Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
MATHS TERM PAPER TOPIC:-USE OF COMLEX NUMBERS IN ENGINEERING SUBMITTED BY:NAME:-TUSHAR ATTRAY SUBMITTED TO:MR. GURPREET SHINGH BHATIA AKNOWLEDGEMENT I TUSHAR ATTRAY from B.Tech. (M.E.) of section B4903 thanks my maths faculty Mr. Gurpreet Singh Bhatia that he gave me an opportunity to work on the topic. I also thank him to help me in making my term paper by giving me new ideas. I also thank author of my engineering mathematics book by N.P. Bali from where I got suffceint information related to my topic.once again thanking to all. INTRODUCTION TO 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. The solution of a general cubic equation in radicals (without trigonometric functions) may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. 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, multiplication, and division 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, who extended this abstraction to the theory of quaternions. Complex numbers are used in a number of fields, including: engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial, and complex Lie algebra. DEFINITION OF COMPLEX NUMBERS:- A complex number is a number comprising a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −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. HOW TO DONATE COMPLEX NUMBERS:The set of all complex numbers is usually denoted by C, or in blackboard bold by . Although other notations can be used, complex numbers are usually written in the form where a and b are real numbers, and i is the imaginary unit, which has the property i 2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. If z = a + bi, the real part a is denoted Re(z) or ℜ(z), and the imaginary part b is denoted Im(z) or ℑ(z SOME BASIC PROPERTIES OF COMPLEX NUMBERS:- EQUALITY OF A COMLEX NUMBER:Two complex numbers are said to be equal if and only if their real parts are equal and their imaginary parts are equal. In other words, if the two complex numbers are written as a + bi and c + di with a, b, c, and d real, then they are equal if and only if a = c and b = d. SOME BASIC OPERATIONS IN COMPLEX NUMBERS:Complex numbers are added, subtracted, multiplied, and divided by formally applying the associative, commutative and distributive laws of algebra, together with the equation i 2 = −1: Addition: Subtraction: Multiplication: Division: where c and d are not both zero. This is obtained by multiplying both the numerator and the denominator by (c − di). CONJUGATE OF A COMPLEX NUMBER:- Geometric representation of z and its conjugate in the complex plane The complex conjugate of the complex number z = x + yi is defined to be x − yi, written as or . As seen in the figure, is the "reflection" of z about the real axis, and so both and are real numbers. Many identities relate complex numbers and their conjugates. Conjugating twice gives the original complex number: The square of the absolute value is obtained by multiplying a complex number by its conjugate: 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. Conjugation distributes over the standard arithmetic operations: That conjugation distributes over all the algebraic operations and many functions, e.g. is rooted in the ambiguity in choice of i (−1 has two square roots). It is important to note, however, that the function is not complex-differentiable (see holomorphic function). The real and imaginary parts of a complex number can be extracted using the conjugate: if and only if z is real if and only if z is purely imaginary APPLICATION OF COMPLEX NUMBER:1. IN ELECTRICAL ENGINEERING: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! Once you get used to them, however, they do make a lot of things clearer. The problem is understanding what they 'mean' and how to use them in the first place. We can now reverse the above argument when considering a.c. (sinewave) 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 a phase angle which varies smoothly with time 2. IN CONTROLL THEORY:In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's 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 are in 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. 3. IN SIGNAL ANALYSIS:Complex numbers are used in signal 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 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 ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.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, frequencydependent 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. 4. 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. 5. IN RELATIVITY:In 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 longer 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. 6. 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. 7. IN FLUID MECHANICS:In fluid dynamics, complex functions are used to describe potential flow in two dimensions. Submitted for – www.mycollegebag.in