Application And Use Of Complex Numbers

Application And Use Of Complex NumbersHISTORY OF COMPLEX NUMBERS-Complex be were kickoff conceived and defined by the Italian mathematician Gerolamo Cardano, who called them fictitious, during his attempts to find solutions to cubic equatings. This ultimately led to the fundamental theorem of algebra, which shows that with mazy gets, a solution exists to every polynomial equation of degree one or higher. Complex be thus form an algebraically unlikable field, where any polynomial equation has a blood line.The rules for addition, fill intraction and contemporaries of complex subjects were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex rounds was further developed by the Irish mathematician William Rowan Hamilton.COMPLEX NUMBER INTERPRETATION-A enumerate in the form of x+iy where x and y argon real rime and i = is called a complex number.Let z= x+iyX is called real part of z and is denoted by R (z)Y is called unreal part of z and is denoted by I (z)CONJUGATE OF A COMPLEX NUMBER A pair of complex numbers x+iy and x-iy atomic number 18 said to be fuse of each other.PROPERTIES OF COMPLEX NUMBERS ARE-1) If + = + then =2) Two complex numbers + and + be said to be equalIf R (+) = R ( +)I (+) = I ( +)3) Sum of the devil complex numbers is( +) +( + = (+ ) + (+)4) Difference of two complex numbers is( +) ( + = () + ()5) Product of two complex numbers is( +) ( + = +( )6) Division of two complex numbers is= +7) Every complex number faecal matter be expressed in terms of r ( romaine lettuce + sin)R (x+) = r cosI (x+) = r sinr = and =REPRESENTATION OF COMPLEX NUMBERS IN PLANEThe set of complex numbers is two-dimensional, and a coordinate flavour little is required to illustrate them graphically. This is in contrast to the real numbers, which are one-dimensional, and can be illustrated by a naive number line. The rectangular complex number flavourless is constructed by arranging the real numbers along th e horizontal axis, and the complex measuring rod numbers along the vertical axis. Each bit 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 = is called modulus of x+ and is written by mod (x+) or = is called amplitude or pipeline of x+ and is written by amp (x+) or arg (x+)Application of imaginary numbers For most human tasks, real numbers (or even rational numbers) leave an adequate description of data. Fractions such as and are meaningless to a person counting stones, but essential to a person comparing the sizes of varied 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 bring forth essential concrete drills in a variety of scientific disciplines and related areas such as signal touch, control theory, electromagnetism, quantum mechanics, cartography, vibration analysis, and some(prenominal) others.APPLICATION OF COMPLEX NO IN ENGINEERING-Control TheoryIn control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The systems 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 arein the right half plane, it will be unstable,all in the left half plane, it will be stable,on the imaginary axis, it will have marginal stability.If a system has zeros in the right half plane, it is a nonminimum phase system.Signal analysisComplex numbers are utilise in signal analysis and other fields for a convenient description for per iodically varying signals. For given real functions representing developed physical quantities, often in terms of sines and cosines, corresponding complex functions are get byed of which the real parts are the accredited 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 analysis is 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 formwhere represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.Improper integralsIn applied fields, complex numbers are often utilize to compute certain real-valued wrong integrals, by means of complex-valued functions. Several methods exist to do this see methods of contour integration.Residue theoremThe balance wheel theorem in complex analysis is a powerful pecker to evaluate path int egrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy and Cauchys integral formula.The statement is as follows. speak up 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 is a rectifiable curve in U which doesnt meet any of the points ak and whose start point equals its endpoint, thenHere, Res(f,ak) denotes the residue of f at ak, and n(,ak) is the winding number of the curve about the point ak. This winding number is an integer which intuitively measures how often the curve winds around the point ak it is positive if moves in a counter clockwise (mathematically positive) manner around ak and 0 if doesnt 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 com puted (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 interestedQuantum mechanicsThe 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 Schrdinger equation and Heisenbergs 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 forSome observable physical quantities, such as the total energy of a bla ckbody, take on discrete rather than consecutive 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 observational 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 of space), such as scattering. This phenomenon is known as wave-particle duality.Application of complex number in Computer Science.1) Arithmetic and logic in computer systemArithmetic and Logic in Computer Systems provides a useful guide to a fundamental subject of computer science and plan. Algorithms for performing operations like addition, subtraction, multiplication, and division in digital computer systems are presented, with the goal of explaining the excogitations behind the algorithms, rather than addressing any direct applications. Alternative methods are examined, and explanations are supplied of the fundamental materials and causaing behind theories and examples.2) Recticing Software engineering in 21st centuryThis 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.In Electrical engine room 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 in electronics 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 telegraph (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 interpreted 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 and complex numbers were 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 electrical energy and electronics, j is used to represent imaginary numbers so that there is no confusion with i, which in electr onics 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, and video signals.Introduce the formula E = I 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 theyre used and what they mean we can look at a mechanical exampleWe can now reverse the above argument when considering a.c. (sine wave) oscillations in electronic circuits. Here we can bear on 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 hypothesise 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 a phase angle which varie s 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 cant. At first it seems pointless to create something we cant see or measure, but it turns out to be useful in a number of ways.1) It helps us understand the demeanour of circuits which contain reactance (produced by capacitors or inductors) when we apply a.c. signals.2) It gives us a new way to think about oscillations. This is useful when we want to apply concepts like the saving of energy to understanding the behaviour of systems which range from simple a mechanical pendulums to a quartz-crystal oscillator.Applications in Fluid DynamicsIn fluid 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 setFluid Dynamics and its sub disciplines aerodynamics, hydrodynamics, and hydraulics have a wide range of appl ications. For example, they are used in calculating forces and moments on aircraft, the mass flow of petroleum through pipelines, and vaticination 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.RelativityIn special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary. (This is no durable standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.Applied mathematicsIn 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.In Electromagnetism Instead of tak ing electrical and magnetic part as a two different real numbers, we can represent it as in one complex numberIN 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.