A complex number is made up using two numbers combined together. The information has been put together for students of complex analysis who. Lecture 4 roots of complex numbers characterization of a. This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. Duality is a famous concept in physics wavematter duality etc. Lecture notes, lecture 2 complex numbers math1054 studocu. The imaginary part of a complex number contains the imaginary unit, this number is called imaginary because it is equal to the square root of negative one. Imaginary numbers a number whose square is less than zero negative imaginary number 1. 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. Notes on complex numbers university of pennsylvania. Graphically the absolute value of complex number is the distance from the origin to the complex point in the complex plane.
Lecture notes for complex analysis lsu mathematics. The plane representing complex numbers as points is called complex. Vii given any two real numbers a,b, either a b or a 0. Imaginary numbers a number whose square is less than zero negative imaginary number 1 is called i other imaginary numbers write using i notation. In other words, a real number is just a complex number with vanishing imaginary part. The union of the set of all imaginary numbers and the set of all real numbers is the set of complex numbers. A complex number z is a purely real if its imaginary part is 0. Class 11 maths revision notes for chapter5 complex. Mathematical institute, oxford, ox1 2lb, july 2004 abstract this article discusses some introductory ideas associated with complex numbers, their algebra and geometry. Complex numbers are awesome 11 2 basic operations involving complex numbers 15 2.
We sketch a vector with initial point 0,0 and terminal point p x,y. The resultant complex number is therefore 1 2 rrei. Hamilton 18051865 mathematics is the queen of sciences and arithmetic is the queen of mathematics. The numbers x and y are called respectively real and imaginary parts of complex number z. These notes are based on a course of lectures given by prof. Similarly, the representation of complex numbers as points in the plane is known as.
Note that real numbers are complex a real number is simply a complex number with zero imaginary part. Download revision notes for complex numbers and quadratic equations class 11 notes mathematics and score high in exams. The complex numbers may be represented as points in the plane sometimes called the argand diagram. Alevel mathematics9709 organic chemistry for as level 9701 about me. These are the complex numbers and quadratic equations class 11 notes mathematics prepared by team of expert teachers. Well, complex numbers are the best way to solve polynomial equations, and thats what we sometimes need for solving certain kinds of di. What are complex numbers, how do you represent and operate using then. The second part of a complex number is an imaginary number. The relationship between exponential and trigonometric functions.
You also learn how to represent complex numbers as points in the plane. Next, lets take a look at a complex number that has a zero imaginary part. Introduction to complex numbers introduction to the introduction. The sum of two complex numbers is a complex number. Complex numbers of the form x 0 0 x are scalar matrices and are called. Complex numbers are built on the concept of being able to define the square root of negative one. The angle made by the line joining point z to the origin, with the xaxis is called argument of that complex number. A short history of complex numbers home department of. The set of all the complex numbers are generally represented by c. We then proceed to develop the theory and applications of conformal mappings. We will discuss a sketch of the proof, and note some remaining questions and future plans. Notice that the conjugate of a real number is just itself with no changes. We can let the imaginary unit take part in the usual arithmetic operations of addition. The representation is known as the argand diagram or complex plane.
Consider a complex number z 1 1 re i if it is multiplied by another complex number w 2 2 rei. I can add, subtract, multiply, and divide with complex numbers. By doing so, it unexpectedly brings the property of duality to mathematics. Notes on complex numbers university of british columbia, vancouver yuexian li march 17, 2015 1. Introduction to complex numbers introduction to the. The final topic in this section involves procedures for. A line that bisects the cord joining complex numbers a and b in a perpendicular fashion im b re a iii argz. I have collected these notes from various websites. 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. Use pythagorean theorem to determine the absolute value of this point. Use the commutative, assoc iative, and distributive properties to add and subtract complex numbers. Lecture notes for complex analysis frank neubrander fall 2003 analysis does not owe its really signi. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they.
Complex numbers exercises with detailed solutions 1. A short history of complex numbers orlando merino university of rhode island january, 2006 abstract this is a compilation of historical information from various sources, about the number i v. When the points of the plane represent complex numbers in this way, the plane is called the complexplane. 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. Set of variable points denoted by zwhich will form an argument of. Imaginary form, complex number, i, standard form, pure imaginary number, complex. Investigating investigating cardanos formula, which gives a solution of the cubic equation x 3.
Complex numbers study material for iit jee askiitians. Here we introduce a number symbol i v1 or i2 1 and we may deduce i3 i i4 1. A complex number is a number that contains a real part and an imaginary part. Express each expression in terms of i and simplify. They constitute a number system which is an extension of the wellknown real number system. The set of real algebraic numbers, a, consists of the real numbers that are solution to polynomial equations with rational coefficients. The complex plane the real number line below exhibits a linear ordering of the real numbers. Algebra revision notes on complex numbers for iit jee.
The second reason is complex analysis has a large number of applications in both the pure math and applied math senses of the word to things that seem like they ought to have little to do with complex numbers. From this starting point evolves a rich and exciting world of the number system that encapsulates everything we have known before. The real complex numbers lie on the xaxis, which is then called the real axis, while the imaginary numbers lie on the. A real number is algebraic if it is a root of some polynomial with integer or, equivalentl,y. Here are some examples of complex numbers and their conjugates. Although it is rarely, if ever, used in some fields of math, it comes in very handy. Notes for a2level mathematics9709 complex number hola amigos i have got few notes for you guys that really helped me. Complex analysis indian institute of technology guwahati. Notes on complex numbers ubc math university of british. General topology, addisonwesley 1966 translated from french mr0205211 mr0205210 zbl 0301. The complex number system is an extension of the real number system. Complex number simple english wikipedia, the free encyclopedia. The first section is a more mathematical definition of complex numbers and is not really required for understanding the remainder of the document.
We assume the reader is familiar with the basics of complex numbers and complex arithmetic, as in 18. In these cases, we call the complex number a pure imaginary number. The complex numbers may be represented as points in the plane, with the real number 1 represented by the point 1. We can take the square root of positive numbers like. Note that, when cardano stated his problem about dividing ten.
The real number 1 is represented by the point 1,0, and the complex number i is represented by the point 0,1. Every z 2 chas n distinct roots of order n, which correspond in the complex plane to the vertices of a regular nagon inscribed in the circle of radius n p. The revision notes help you revise the whole chapter in minutes. Youtube workbook 4 contents contents how to use this workbook 8 about the author 9 acknowledgments 10 1 what is a complex number.
Gowers in part ia of the mathematicalriptos at the university of cambridge in the academic year 2004 2005. Complex numbers are represented geometrically by points in the plane. Well also be seeing a slightly different way of looking at some of the basics that you probably didnt see when you were first introduced to complex numbers and proving some of the basic facts. In spite of this it turns out to be very useful to assume that there is a number ifor which one has 1 i2. Appendix a, and commence our exposition with the basics of complex functions and their di. Complex numbers are added or subtracted by adding or subtracting on their real parts and also their imaginary parts. Similarly, the representation of complex numbers as points in the plane is known as argand diagram. In fact, for any complex number z, its conjugate is given by z rez imz. Im a little less certain that you remember how to divide them.
Traditionally the letters z and w are used to stand for complex numbers. In spite of this it turns out to be very useful to assume that there is a number. Use the relation i 2 1 to multiply two imaginary numbers to get a real number. Download englishus transcript pdf i assume from high school you know how to add and multiply complex numbers using the relation i squared equals negative one. In other words, it is the original complex number with the sign on the imaginary part changed. Given two complex numbers in polar form and the product and quotient of the numbers are as follows.
Chalkboard photos, reading assignments, and exercises pdf 1. Mathematical institute, oxford, ox1 2lb, november 2003 abstract cartesian and polar form of a complex number. Derive the equation of a parabola given the focus and directrix 10. Two complex numbers are said to be equal if they have the same real and imaginary parts. Complex numbers extends the concept of one dimensional real numbers to the two dimensional complex numbers in which two dimensions comes from real part and the imaginary part. Complex number can be considered as the superset of all the other different types of number. Real axis, imaginary axis, purely imaginary numbers. By switching to polar coordinates, we can write any nonzero complex number in an alternative. When d 0, roots of the quadratic equation are real and equal. Postscript or pdf produced by some word processors. A complex number is a number, but is different from common numbers in many ways. We can plot complex numbers on the complex plane, where the xaxis is the real part.