The representation is known as the argand diagram or complex plane. General topology, addisonwesley 1966 translated from french mr0205211 mr0205210 zbl 0301. The sum of two complex numbers is a complex number. 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. Derive the equation of a parabola given the focus and directrix 10. Mathematical institute, oxford, ox1 2lb, november 2003 abstract cartesian and polar form of a complex number. Similarly, the representation of complex numbers as points in the plane is known as argand diagram. Use the relation i 2 1 to multiply two imaginary numbers to get a real number. Complex analysis indian institute of technology guwahati.
Alevel mathematics9709 organic chemistry for as level 9701 about me. We then proceed to develop the theory and applications of conformal mappings. A complex number is a number that contains a real part and an imaginary part. You also learn how to represent complex numbers as points in the plane. Imaginary numbers a number whose square is less than zero negative imaginary number 1. But for complex numbers we do not use the ordinary planar coordinates x,ybut. In spite of this it turns out to be very useful to assume that there is a number.
The real complex numbers lie on the xaxis, which is then called the real axis, while the imaginary numbers lie on the. Traditionally the letters z and w are used to stand for complex numbers. Notes on complex numbers university of british columbia, vancouver yuexian li march 17, 2015 1. Appendix a, and commence our exposition with the basics of complex functions and their di. Similarly, the representation of complex numbers as points in the plane is known as. Notes for a2level mathematics9709 complex number hola amigos i have got few notes for you guys that really helped me. Gowers in part ia of the mathematicalriptos at the university of cambridge in the academic year 2004 2005. Lecture notes, lecture 2 complex numbers math1054 studocu.
When the points of the plane represent complex numbers in this way, the plane is called the complexplane. From this starting point evolves a rich and exciting world of the number system that encapsulates everything we have known before. In these cases, we call the complex number a pure imaginary number. We sketch a vector with initial point 0,0 and terminal point p x,y. Class 11 maths revision notes for chapter5 complex.
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. Next, lets take a look at a complex number that has a zero imaginary part. The complex numbers may be represented as points in the plane, with the real number 1 represented by the point 1. Note that real numbers are complex a real number is simply a complex number with zero imaginary part. Although it is rarely, if ever, used in some fields of math, it comes in very handy. Postscript or pdf produced by some word processors. In other words, it is the original complex number with the sign on the imaginary part changed. The union of the set of all imaginary numbers and the set of all real numbers is the set of complex numbers.
The numbers x and y are called respectively real and imaginary parts of complex number z. Notes on complex numbers university of pennsylvania. Express each expression in terms of i and simplify. 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. The second part of a complex number is an imaginary number. The complex plane the real number line below exhibits a linear ordering of the real numbers. The real number 1 is represented by the point 1,0, and the complex number i is represented by the point 0,1. Well, complex numbers are the best way to solve polynomial equations, and thats what we sometimes need for solving certain kinds of di. In fact, for any complex number z, its conjugate is given by z rez imz.
Here we introduce a number symbol i v1 or i2 1 and we may deduce i3 i i4 1. The information has been put together for students of complex analysis who. A real number is algebraic if it is a root of some polynomial with integer or, equivalentl,y. Lecture notes for complex analysis lsu mathematics. The final topic in this section involves procedures for.
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 plane representing complex numbers as points is called complex. The revision notes help you revise the whole chapter in minutes. Consider a complex number z 1 1 re i if it is multiplied by another complex number w 2 2 rei. Imaginary form, complex number, i, standard form, pure imaginary number, complex. 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. Real axis, imaginary axis, purely imaginary numbers. Complex numbers are built on the concept of being able to define the square root of negative one. By switching to polar coordinates, we can write any nonzero complex number in an alternative. They constitute a number system which is an extension of the wellknown real number system. Chalkboard photos, reading assignments, and exercises pdf 1. Introduction to complex numbers introduction to the introduction.
Complex numbers are represented geometrically by points in the plane. Lecture 4 roots of complex numbers characterization of a. 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. A complex number z is a purely real if its imaginary part is 0. The set of real algebraic numbers, a, consists of the real numbers that are solution to polynomial equations with rational coefficients. Two complex numbers are said to be equal if they have the same real and imaginary parts.
We can take the square root of positive numbers like. The resultant complex number is therefore 1 2 rrei. Complex numbers and quadratic equations class 11 notes. Complex numbers exercises with detailed solutions 1. Algebra revision notes on complex numbers for iit jee. The first section is a more mathematical definition of complex numbers and is not really required for understanding the remainder of the document. Investigating investigating cardanos formula, which gives a solution of the cubic equation x 3. I have collected these notes from various websites.
Youtube workbook 4 contents contents how to use this workbook 8 about the author 9 acknowledgments 10 1 what is a complex number. Vii given any two real numbers a,b, either a b or a 0. In spite of this it turns out to be very useful to assume that there is a number ifor which one has 1 i2. Note that, when cardano stated his problem about dividing ten. Given two complex numbers in polar form and the product and quotient of the numbers are as follows. Complex number simple english wikipedia, the free encyclopedia. A line that bisects the cord joining complex numbers a and b in a perpendicular fashion im b re a iii argz. A short history of complex numbers home department of. Duality is a famous concept in physics wavematter duality etc. Complex numbers of the form x 0 0 x are scalar matrices and are called. Use pythagorean theorem to determine the absolute value of this point.
By doing so, it unexpectedly brings the property of duality to mathematics. Download revision notes for complex numbers and quadratic equations class 11 notes mathematics and score high in exams. When d 0, roots of the quadratic equation are real and equal. A complex number is made up using two numbers combined together. Complex numbers are added or subtracted by adding or subtracting on their real parts and also their imaginary parts. Lecture notes for complex analysis frank neubrander fall 2003 analysis does not owe its really signi. The angle made by the line joining point z to the origin, with the xaxis is called argument of that complex number.
A complex number is a number, but is different from common numbers in many ways. Hamilton 18051865 mathematics is the queen of sciences and arithmetic is the queen of mathematics. 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. Here are some examples of complex numbers and their conjugates. Graphically the absolute value of complex number is the distance from the origin to the complex point in the complex plane. 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. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they. Im a little less certain that you remember how to divide them. Use the commutative, assoc iative, and distributive properties to add and subtract complex numbers. These are the complex numbers and quadratic equations class 11 notes mathematics prepared by team of expert teachers. Complex numbers are awesome 11 2 basic operations involving complex numbers 15 2. 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. We will discuss a sketch of the proof, and note some remaining questions and future plans. Complex numbers study material for iit jee askiitians.
The complex numbers may be represented as points in the plane sometimes called the argand diagram. Complex number can be considered as the superset of all the other different types of number. Introduction to complex numbers introduction to the. We can let the imaginary unit take part in the usual arithmetic operations of addition. Mathematical institute, oxford, ox1 2lb, july 2004 abstract this article discusses some introductory ideas associated with complex numbers, their algebra and geometry. I can add, subtract, multiply, and divide with complex numbers. The complex number system is an extension of the real number system. Notice that the conjugate of a real number is just itself with no changes. The set of all the complex numbers are generally represented by c. Notes on complex numbers ubc math university of british. This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. What are complex numbers, how do you represent and operate using then.
202 17 418 427 1458 1308 777 872 504 180 206 336 332 1272 1164 773 954 746 779 39 764 550 309 324 1438 225 148 863 435 1260 826 722 899 937 871