The complex numbers may be represented as points in the plane, with. Everyone knew that certain quadratic equations, like x2. What do solutions of third degree equations, electrical impedance, electromagnetic fields, light beams, and the imaginary unit have in common. This is a short introduction to complex numbers written primarily for students aged from about 14 or 15 to 18 or 19. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they. Introduction to complex numbers introduction to the introduction. If we multiply a real number by i, we call the result an imaginary number. Euler, has become one of the most utilized ideas in mathematics the exposition concentrates on key concepts and then elementary results concerning these numbers. The second edition of complex numbers from a to z introduces the reader to this fascinating subject that, from the time of l. This is just unity but, as we shall see, it is needed to obtain the correct number of roots. In the following discussion, we will use the same letter for a. It is impossible to imagine modern mathematics without complex numbers.
Thus to determine the limiting behavior of a sequence z n of complex numbers, we need only consider the behavior of the two sequences of real numbers, complex numbers 17 3 complex numbers law and order life is unfair. Complex numbers can be plotted on the complex plane. A frequently used property of the complex conjugate is the following formula 2 ww. Complex numbers may be ordered in size only by using a true 1d metric. Next we investigate the values of the exponential function with complex arguments. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. The complex logarithm, exponential and power functions in these notes, we examine the logarithm, exponential and power functions, where the arguments.
They constitute a number system which is an extension of the wellknown real number system. Well, complex numbers are the best way to solve polynomial equations, and thats what we sometimes need for solving certain kinds of di. I can add, subtract, multiply, and divide with complex numbers. We sketch a vector with initial point 0,0 and terminal point p x,y. We now consider another method of representing complex numbers. It includes the mathematics and a little bit of history as well. Mathematical institute, oxford, ox1 2lb, july 2004 abstract this article discusses some introductory ideas associated with complex numbers, their algebra and geometry.
The multiplication of complex numbers possesses the following properties, which we state without proofs. It is important to know that the collection of all complex numbers of the form z ei form a circle of radius one unit circle in the complex plane centered at. In complex analysis, a complex logarithm of the nonzero complex number z, denoted by w log z, is defined to be any complex number w for which e w z. Complex numbers and vectors 4 in various oscillation and wave problems you are likely to come across this sort of analysis, where the argument of the complex number represents the phase of the wave and the modulus of the complex number the amplitude. It can also be graphed on the cartesian complex plane with the coordinate. In this article we will look at some applications of complex numbers t o solving geometry problems. The calculus of complex functions in this section we will discuss limits, continuity, di.
Throughout this handout, we use a lowercase letter to denote the complex number that. Why should we convert a complex number to its exponential form. Complex numbers are awesome 11 2 basic operations involving complex numbers 15 2. It requires no brilliance that taking log of antilog gives us the log of that number. Then the residue of fz at z0 is the integral resz0 1 2. The following notation is used for the real and imaginary parts of a complex number z. These are called numbers but they have nothing to do with the concept of counting. Complex logarithm function lnz is a multivalued function. The real and imaginary parts of a complex number are given by re3. In the following discussion, we will use the same letter for a point to.
Complex numbers of the form x 0 0 x are scalar matrices and are called. Taylor and laurent series complex sequences and series. To restore justice one introduces new number i, the imaginary unit. The complex logarithm, exponential and power functions. Hamilton 18051865 mathematics is the queen of sciences and arithmetic is the queen of mathematics. In spite of this it turns out to be very useful to assume that there is a number ifor which one has 1 i2. This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. Moreover, we will introduce complex extensions of a number of familiar functions. This construction is analogous to the real logarithm function ln, which is the inverse of the real exponential function e y, satisfying e lnx x for positive real numbers x.
Taylor and laurent series complex sequences and series an in. Lecture notes for complex analysis frank neubrander fall 2003 analysis does not owe its really signi. Complex numbers solutions joseph zoller february 7, 2016 solutions 1. Lecture notes for complex analysis lsu mathematics. To each point in vector form, we associate the corresponding complex number. University of british columbia, vancouver yuexian li march 17, 2015 1. C has two complex solutions given by the quadratic formula see above, because according to example a, the square root of a complex. The first section is a more mathematical definition of complex numbers and is not really required for understanding the remainder of the document.
This will leaf to the wellknown euler formula for complex numbers. Domain of individual complex numbers is equivalent to 2d domain of real numbers set of individual complex numbers a onedimensional set does not exhibit the property. To understand the first few sections, it would be helpful to be familiar with polynomial equations for example. Notes on complex numbers university of british columbia, vancouver yuexian li march 17, 2015 1. A complex number is noted as, is the real part of the complex number a number as we know it and is the imaginary part of the complex number where is a real number. Powers and roots of complex numbers to nd powers and root of complex numbers it is almost always easiest to write them as complex exponentials and it is often important to include a factor e2. Recall that there is no real number z such that z2. Complex numbers solutions carnegie mellon university.
As a result, applying methods involving complex numbers can simplify calculations, removing a lot of the boring and tedious parts of mathematical work. You also learn how to represent complex numbers as points in the plane. Mathematics, mathematical research, mathematical modeling, mathematical programming, math tutorial, applied math. An introduction to complex numbers the complex numbers are the real numbers not sufficient. The logarithm of a complex number can be a real number only if. Youtube workbook 4 contents contents how to use this workbook 8 about the author 9 acknowledgments 10 1 what is a complex number. The complex numbers c are important in just about every branch of mathematics. But for complex numbers we do not use the ordinary planar coordinates x,ybut.
If a problem involves points and chords on a circle, often we can with out loss of generality assume it is the unit circle. This has proved extremely useful in pure mathematics and especially in physics where the solutions of many fundamental equations involve complex numbers. Complex numbers in rectangular and polar form to represent complex numbers x yi geometrically, we use the rectangular coordinate system with the horizontal axis representing the real part and the vertical axis representing the imaginary part of the complex number. The complex plane the real number line below exhibits a linear ordering of the real numbers.
It is presented solely for those who might be interested. The necessary background in a familiarity with ordinary real numbers all positive and negative numbers and zero and algebra. In particular, we are interested in how their properties di. We can take the square root of positive numbers like. Use eulers theorem to rewrite complex number in polar form to exponential form. Multiplying a complex z by i is the equivalent of rotating. They all have the same real part, but their imaginary parts di. Addition and subtraction of complex numbers follow the same rules as for ordinary numbers except that the real and imaginary parts are treated separately. Complex numbers in geometry yi sun mop 2015 1 how to use complex numbers in this handout, we will identify the two dimensional real plane with the one dimensional complex plane. Real numbers are the usual positive and negative numbers. Quiz on complex numbers solutions to exercises solutions to quizzes the full range of these packages and some instructions, should they be required, can be obtained from our web. Complex numbers are awesome because they provide a simpler framework from which we can view and do mathematics. In spite of this it turns out to be very useful to assume that there is a number ifor which one has.
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