Chapter 9: Complex Numbers


 Barbara Gibson
 1 years ago
 Views:
Transcription
1 Chapter 9: Comple Numbers 9.1 Imaginary Number 9. Comple Number  definition  argand diagram  equality of comple number 9.3 Algebraic operations on comple number  addition and subtraction  multiplication  comple conjugate  division 9.4 Polar form of comple number  modulus and argument  multiplication and division of comple number in polar form 9.5 De Moivre s Theorem  definition  n th power of comple number  n th roots of comple number  proving some trigonometric identities 9.6 Euler s formula  definition  n th power of comple number  n th roots of comple number  relationship between circular and hyperbolic functions 1
2 9.1 Imaginary numbers Consider: 4 This equation has no real solution. To solve the equation, we will introduce an imaginary number. Definition 9.1 (Imaginary Number) The imaginary number i is defined as: i 1 Therefore, using the definition, we will get, 4 4 4( 1) 4i i Eample: Epress the following as imaginary numbers a) 5 b) 8
3 9. Comple Numbers Definition 9. (Comple Numbers) If is a comple number, then it can be epressed in the form : iy, where, y R and i 1. : real part y : imaginary part Or frequently represented as : Re() = and Im() = y Eample: Find the real and imaginary parts of the following comple numbers (a) 1 3i (b) 4i i i 3 3
4 9..1 Argand Diagram We can graph comple numbers using an Argand Diagram. y Re( ) Eample: Sketch the following comple numbers on the same aes. (a) (c) 3 i (b) 3 i 1 3 i (d) 3 i 3 4 4
5 9.. Equality of Two Comple Numbers Given that where 1 a bi and c di 1, C. Two comple numbers are equal iff the real parts and the imaginary parts are respectively equal. So, if 1, then a c and b d. Eample 1: Solve for and y if given Eample : Solve and y. 3 4i (y ) i. for real numbers Eample 3: Solve the following equation for and y where 5
6 9.3 Algebraic Operations on Comple Numbers Addition and subtraction If 1 a bi and c di are two comple numbers, then 1 a c b di Eample: Given. Find a) Z 1 Z b) Z 1 + Z Multiplication If 1 a bi and c di are two comple numbers, and k is a constant, then (i) a bic di (ii) k1 1 ka kbi ac bd ad bci Eample: Given. Find Z 1 Z. 6
7 9.3.3 Comple Conjugate If Note that a bi then the conjugate of is denoted as Division If we are dividing with a comple number, the denominator must be converted to a real number. In order to do that, multiply both the denominator and numerator by comple conjugate of the denominator. 1 1 iy iy 1 iy iy 7
8 Eample 1: Given that 1 1 i, 3 4i. Find 1, and epress it in a bi form. Eample : Given 1 = + i and = 3 4i, find in the form of a + ib. Eample 3: Given Z =. Find the comple conjugate, Write your answer in a + ib form. Eample 4: Given. Find. 8
9 9.4 Polar Form of Comple Numbers P r y O Modulus of, r y. Argument of, arg () = where y tan. From the diagram above, we can see that r cos y r sin Then, can be written as iy r cos irsin r(cos isin ) rcis ( inpolarform) 9
10 Eample 1: Epress i in polar form. Eample : Epress in polar form. Eample 3 Given that 1 = + i and =  + 4i, find such that. Give your answer in the form of a + ib. Hence, find the modulus and argument of. 10
11 9.5 De Moivre s Theorem The nth Power Of A Comple Number Definition 9.5 ( De Moivre s Theorem) and n R, then If rcos isin n r n cosn isinn Eample 1: a) Write = 1 i in the polar form. Then, using De Moivre s theorem, find 4. b) Use D Moivre s formula to write (1 i) 1 in the form of a + ib. 11
12 9.5. The nth Roots of a Comple Number A comple number w is a nth root of the comple number if w n = or w =. Hence w = 1 = n [ r(cos isin )], 1/ r n k k cos isin n n for k 0,1,,, n 1 Substituting k 0,1,,, n 1 yields the nth roots of the given comple number. Eample 1: Find all the roots for the following equations: (a) (b) 3 i. Eample : Solve and epress them in a + ib form. 1
13 Eample 3: Find all cube roots of. Eample 4: Solve = 0. Sketch the roots on the argand diagram. 13
14 9.5.3 De Moivre s Theorem to Prove Trigonometric Identities DeMoivre s theorem can be used to prove some trigonometric identities. (with the help of Binomial theorem or Pascal triangle.) Eample: Prove that 5 3 cos5 16cos 0cos 5cos and 5 3 sin 5 = 16 sin  0 sin + 5 sin. Solution: The idea is to write (cos θ + i sin θ) 5 in two different ways. We use both the Pascal triangle and De Moivre s theorem, and compare the results. From Pascal triangle, (cos θ + i sinθ) 5 = cos 5 θ + i 5 cos 4 θ sin θ 10 cos θ sin θ i 10 cos θ sin 3 θ + 5 cosθ sin 4 θ + i sin 5 θ. = (cos 5 θ 10 cos 3 θ sin θ + 5 cos θ sin 4 θ) + i(5 cos 4 θ sin θ 10 cos θ sin 3 θ + sin 5 θ). 14
15 Also, by De Moivre s Theorem, we have (cos θ + i sin θ) 5 = cos 5θ + i sin 5θ. and so cos 5θ + i sin 5θ = (cos 5 θ 10 cos 3 θ sin θ + 5 cos θ sin 4 θ) + i(5 cos 4 θ sin θ 10 cos θ sin 3 θ + sin 5 θ). Equating the real parts gives cos 5θ = cos 5 θ 10 cos 3 θ sin θ + 5 cos θ sin 4 θ. = cos 5 θ 10 cos 3 θ(1 cos θ) + 5 cos θ(1 cos θ) = cos 5 θ 10 cos 3 θ + 10 cos 5 θ + 5 cos θ 10 cos 3 θ + 5 cos 5 θ. = 16 cos 5 θ 0 cos 3 θ + 5 cos θ. (proved) Equating the imaginary parts gives sin 5θ = 5 cos 4 θ sin θ 10 cos θ sin 3 θ + sin 5 θ = 5(1 sin θ) sin θ 10(1 sin θ) sin 3 θ + sin 5 θ = 5(1 sin θ + sin 4 θ) sin θ 10 sin 3 θ + 10 sin 5 θ + sin 5 θ = 5 sin θ 10 sin 3 θ + 5 sin 5 θ 10 sin 3 θ + 10 sin 5 θ + sin 5 θ = 16 sin 5 θ 0 sin 3 θ + 5 sin θ (proved). 15
16 9.6 Eulers s Formula Definition 9.6 Euler s formula states that e i It follows that e in cos isin cosn isinn From the definition, if is a comple number with modulus r and Arg(), ; then r(cos isin ) i re ( ineuler form) Eample: Epress the following comple numbers in the form of (a) 3 + i (b) 4i i re 16
17 9.6.1 The nth Power Of A Comple Number We know that a comple number can be epress as then i re, i r e 3 3 i 3 r e 4 4 i4 r n r n e e in Eample 1: Given. Find the modulus and argument of 5. Eample : Find in the form of a + ib. Eample 3: Epress the comple number = Then find 1 3i in the form of i re. (a) (b) 3 (c) 7 17
18 9.6. The nth Roots Of A Comple Number The nth roots of a comple number can be found using the Euler s formula. Note that: Then, i re ( k ) 1 1 k i r e, k = 0,1 1 1 k i r e, k = 0,1, 1 n r 1 n e k i n, k = 0,1,,,n 1 Eample 1: Find the cube roots of 1 i. Eample : Given. Find all roots of in Euler form. Eample 3: Solve 3 + 8i = 0 and sketch the roots on an Argand diagram. 18
19 9.6.3 Relationship between circular and hyperbolic functions. Euler s formula provides the theoretical link between circular and hyperbolic functions. Since e i cos isin and i e cos isin we deduce that i i i i e e e e cos and sin. i (1) In Chapter 8, we defined the hyperbolic function by cosh e e and sinh e e () Comparing (1) and (), we have so that e coshi tanh i i tan i e i cos i i e e sinhi isin i. 19
20 Also, so that cosi sini i i e e e e cosh i i e e e e sinh i tani i tanh. Using these results, we can evaluate functions such as sin, cos, tan, sinh, cosh and tanh. For eample, to evaluate cos cos( iy) we use the identity cos( A B) cos Acos B sin Asin B and obtain cos cos cosiy sin siniy Using results in (3), this gives cos cos cosh y isin sinh y. (3) 0
21 Eample: Find the values of a) sinh(3 4 i) b) tan 3i 4 c) sin (1 i) 4 (Ans: a) i b) i c) i) 1
22 Pascal s Triangle In general:
AH Complex Numbers.notebook October 12, 2016
Complex Numbers Complex Numbers Complex Numbers were first introduced in the 16th century by an Italian mathematician called Cardano. He referred to them as ficticious numbers. Given an equation that does
More informationSome commonly encountered sets and their notations
NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS (This notes are based on the book Introductory Mathematics by Ng Wee Seng ) LECTURE SETS & FUNCTIONS Some commonly encountered sets and their
More informationThe modulus, or absolute value, of a complex number z a bi is its distance from the origin. From Figure 3 we see that if z a bi, then.
COMPLEX NUMBERS _+i _i FIGURE Complex numbers as points in the Arg plane i _i +i i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with
More informationComplex Numbers. 1 Introduction. 2 Imaginary Number. December 11, Multiplication of Imaginary Number
Complex Numbers December, 206 Introduction 2 Imaginary Number In your study of mathematics, you may have noticed that some quadratic equations do not have any real number solutions. For example, try as
More information) z r θ ( ) ( ) ( ) = then. Complete Solutions to Examination Questions Complete Solutions to Examination Questions 10.
Complete Solutions to Examination Questions 0 Complete Solutions to Examination Questions 0. (i We need to determine + given + j, j: + + j + j (ii The product ( ( + j6 + 6 j 8 + j is given by ( + j( j
More informationLesson 73 Polar Form of Complex Numbers. Relationships Among x, y, r, and. Polar Form of a Complex Number. x r cos y r sin. r x 2 y 2.
Lesson 7 Polar Form of Complex Numbers HL Math  Santowski Relationships Among x, y, r, and x r cos y r sin r x y tan y x, if x 0 Polar Form of a Complex Number The expression r(cos isin ) is called the
More informationSolution Sheet 1.4 Questions 2631
Solution Sheet 1.4 Questions 2631 26. Using the Limit Rules evaluate i) ii) iii) 3 2 +4+1 0 2 +4+3, 3 2 +4+1 2 +4+3, 3 2 +4+1 1 2 +4+3. Note When using a Limit Rule you must write down which Rule you
More informationMTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.
MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:
More informationComplex Numbers Introduction. Number Systems. Natural Numbers ℵ Integer Z Rational Q Real Complex C
Number Systems Natural Numbers ℵ Integer Z Rational Q R Real Complex C Number Systems Natural Numbers ℵ Integer Z Rational Q R Real Complex C The Natural Number System All whole numbers greater then zero
More informationMAT01A1: Complex Numbers (Appendix H)
MAT01A1: Complex Numbers (Appendix H) Dr Craig 14 February 2018 Announcements: equiz 1 is live. Deadline is Wed 21 Feb at 23h59. equiz 2 (App. A, D, E, H) opens tonight at 19h00. Deadline is Thu 22 Feb
More informationComplex Numbers and the Complex Exponential
Complex Numbers and the Complex Exponential φ (2+i) i 2 θ φ 2+i θ 1 2 1. Complex numbers The equation x 2 + 1 0 has no solutions, because for any real number x the square x 2 is nonnegative, and so x 2
More informationComplex Numbers. Basic algebra. Definitions. part of the complex number a+ib. ffl Addition: Notation: We i write for 1; that is we
Complex Numbers Definitions Notation We i write for 1; that is we define p to be p 1 so i 2 = 1. i Basic algebra Equality a + ib = c + id when a = c b = and d. Addition A complex number is any expression
More informationPart D. Complex Analysis
Part D. Comple Analsis Chapter 3. Comple Numbers and Functions. Man engineering problems ma be treated and solved b using comple numbers and comple functions. First, comple numbers and the comple plane
More informationUnit 3 Specialist Maths
Unit 3 Specialist Maths succeeding in the vce, 017 extract from the master class teaching materials Our Master Classes form a component of a highly specialised weekly program, which is designed to ensure
More informationCHAPTER 1 COMPLEX NUMBER
BA0 ENGINEERING MATHEMATICS 0 CHAPTER COMPLEX NUMBER. INTRODUCTION TO COMPLEX NUMBERS.. Quadratic Equations Examples of quadratic equations:. x + 3x 5 = 0. x x 6 = 0 3. x = 4 i The roots of an equation
More information1. COMPLEX NUMBERS. z 1 + z 2 := (a 1 + a 2 ) + i(b 1 + b 2 ); Multiplication by;
1. COMPLEX NUMBERS Notations: N the set of the natural numbers, Z the set of the integers, R the set of real numbers, Q := the set of the rational numbers. Given a quadratic equation ax 2 + bx + c = 0,
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationRe(z) = a, For example, 3 + 2i = = 13. The complex conjugate (or simply conjugate") of z = a + bi is the number z defined by
F COMPLEX NUMBERS In this appendi, we review the basic properties of comple numbers. A comple number is a number z of the form z = a + bi where a,b are real numbers and i represents a number whose square
More informationSolutions to Exercises 1.1
Section 1.1 Complex Numbers 1 Solutions to Exercises 1.1 1. We have So a 0 and b 1. 5. We have So a 3 and b 4. 9. We have i 0+ 1i. i +i because i i +i 1 {}}{ 4+4i + i 3+4i. 1 + i 3 7 i 1 3 3 + i 14 1 1
More informationComplex Numbers. Introduction
10 Assessment statements 1.5 Complex numbers: the number i 5 1 ; the term s real part, imaginary part, conjugate, modulus and argument. Cartesian form z 5 a 1 ib. Sums, products and quotients of complex
More informationPrecalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers
Syllabus Objectives: 5.1 The student will eplore methods of vector addition and subtraction. 5. The student will develop strategies for computing a vector s direction angle and magnitude given its coordinates.
More informationFurther Mathematics SAMPLE. Marking Scheme
Further Mathematics SAMPLE Marking Scheme This marking scheme has been prepared as a guide only to markers. This is not a set of model answers, or the exclusive answers to the questions, and there will
More informationIntroduction. The first chapter of FP1 introduces you to imaginary and complex numbers
Introduction The first chapter of FP1 introduces you to imaginary and complex numbers You will have seen at GCSE level that some quadratic equations cannot be solved Imaginary and complex numbers will
More informationComplex Numbers. April 10, 2015
Complex Numbers April 10, 2015 In preparation for the topic of systems of differential equations, we need to first discuss a particularly unusual topic in mathematics: complex numbers. The starting point
More informationMAT01A1: Complex Numbers (Appendix H)
MAT01A1: Complex Numbers (Appendix H) Dr Craig 13 February 2019 Introduction Who: Dr Craig What: Lecturer & course coordinator for MAT01A1 Where: CRing 508 acraig@uj.ac.za Web: http://andrewcraigmaths.wordpress.com
More informationPURE MATHEMATICS AM 27
AM Syllabus (014): Pure Mathematics AM SYLLABUS (014) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (014): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)
More informationMath Spring 2014 Solutions to Assignment # 6 Completion Date: Friday May 23, 2014
Math 11  Spring 014 Solutions to Assignment # 6 Completion Date: Friday May, 014 Question 1. [p 109, #9] With the aid of expressions 15) 16) in Sec. 4 for sin z cos z, namely, sin z = sin x + sinh y cos
More information3 COMPLEX NUMBERS. 3.0 Introduction. Objectives
3 COMPLEX NUMBERS Objectives After studying this chapter you should understand how quadratic equations lead to complex numbers and how to plot complex numbers on an Argand diagram; be able to relate graphs
More informationFY B. Tech. Semester II. Complex Numbers and Calculus
FY B. Tech. Semester II Comple Numbers and Calculus Course Code FYT Course Comple numbers and Calculus (CNC) Prepared by S M Mali Date 6//7 Prerequisites Basic knowledge of results from Algebra. Knowledge
More informationLecture 4. Properties of Logarithmic Function (Contd ) y Log z tan constant x. It follows that
Lecture 4 Properties of Logarithmic Function (Contd ) Since, Logln iarg u Re Log ln( ) v Im Log tan constant It follows that u v, u v This shows that Re Logand Im Log are (i) continuous in C { :Re 0,Im
More informationMathematics Extension 2 HSC Examination Topic: Polynomials
by Topic 995 to 006 Polynomials Page Mathematics Etension Eamination Topic: Polynomials 06 06 05 05 c Two of the zeros of P() = + 59 8 + 0 are a + ib and a + ib, where a and b are real and b > 0. Find
More informationSolutions to Problem Sheet for Week 6
THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS Solutions to Problem Sheet for Week 6 MATH90: Differential Calculus (Advanced) Semester, 07 Web Page: sydney.edu.au/science/maths/u/ug/jm/math90/
More informationCollege Trigonometry
College Trigonometry George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 131 George Voutsadakis (LSSU) Trigonometry January 2015 1 / 25 Outline 1 Functions
More informationSYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS ALEVEL MATHEMATICS
SYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS ALEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be one hour paper consisting of 4 questions..
More informationPolar Form of Complex Numbers
OpenStaxCNX module: m49408 1 Polar Form of Complex Numbers OpenStax College This work is produced by OpenStaxCNX and licensed under the Creative Commons Attribution License 4.0 In this section, you will:
More informationChapter 7 PHASORS ALGEBRA
164 Chapter 7 PHASORS ALGEBRA Vectors, in general, may be located anywhere in space. We have restricted ourselves thus for to vectors which are all located in one plane (co planar vectors), but they may
More informationFind the common ratio of the geometric sequence. (2) 1 + 2
. Given that z z 2 = 2 i, z, find z in the form a + ib. (Total 4 marks) 2. A geometric sequence u, u 2, u 3,... has u = 27 and a sum to infinity of 8. 2 Find the common ratio of the geometric sequence.
More informationComplex Analysis Homework 1: Solutions
Complex Analysis Fall 007 Homework 1: Solutions 1.1.. a) + i)4 + i) 8 ) + 1 + )i 5 + 14i b) 8 + 6i) 64 6) + 48 + 48)i 8 + 96i c) 1 + ) 1 + i 1 + 1 i) 1 + i)1 i) 1 + i ) 5 ) i 5 4 9 ) + 4 4 15 i ) 15 4
More informationThe number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.
ADVANCED GCE UNIT / MATHEMATICS (MEI Further Methods for Advanced Mathematics (FP THURSDAY JUNE Additional materials: Answer booklet (8 pages Graph paper MEI Eamination Formulae and Tables (MF Morning
More informationMTH 362: Advanced Engineering Mathematics
MTH 362: Advanced Engineering Mathematics Lecture 1 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 7, 2017 Course Name and number: MTH 362: Advanced Engineering
More informationComplex Numbers, Polar Coordinates, and Parametric Equations
8 Complex Numbers, Polar Coordinates, and Parametric Equations If a golfer tees off with an initial velocity of v 0 feet per second and an initial angle of trajectory u, we can describe the position of
More informationChapter 3: Complex Numbers
Chapter 3: Complex Numbers Daniel Chan UNSW Semester 1 2018 Daniel Chan (UNSW) Chapter 3: Complex Numbers Semester 1 2018 1 / 48 Philosophical discussion about numbers Q In what sense is 1 a number? DISCUSS
More informationTopic 1 Part 3 [483 marks]
Topic Part 3 [483 marks] The complex numbers z = i and z = 3 i are represented by the points A and B respectively on an Argand diagram Given that O is the origin, a Find AB, giving your answer in the form
More informationSyllabus: for Complex variables
EE2020, Spring 2009 p. 1/42 Syllabus: for omplex variables 1. Midterm, (4/27). 2. Introduction to Numerical PDE (4/30): [Ref.num]. 3. omplex variables: [Textbook]h.13h.18. omplex numbers and functions,
More informationComplex Numbers. Integers, Rationals, and Reals. The natural numbers: The integers:
Complex Numbers Integers, Rationals, and Reals The natural numbers: N {... 3, 2,, 0,, 2, 3...} The integers: Z {... 3, 2,, 0,, 2, 3...} Note that any two integers added, subtracted, or multiplied together
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
Chapter 5 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 5. Overview We know that the square of a real number is always nonnegative e.g. (4) 6 and ( 4) 6. Therefore, square root of 6 is ± 4. What about the square
More informationMathematics Specialist Units 3 & 4 Program 2018
Mathematics Specialist Units 3 & 4 Program 018 Week Content Assessments Complex numbers Cartesian Forms Term 1 3.1.1 review real and imaginary parts Re(z) and Im(z) of a complex number z Week 1 3.1. review
More informationPrecalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers. A: Initial Point (start); B: Terminal Point (end) : ( ) ( )
Syllabus Objectives: 5.1 The student will explore methods of vector addition and subtraction. 5. The student will develop strategies for computing a vector s direction angle and magnitude given its coordinates.
More information4.1 Exponential and Logarithmic Functions
. Exponential and Logarithmic Functions Joseph Heavner Honors Complex Analysis Continued) Chapter July, 05 3.) Find the derivative of f ) e i e i. d d e i e i) d d ei ) d d e i ) e i d d i) e i d d i)
More informationFurther mathematics. AS and A level content
Further mathematics AS and A level content December 2014 s for further mathematics AS and A level for teaching from 2017 3 Introduction 3 Purpose 3 Aims and objectives 3 Subject content 5 Structure 5 Background
More information3 + 4i 2 + 3i. 3 4i Fig 1b
The introduction of complex numbers in the 16th century was a natural step in a sequence of extensions of the positive integers, starting with the introduction of negative numbers (to solve equations of
More informationMATH 135: COMPLEX NUMBERS
MATH 135: COMPLEX NUMBERS (WINTER, 010) The complex numbers C are important in just about every branch of mathematics. These notes 1 present some basic facts about them. 1. The Complex Plane A complex
More informationHyperbolic functions
Hyperbolic functions Attila Máté Brooklyn College of the City University of New York January 5, 206 Contents Hyperbolic functions 2 Connection with the unit hyperbola 2 2. Geometric meaning of the parameter...........................
More informationIn Z: x + 3 = 2 3x = 2 x = 1 No solution In Q: 3x = 2 x 2 = 2. x = 2 No solution. In R: x 2 = 2 x = 0 x = ± 2 No solution Z Q.
THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH 1141 HIGHER MATHEMATICS 1A ALGEBRA. Section 1:  Complex Numbers. 1. The Number Systems. Let us begin by trying to solve various
More informationCM2202: Scientific Computing and Multimedia Applications General Maths: 3. Complex Numbers
CM2202: Scientific Computing and Multimedia Applications General Maths: 3. Complex Numbers Prof. David Marshall School of Computer Science & Informatics A problem when solving some equations There are
More information( ) ( ) ( ) 2 6A: Special Trig Limits! Math 400
2 6A: Special Trig Limits Math 400 This section focuses entirely on the its of 2 specific trigonometric functions. The use of Theorem and the indeterminate cases of Theorem are all considered. a The it
More informationGetting Ready to Teach Online course Core Pure
Getting Ready to Teach Online course Core Pure GCE Further Mathematics (2017) Poll 1 Which boards do you have experience of teaching Further Maths with? GCE Further Mathematics (2017) Poll 2 Which Edexcel
More informationor i 2 = 1 i 4 = 1 Example : ( i ),, 7 i and 0 are complex numbers. and Imaginary part of z = b or img z = b
1 A LEVEL MATHEMATICS P 3 Complex Numbers (NOTES) 1. Given a quadratic equation : x 2 + 1 = 0 or ( x 2 = 1 ) has no solution in the set of real numbers, as there does not exist any real number whose
More information1 Functions and Inverses
October, 08 MAT86 Week Justin Ko Functions and Inverses Definition. A function f : D R is a rule that assigns each element in a set D to eactly one element f() in R. The set D is called the domain of f.
More informationP3.C8.COMPLEX NUMBERS
Recall: Within the real number system, we can solve equation of the form and b 2 4ac 0. ax 2 + bx + c =0, where a, b, c R What is R? They are real numbers on the number line e.g: 2, 4, π, 3.167, 2 3 Therefore,
More informationQuick Overview: Complex Numbers
Quick Overview: Complex Numbers February 23, 2012 1 Initial Definitions Definition 1 The complex number z is defined as: z = a + bi (1) where a, b are real numbers and i = 1. Remarks about the definition:
More informationPURE MATHEMATICS AM 27
AM SYLLABUS (2020) PURE MATHEMATICS AM 27 SYLLABUS 1 Pure Mathematics AM 27 (Available in September ) Syllabus Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics
More informationMA40S Precalculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically
1 MA40S Precalculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically Definition Trigonometric identity Investigate 1. Using the diagram
More informationOverview of Complex Numbers
Overview of Complex Numbers Definition 1 The complex number z is defined as: z = a+bi, where a, b are real numbers and i = 1. General notes about z = a + bi Engineers typically use j instead of i. Examples
More informationz = a + ib (4.1) i 2 = 1. (4.2) <(z) = a, (4.3) =(z) = b. (4.4)
Chapter 4 Complex Numbers 4.1 Definition of Complex Numbers A complex number is a number of the form where a and b are real numbers and i has the property that z a + ib (4.1) i 2 1. (4.2) a is called the
More informationMATHEMATICS. Higher 2 (Syllabus 9740)
MATHEMATICS Higher (Syllabus 9740) CONTENTS Page AIMS ASSESSMENT OBJECTIVES (AO) USE OF GRAPHING CALCULATOR (GC) 3 LIST OF FORMULAE 3 INTEGRATION AND APPLICATION 3 SCHEME OF EXAMINATION PAPERS 3 CONTENT
More informationMTH3101 Spring 2017 HW Assignment 4: Sec. 26: #6,7; Sec. 33: #5,7; Sec. 38: #8; Sec. 40: #2 The due date for this assignment is 2/23/17.
MTH0 Spring 07 HW Assignment : Sec. 6: #6,7; Sec. : #5,7; Sec. 8: #8; Sec. 0: # The due date for this assignment is //7. Sec. 6: #6. Use results in Sec. to verify that the function g z = ln r + iθ r >
More informationComplex Trigonometric and Hyperbolic Functions (7A)
Complex Trigonometric and Hyperbolic Functions (7A) 07/08/015 Copyright (c) 011015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationPure Further Mathematics 2. Revision Notes
Pure Further Mathematics Revision Notes October 016 FP OCT 016 SDB Further Pure 1 Inequalities... 3 Algebraic solutions... 3 Graphical solutions... 4 Series Method of Differences... 5 3 Comple Numbers...
More informationB Elements of Complex Analysis
Fourier Transform Methods in Finance By Umberto Cherubini Giovanni Della Lunga Sabrina Mulinacci Pietro Rossi Copyright 21 John Wiley & Sons Ltd B Elements of Complex Analysis B.1 COMPLEX NUMBERS The purpose
More informationPreCalculus Assignment Sheet Unit 83rd term January 20 th to February 6 th 2015 Polynomials
PreCalculus Assignment Sheet Unit 8 rd term January 0 th to February 6 th 01 Polynomials Date Topic Assignment Calculator Did it Tuesday Multiplicity of zeroes of 1/0/1 a function TInspire activity
More informationA Learning Progression for Complex Numbers
A Learning Progression for Complex Numbers In mathematics curriculum development around the world, the opportunity for students to study complex numbers in secondary schools is decreasing. Given that the
More informationTopic 4 Notes Jeremy Orloff
Topic 4 Notes Jeremy Orloff 4 Complex numbers and exponentials 4.1 Goals 1. Do arithmetic with complex numbers.. Define and compute: magnitude, argument and complex conjugate of a complex number. 3. Euler
More informationMTH4101 Calculus II. Carl Murray School of Mathematical Sciences Queen Mary University of London Spring Lecture Notes
MTH40 Calculus II Carl Murray School of Mathematical Sciences Queen Mary University of London Spring 20 Lecture Notes Complex Numbers. Introduction We have already met several types of numbers. Natural
More information= + then for all n N. n= is true, now assume the statement is. ) clearly the base case 1 ( ) ( ) ( )( ) θ θ θ θ ( θ θ θ θ)
Complex numbers mixed exercise i a We have e cos + isin hence i i ( e + e ) ( cos + isin + cos + isin ) ( cos + isin + cos sin) cos Where we have used the fact that cos cos sin sin b We have ia ia ib ib
More informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India Email : sahoopulak1@gmail.com 1 Module1: Basic Ideas 1 Introduction
More informationComplex Numbers and Exponentials
omple Numbers and Eponentials Definition and Basic Operations comple number is nothing more than a point in the plane. The sum and product of two comple numbers ( 1, 1 ) and ( 2, 2 ) is defined b ( 1,
More information1,cost 1 1,tant 0 1,cott ,cost 0 1,tant 0. 1,cott 1 0. ,cost 5 6,tant ,cott x 2 1 x. 1 x 2. Name: Class: Date:
Class: Date: Practice Test (Trigonometry) Instructor: Koshal Dahal Multiple Choice Questions SHOW ALL WORK, EVEN FOR MULTIPLE CHOICE QUESTIONS, TO RECEIVE CREDIT. 1. Find the values of the trigonometric
More informationALGEBRAIC LONG DIVISION
QUESTIONS: 2014; 2c 2013; 1c ALGEBRAIC LONG DIVISION x + n ax 3 + bx 2 + cx +d Used to find factors and remainders of functions for instance 2x 3 + 9x 2 + 8x + p This process is useful for finding factors
More informationUNIT TWO POLAR COORDINATES AND COMPLEX NUMBERS MATH 611B 15 HOURS
UNIT TWO POLAR COORDINATES AND COMPLEX NUMBERS MATH 611B 15 HOURS Revised Dec 10, 02 38 SCO: By the end of grade 12, students will be expected to: C97 construct and examine graphs in the polar plane Elaborations
More informationUNIVERSITY OF SOUTHAMPTON
UNIVERSITY OF SOUTHAMPTON MATH03W SEMESTER EXAMINATION 0/ MATHEMATICS FOR ELECTRONIC & ELECTRICAL ENGINEERING Duration: 0 min This paper has two parts, part A and part B. Answer all questions from part
More informationComplex Numbers. Copyright Cengage Learning. All rights reserved.
4 Complex Numbers Copyright Cengage Learning. All rights reserved. 4.1 Complex Numbers Copyright Cengage Learning. All rights reserved. Objectives Use the imaginary unit i to write complex numbers. Add,
More informationIntegrating Algebra and Geometry with Complex Numbers
Integrating Algebra and Geometry with Complex Numbers Complex numbers in schools are often considered only from an algebraic perspective. Yet, they have a rich geometric meaning that can support developing
More informationLecture 1 Complex Numbers. 1 The field of complex numbers. 1.1 Arithmetic operations. 1.2 Field structure of C. MATHGA Complex Variables
Lecture Complex Numbers MATHGA 245.00 Complex Variables The field of complex numbers. Arithmetic operations The field C of complex numbers is obtained by adjoining the imaginary unit i to the field R
More informationWith this expanded version of what we mean by a solution to an equation we can solve equations that previously had no solution.
M 74 An introduction to Complex Numbers. 1 Solving equations Throughout the calculus sequence we have limited our discussion to real valued solutions to equations. We know the equation x 1 = 0 has distinct
More informationCandidates sitting FP2 may also require those formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1 C4. e π.
F Further IAL Pure PAPERS: Mathematics FP 046 AND SPECIMEN Candidates sitting FP may also require those formulae listed under Further Pure Mathematics FP and Core Mathematics C C4. Area of a sector A
More informationThis leaflet describes how complex numbers are added, subtracted, multiplied and divided.
7. Introduction. Complex arithmetic This leaflet describes how complex numbers are added, subtracted, multiplied and divided. 1. Addition and subtraction of complex numbers. Given two complex numbers we
More informationC. Complex Numbers. 1. Complex arithmetic.
C. Complex Numbers. Complex arithmetic. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they first appeared.
More informationSection 4.3: Quadratic Formula
Objective: Solve quadratic equations using the quadratic formula. In this section we will develop a formula to solve any quadratic equation ab c 0 where a b and c are real numbers and a 0. Solve for this
More informationMATH 100 REVIEW PACKAGE
SCHOOL OF UNIVERSITY ARTS AND SCIENCES MATH 00 REVIEW PACKAGE Gearing up for calculus and preparing for the Assessment Test that everybody writes on at. You are strongly encouraged not to use a calculator
More information8. Complex Numbers. sums and products. basic algebraic properties. complex conjugates. exponential form. principal arguments. roots of complex numbers
EE202  EE MATH II 8. Complex Numbers Jitkomut Songsiri sums and products basic algebraic properties complex conjugates exponential form principal arguments roots of complex numbers regions in the complex
More informationChapter 8: Polar Coordinates and Vectors
Chapter 8: Polar Coordinates and Vectors 8.1 Polar Coordinates This is another way (in addition to the xy system) of specifying the position of a point in the plane. We give the distance r of the point
More informationPreCalculus Notes. MAT 129 Chapter 10: Polar Coordinates; Vectors. David J. Gisch. Department of Mathematics Des Moines Area Community College
PreCalculus Notes MAT 129 Chapter 10: Polar Coordinates; Vectors David J. Gisch Department of Mathematics Des Moines Area Community College October 25, 2011 1 Chapter 10 Section 10.1: Polar Coordinates
More information10.1 Complex Arithmetic Argand Diagrams and the Polar Form The Exponential Form of a Complex Number De Moivre s Theorem 29
10 Contents Complex Numbers 10.1 Complex Arithmetic 2 10.2 Argand Diagrams and the Polar Form 12 10.3 The Exponential Form of a Complex Number 20 10.4 De Moivre s Theorem 29 Learning outcomes In this Workbook
More informationSolutions to Selected Exercises. Complex Analysis with Applications by N. Asmar and L. Grafakos
Solutions to Selected Exercises in Complex Analysis with Applications by N. Asmar and L. Grafakos Section. Complex Numbers Solutions to Exercises.. We have i + i. So a and b. 5. We have So a 3 and b 4.
More information18.04 Practice problems exam 1, Spring 2018 Solutions
8.4 Practice problems exam, Spring 8 Solutions Problem. omplex arithmetic (a) Find the real and imaginary part of z + z. (b) Solve z 4 i =. (c) Find all possible values of i. (d) Express cos(4x) in terms
More informationThe number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.
ADVANCED GCE UNIT 7/ MATHEMATICS (MEI Further Methods for Advanced Mathematics (FP THURSDAY 7 JUNE 7 Additional materials: Answer booklet (8 pages Graph paper MEI Examination Formulae and Tables (MF Morning
More informationP.6 Complex Numbers. 6, 5i, 25, 7i, 5 2 i + 2 3, i, 53i, i. DEFINITION Complex Number. Operations with Complex Numbers
SECTION P.6 Complex Numbers 49 P.6 Complex Numbers What you ll learn about Complex Numbers Operations with Complex Numbers Complex Conjugates and Division Complex Solutions of Quadratic Equations... and
More informationFurther Pure Mathematics 1
Complex Numbers Section Supplementary Notes and Examples The modulusargument form of complex numbers You are familiar with describing a point in the plane using Cartesian coordinates. However, this is
More informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.4 Complex Numbers Copyright Cengage Learning. All rights reserved. What You Should Learn Use the imaginary unit i
More information