t) = z

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

Improper integrals

In applied fields, complex numbers are 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.

Residue theorem

The residue theorem in complex analysisis a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy and Cauchy’s integral formula.

The statement is as follows. Suppose U is a simply connected open subset of the complex plane C, a1,…, an are finitely many points of U and f is a function which is defined and holomorphic on U{a1,…,an}. If γ is a rectifiable curve in which doesn’t meet any of the points ak and whose start point equals its endpoint, then

Here, Res(f,ak) denotes the residue off at ak, and n(γ,ak) is the winding number of the curve γ about the point ak. This winding number is an integer which intuitively measures how often the curve γ winds around the point ak; it is positive if γ moves in a counter clockwise (“mathematically positive”) manner around ak and 0 if γ doesn’t move around ak at all.

In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed (which is usually easy), and a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane. The integral over this curve can then be computed using the residue theorem. Often, the half-circle part of the integral will tend towards zero if it is large enough, leaving only the real-axis part of the integral, the one we were originally interested

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Applications of Complex Numbers in Quantum mechanics

The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg’s matrix mechanics – make use of complex numbers.

The quantum theory provides a quantitative explanation for two types of phenomena that classical mechanics and classical electrodynamics cannot account for:

Some observable physical quantities, such as the total energy of a black body, take on discrete rather than continuous values. This phenomenon is called quantization, and the smallest possible intervals between the discrete values are called quanta (singular:quantum, from the Latin word for “quantity”, hence the name “quantum mechanics.”) The size of the quanta typically varies from system to system.
Under certain experimental conditions, microscopic objects like atoms or electrons exhibit wave-like behavior, such as interference. Under other conditions, the same species of objects exhibit particle-like behavior (“particle” meaning an object that can be localized to a particular region ofspace), such as scattering. This phenomenon is known as wave-particle duality.

Application of complex numbers in Computer Science.

Arithmetic and Logic in Computer Systems provides a useful guide to a fundamental subject of computer science and engineering. Algorithms for performing operations like addition, subtraction, multiplication, and division in digital computer systems are presented, with the goal of explaining the concepts behind the algorithms, rather than addressing any direct applications. Alternative methods are examined, and explanations are supplied of the fundamental materials and reasoning behind theories and examples.

This technological manual explores how software engineering principles can be used in tandem with software development tools to produce economical and reliable software that is faster and more accurate. Tools and techniques provided include the Unified Process for GIS application development, service-based approaches to business and information technology alignment, and an integrated model of application and software security. Current methods and future possibilities for software design are covered.

Application of complex numbers in Electrical Engineering:

The voltage produced by a battery is characterized by one real number (called potential), such as +12 volts or -12 volts. But the “AC” voltage in a home requires two parameters. One is a potential, such as 120 volts, and the other is an angle (called phase). The voltage is said to have two dimensions. A 2-dimensional quantity can be represented mathematically as either a vector or as a complex number (known in the engineering context as phasor). In the vector representation, the rectangular coordinates are typically referred to simply as X and Y. But in the complex number representation, the same components are referred to as real and imaginary. When the complex number is purely imaginary, such as a real part of 0 and an imaginary part of 120, it means the voltage has a potential of 120 volts and a phase of 90°, which is physically very real.

Application of complex numbers in electronic engineering

Information that expresses a single dimension, such as linear distance, is called a scalar quantity in mathematics. Scalar numbers are the kind of numbers students use most often. In relation to science, the voltage produced by a battery, the resistance of a piece of wire (ohms), and current through a wire (amps) are scalar quantities.

When electrical engineers analyzed alternating current circuits, they found that quantities of voltage, current and resistance (called impedance in AC) were not the familiar one-dimensional scalar quantities that are used when measuring DC circuits. These quantities which now alternate in direction and amplitude possess other dimensions (frequency and phase shift) that must be taken into account.

In order to analyze AC circuits, it became necessary to represent multi-dimensional quantities. In order to accomplish this task, scalar numbers were abandoned andcomplex numberswere used to express the two dimensions of frequency and phase shift at one time.

In mathematics, i is used to represent imaginary numbers. In the study of electricity and electronics, j is used to represent imaginary numbers so that there is no confusion with i, which in electronics represents current. It is also customary for scientists to write the complex number in the form a+jb.

In electrical engineering, the Fourier transform is used to analyze 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 and may come into conflict with i.) This approach is called phasor calculus. This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, andvideosignals.

Introduce the formula E = I â€¢ Z where E is voltage, I is current, and Z is impedance.

Complex numbers are used a great deal in electronics. The main reason for this is they make the whole topic of analyzing and understanding alternating signals much easier. This seems odd at first, as the concept of using a mix of real and ‘imaginary’ numbers to explain things in the real world seem crazy!. To help you get a clear picture of how they’re used and what they mean we can look at a mechanical example…

We can now reverse the above argument when considering a.c. (sine wave) oscillations in electronic circuits. Here we can regard the oscillating voltages and currents as ‘side views’ of something which is actually ‘rotating’ at a steady rate. We can only see the ‘real’ part of this, of course, so we have to ‘imagine’ the changes in the other direction. This leads us to the idea that what the oscillation voltage or current that we see is just the ‘real’ portion’ of a ‘complex’ quantity that also has an ‘imaginary’ part. At any instant what we see is determined by aphase anglewhich varies smoothly with time.

We can now consider oscillating currents and voltages as being complex values that have a real part we can measure and an imaginary part which we can’t. At first it seems pointless to create something we can’t see or measure, but it turns out to be useful in a number of ways.

Applications in Fluid Dynamics

Influid dynamics, complex functions are used to describe potential flow in two dimensions. Fractals.

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

Fluid Dynamics and its sub disciplines aerodynamics, hydrodynamics, and hydraulics have a wide range of applications. For example, they are used in calculating forces and moments onaircraft, the mass flow of petroleum through pipelines, and prediction of weather patterns.

The concept of a fluid is surprisingly general. For example, some of the basic mathematical concepts in traffic engineering are derived from considering traffic as a continuous fluids.

Application of complex numbers in Relativity

In special and general relativity, some formulas for the metric on space time become simpler if one takes the time variable to be imaginary. (This is no longer standard in classical relativity, but isused in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.

Application of complex numbers in 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) = ert.

Application of complex numbers in Electromagnetism:

Instead of taking electrical and magnetic part as a two different real numbers, we can represent it as in one complex number

Application of complex numbers in Civil and Mechanical Engineering:

The concept of complex geometry and Argand plane is very much useful in constructing buildings and cars. This concept is used in 2-D designing of buildings and cars. It is also very useful in cutting of tools. Another possibility to use complex numbers in simple mechanics might be to use them to represent rotations.