2.0 COMPLEX NUMBER SYSTEM. Bakiss Hiyana bt Abu Bakar JKE, POLISAS BHAB 1
|
|
- Dana Carter
- 6 years ago
- Views:
Transcription
1 2.0 COMPLEX NUMBER SYSTEM Bakiss Hiyana bt Abu Bakar JKE, POLISAS BHAB 1
2 COURSE LEARNING OUTCOME 1. Explain AC circuit concept and their analysis using AC circuit law. 2. Apply the knowledge of AC circuit in solving problem related to AC electrical circuit. BHAB 2
3 CHAPTER CONTENT Understand the complex plane Understand real and imaginary numbers COMPLEX NUMBER SYSTEM Understand phasor quantities in both rectangular and polar forms Understand rectangular form and polar form Understand arithmetic operations with complex numbers BHAB 3
4 2.1 UNDERSTAND THE COMPLEX PLANE LABEL POSITIVE AND NEGATIVE NUMBERS BHAB 4
5 2.1 UNDERSTAND THE COMPLEX PLANE LABEL POSITIVE AND NEGATIVE NUMBERS On the Argand diagram, the horizontal axis represents all positive real numbers to the right of the vertical imaginary axis and all negative real numbers to the left of the vertical imaginary axis. All positive imaginary numbers are represented above the horizontal axis while all the negative imaginary numbers are below the horizontal real axis. This then produces a two dimensional complex plane with four distinct quadrants labelled, QI, QII, QIII, and QIV BHAB 5
6 2.1.2 CONSTRUCT A COMPLEX PLANE - A two-dimensional graph where the horizontal axis maps is the real part and the vertical axis maps is the imaginary part of any complex number or function. - The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. - A complex number can be viewed as a point or position vector in a twodimensional Cartesian coordinate system called the complex plane or Argand diagram BHAB 6
7 2.1.3 DRAW THE ANGULAR POSITION ON THE COMPLEX PLANE - A complex number can be visually represented as a pair of numbers (a,b) forming a vector on a diagram called an Argand diagram, representing the complex plane. BHAB 7
8 2.1.3 DRAW THE ANGULAR POSITION ON THE COMPLEX PLANE - Complex numbers can also be expressed in polar coordinates as r θ. BHAB 8
9 2.2 UNDERSTAND REAL AND IMAGINARY NUMBERS DEFINE A REAL NUMBER AND IMAGINARY NUMBER - A complex number has a real part & imaginary part. - Standard form is: a + bj Real part Imaginary part BHAB 9
10 2.2.1 DEFINE A REAL NUMBER AND IMAGINARY NUMBER Z = x + jy Where: Z = is the complex number representing the vector x = is the Real part or the active component y = is the Imaginary part or the reactive component j = is define by -1 BHAB 10
11 2.2.1 DEFINE A REAL NUMBER AND IMAGINARY NUMBER A complex number is an expression in the form: a + bj where a and b are real numbers. The symbol j is defined as j = -1 : j is the imaginary unit. a is the real part of the complex number, and b is the complex part of the complex number. Then a + bj is called complex number. BHAB 11
12 2.2.2 DETERMINE A VALUE OF J The imaginary unit, j, is defined as j = Therefore, j 2 = -1 we can notice that: j 3 = j 2 x j = -1 x j = -j j 4 = j 2 x j 2 = -1 x -1 = 1 Example: Simplify j 12 By what we saw above we can simply write j 12 = (j 4 ) 3 = ( j 2 x j 2 ) 3 = ( 1 ) 3 Therefore, j 12 = 1 BHAB 12
13 2.3 UNDERSTAND PHASOR QUANTITIES IN BOTH RECTANGULAR AND POLAR FORMS EXPRESS PHASOR QUANTITIES IN BOTH RECTANGULAR AND POLAR FORMS - THEORY: Rectangular coordinates & polar coordinates are 2 different ways of using 2 numbers to locate a point on a plane. - Rectangular: coordinates are in the form (x,y), where x and y are the horizontal & vertical distances from the origin. - Rectangular form : Z = a + bj BHAB 13
14 2.3.1 EXPRESS PHASOR QUANTITIES IN BOTH RECTANGULAR AND POLAR FORMS - Rectangular form : Z = a + bj - Example: identify 1 + 2j and 3 - j graphically BHAB 14
15 2.3.1 EXPRESS PHASOR QUANTITIES IN BOTH RECTANGULAR AND POLAR FORMS - Polar : Coordinates are in the form ( r, θ ) where r is the distance from the origin to the point and θ is the angle measured from the positive x axis to the point. - Polar form : x BHAB 15
16 2.4 UNDERSTAND RECTANGULAR FORM AND POLAR FORMS CONVERT BETWEEN RECTANGULAR AND POLAR FORMS - Example: BHAB 16
17 TRANSFORM A POLAR TO RECTANGULAR FORM - Example: Express = 5.19 in rectangular form Therefore; Z = j 3 BHAB 17
18 2.5 UNDERSTAND ARITHMETIC OPERATIONS WITH COMPLEX NUMBER ADD: PERFORM OPERATION WITH COMPLEX NUMBER - Complex numbers are added by adding the real and imaginary parts of the summands. That is to say: ( a + jb ) + ( c + jd ) = ( a + c ) + j( b + d ) SUBTRACT: - To subtract the complex number from another, we subtract the corresponding real parts & subtract the corresponding imaginary part. - Subtraction is defined by; ( a + jb ) ( c + jd ) = ( a c ) + j( b d ) BHAB 18
19 Example: BHAB 19
20 MULTIPLICATION: - The multiplication of two complex numbers is defined by the following formula: ( a + bj )( c + dj ) = ( ac bd ) + ( bc + ad )j - The preceding definition of multiplication of general complex numbers is the natural way of extending this fundamental property of the imaginary unit. Indeed, treating j as a variable, the formula follows from this ( a + bj )( c + dj ) = ac + bcj + adj + bdj 2 = ac + bdj 2 + ( bc + ad )j = ac + bd (-1) + ( bc + ad )j = ( ac bd ) + ( bc + ad )j - In particular, the square of the imaginary ( j 2 ) unit is 1 - Whenever we multiply a complex number by it conjugate, the answer is a real number - If z = a + bj, then z = a 2 + b 2 BHAB 20
21 Example: Given Z 1 = 2 2j, Z 2 = 3 + 4j Find Z 1.Z2 Z 1.Z 2 = ( 2 2j ).(3 + 4j ) = 6 + 8j 6j 8j 2 = 6 + 2j 8(-1) ; j 2 = -1 = j Example: Given Z = 3 2j, find if Z = 3 2j, then the conjugate is = ( 3 2j ).(3 + 2j ) = 9 + 6j 6j 4j 2 = = 13 = 3 + 2j, therefore, BHAB 21
22 DIVISION: - To divide 2 complex number, it is necessary to make use of the complex conjugate. - We multiply both the numerator & denominator by the conjugate of the denominator & then simplify the result. - Example: Z 1 = 2 + 9j, Z 2 = 5 2j, find BHAB 22
23 MULTIPLICATION & DIVISION IN POLAR FORM MULTIPLICATION: DIVISION: BHAB 23
24 EXAMPLE: MULTIPLICATION & DIVISION IN POLAR FORM Multiplying together 6 30 o and 8 45 o in polar form gives us. Likewise, to divide together two vectors in polar form, we must divide the two modulus and then subtract their angles as shown. BHAB 24
25 SUMMARY Complex number consist of two distinct numbers, a real number an imaginary number. Imaginary number are distinguish from a real number by use of the j operator. A number with letter j in front of it identities is an imaginary number in the complex plane. By definition, the j operator = -1. Imaginary number can be +, -, and The multiplication of j by j gives j 2 = -1 the same as real numbers. In rectangular form a complex number is represented by a point in space on the complex plane. In polar form a complex number is represented by a line whose length is the amplitude and by the phase angle. BHAB 25
26 Rectangular Form Polar Form BHAB 26
27 ADD SUBTRACT MULTIPLICATION DIVISION BHAB 27
EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 2 - COMPLEX NUMBERS
EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME - ALGEBRAIC TECHNIQUES TUTORIAL - COMPLEX NUMBERS CONTENTS Be able to apply algebraic techniques Arithmetic progression (AP):
More informationJUST THE MATHS UNIT NUMBER 6.1. COMPLEX NUMBERS 1 (Definitions and algebra) A.J.Hobson
JUST THE MATHS UNIT NUMBER 6.1 COMPLEX NUMBERS 1 (Definitions and algebra) by A.J.Hobson 6.1.1 The definition of a complex number 6.1.2 The algebra of complex numbers 6.1.3 Exercises 6.1.4 Answers to exercises
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 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 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 informationRevision of Basic A.C. Theory
Revision of Basic A.C. Theory 1 Revision of Basic AC Theory (Much of this material has come from Electrical & Electronic Principles & Technology by John Bird) Electricity is generated in power stations
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 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 informationhp calculators HP 9s Solving Problems Involving Complex Numbers Basic Concepts Practice Solving Problems Involving Complex Numbers
Basic Concepts Practice Solving Problems Involving Complex Numbers Basic concepts There is no real number x such that x + 1 = 0. To solve this kind of equations a new set of numbers must be introduced.
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 informationENGIN 211, Engineering Math. Complex Numbers
ENGIN 211, Engineering Math Complex Numbers 1 Imaginary Number and the Symbol J Consider the solutions for this quadratic equation: x 2 + 1 = 0 x = ± 1 1 is called the imaginary number, and we use the
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 informationLesson 8: Complex Number Division
Student Outcomes Students determine the modulus and conjugate of a complex number. Students use the concept of conjugate to divide complex numbers. Lesson Notes This is the second day of a two-day lesson
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 informationModule 4. Single-phase AC Circuits
Module 4 Single-phase AC Circuits Lesson 13 Representation of Sinusoidal Signal by a Phasor and Solution of Current in R-L-C Series Circuits In the last lesson, two points were described: 1. How a sinusoidal
More informationRational Numbers. a) 5 is a rational number TRUE FALSE. is a rational number TRUE FALSE
Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 1 Chapter 1A - - Real Numbers Types of Real Numbers Name(s) for the set 1, 2,, 4, Natural Numbers Positive Integers Symbol(s) for the set, -,
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 informationNatural Numbers Positive Integers. Rational Numbers
Chapter A - - Real Numbers Types of Real Numbers, 2,, 4, Name(s) for the set Natural Numbers Positive Integers Symbol(s) for the set, -, - 2, - Negative integers 0,, 2,, 4, Non- negative integers, -, -
More informationENGIN 211, Engineering Math. Complex Numbers
ENGIN 211, Engineering Math Complex Numbers 1 Imaginary Number and the Symbol J Consider the solutions for this quadratic equation: x 2 + 1 = 0 x = ± 1 1 is called the imaginary number, and we use the
More informationRev Name Date. . For example: 5x 3x
Name Date TI-84+ GC 7 Testing Polynomial Inequalities in One Variable Objectives: Review algebraic method for solving polynomial inequalities Review the signs of y-coordinates of points in each quadrant
More informationModule 10 Polar Form of Complex Numbers
MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex
More informationChapter 2 Linear Equations and Inequalities in One Variable
Chapter 2 Linear Equations and Inequalities in One Variable Section 2.1: Linear Equations in One Variable Section 2.3: Solving Formulas Section 2.5: Linear Inequalities in One Variable Section 2.6: Compound
More informationMA 180 Lecture. Chapter 0. College Algebra and Calculus by Larson/Hodgkins. Fundamental Concepts of Algebra
0.) Real Numbers: Order and Absolute Value Definitions: Set: is a collection of objections in mathematics Real Numbers: set of numbers used in arithmetic MA 80 Lecture Chapter 0 College Algebra and Calculus
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 informationP arenthesis E xponents M ultiplication D ivision A ddition S ubtraction
Section 1: Order of Operations P arenthesis E xponents M ultiplication D ivision A ddition S ubtraction Simplify the following: (18 + 4) 3(10 2 3 2 6) Work inside first set of parenthesis first = 22 3(10
More informationSECTION 6.3: VECTORS IN THE PLANE
(Section 6.3: Vectors in the Plane) 6.18 SECTION 6.3: VECTORS IN THE PLANE Assume a, b, c, and d are real numbers. PART A: INTRO A scalar has magnitude but not direction. We think of real numbers as scalars,
More informationCHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic
CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic equations. They can be solved using a graph, a perfect square,
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 informationSect Complex Numbers
161 Sect 10.8 - Complex Numbers Concept #1 Imaginary Numbers In the beginning of this chapter, we saw that the was undefined in the real numbers since there is no real number whose square is equal to a
More informationCHAPTER ONE FUNCTIONS AND GRAPHS. In everyday life, many quantities depend on one or more changing variables eg:
CHAPTER ONE FUNCTIONS AND GRAPHS 1.0 Introduction to Functions In everyday life, many quantities depend on one or more changing variables eg: (a) plant growth depends on sunlight and rainfall (b) speed
More informationChapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,
More informationCombinational Logic. By : Ali Mustafa
Combinational Logic By : Ali Mustafa Contents Adder Subtractor Multiplier Comparator Decoder Encoder Multiplexer How to Analyze any combinational circuit like this? Analysis Procedure To obtain the output
More informationChapter. Algebra techniques. Syllabus Content A Basic Mathematics 10% Basic algebraic techniques and the solution of equations.
Chapter 2 Algebra techniques Syllabus Content A Basic Mathematics 10% Basic algebraic techniques and the solution of equations. Page 1 2.1 What is algebra? In order to extend the usefulness of mathematical
More informationDate: Math 7 Final Exam Review Name:
Specific Outcome Achievement Indicators Chapter Got it?? the Solve a given problem involving the addition of two or more decimal numbers. addition, subtraction, multiplication and division of decimals
More informationComplex Numbers and Phasor Technique
A P P E N D I X A Complex Numbers and Phasor Technique In this appendix, we discuss a mathematical technique known as the phasor technique, pertinent to operations involving sinusoidally time-varying quantities.
More informationGEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING ECE 2026 Summer 2018 Problem Set #1 Assigned: May 14, 2018 Due: May 22, 2018 Reading: Chapter 1; App. A on Complex Numbers,
More informationMATH 150 Pre-Calculus
MATH 150 Pre-Calculus Fall, 2014, WEEK 3 JoungDong Kim Week 3: 2B, 3A Chapter 2B. Solving Inequalities a < b a is less than b a b a is less than or equal to b a > b a is greater than b a b a is greater
More informationJUST THE MATHS UNIT NUMBER 6.2. COMPLEX NUMBERS 2 (The Argand Diagram) A.J.Hobson
JUST THE MATHS UNIT NUMBER 6.2 COMPLEX NUMBERS 2 (The Argand Diagram) by A.J.Hobson 6.2.1 Introduction 6.2.2 Graphical addition and subtraction 6.2.3 Multiplication by j 6.2.4 Modulus and argument 6.2.5
More informationQuadratic equations: complex solutions
October 28 (H), November 1 (A), 2016 Complex number system page 1 Quadratic equations: complex solutions An issue that can arise when solving a quadratic equation by the Quadratic Formula is the need to
More informationLesson 8: Complex Number Division
Student Outcomes Students determine the modulus and conjugate of a complex number. Students use the concept of conjugate to divide complex numbers. Lesson Notes This is the second day of a two-day lesson
More informationFind the component form of with initial point A(1, 3) and terminal point B(1, 3). Component form = 1 1, 3 ( 3) (x 1., y 1. ) = (1, 3) = 0, 6 Subtract.
Express a Vector in Component Form Find the component form of with initial point A(1, 3) and terminal point B(1, 3). = x 2 x 1, y 2 y 1 Component form = 1 1, 3 ( 3) (x 1, y 1 ) = (1, 3) and ( x 2, y 2
More informationAlgebra I. abscissa the distance along the horizontal axis in a coordinate graph; graphs the domain.
Algebra I abscissa the distance along the horizontal axis in a coordinate graph; graphs the domain. absolute value the numerical [value] when direction or sign is not considered. (two words) additive inverse
More informationMATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL 3 PERIODIC FUNCTIONS
MATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL 3 PERIODIC FUNCTIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.
More informationEE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 18 121025 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review RMS Values Complex Numbers Phasors Complex Impedance Circuit Analysis
More informationMATHS (O) NOTES. SUBJECT: Maths LEVEL: Ordinary Level TEACHER: Jean Kelly. The Institute of Education Topics Covered: Complex Numbers
MATHS (O) NOTES The Institute of Education 07 SUBJECT: Maths LEVEL: Ordinary Level TEACHER: Jean Kelly Topics Covered: COMPLEX NUMBERS Strand 3(Unit ) Syllabus - Understanding the origin and need for complex
More informationChapter 1 Linear Equations and Graphs
Chapter 1 Linear Equations and Graphs Section R Linear Equations and Inequalities Important Terms, Symbols, Concepts 1.1. Linear Equations and Inequalities A first degree, or linear, equation in one variable
More informationCONTENTS COLLEGE ALGEBRA: DR.YOU
1 CONTENTS CONTENTS Textbook UNIT 1 LECTURE 1-1 REVIEW A. p. LECTURE 1- RADICALS A.10 p.9 LECTURE 1- COMPLEX NUMBERS A.7 p.17 LECTURE 1-4 BASIC FACTORS A. p.4 LECTURE 1-5. SOLVING THE EQUATIONS A.6 p.
More informationStudent s Name Course Name Mathematics Grade 7. General Outcome: Develop number sense. Strand: Number. R D C Changed Outcome/achievement indicator
Strand: Number Specific Outcomes It is expected that students will: 1. Determine and explain why a number is divisible by 2, 3, 4, 5, 6, 8, 9 or 10, and why a number cannot be divided by 0. [C, R] 2. Demonstrate
More informationLesson 5.3. Solving Trigonometric Equations
Lesson 5.3 Solving To solve trigonometric equations: Use standard algebraic techniques learned in Algebra II. Look for factoring and collecting like terms. Isolate the trig function in the equation. Use
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 information3-3 Complex Numbers. Simplify. SOLUTION: 2. SOLUTION: 3. (4i)( 3i) SOLUTION: 4. SOLUTION: 5. SOLUTION: esolutions Manual - Powered by Cognero Page 1
1. Simplify. 2. 3. (4i)( 3i) 4. 5. esolutions Manual - Powered by Cognero Page 1 6. 7. Solve each equation. 8. Find the values of a and b that make each equation true. 9. 3a + (4b + 2)i = 9 6i Set the
More informationMathematical Focus 1 Complex numbers adhere to certain arithmetic properties for which they and their complex conjugates are defined.
Situation: Complex Roots in Conjugate Pairs Prepared at University of Georgia Center for Proficiency in Teaching Mathematics June 30, 2013 Sarah Major Prompt: A teacher in a high school Algebra class has
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational
More informationLesson 1: Phasors and Complex Arithmetic
//06 Lesson : Phasors an Complex Arithmetic ET 33b Ac Motors, Generators an Power Systems lesson_et33b.pptx Learning Objectives After this presentation you will be able to: Write time equations that represent
More informationVectors. Introduction. Prof Dr Ahmet ATAÇ
Chapter 3 Vectors Vectors Vector quantities Physical quantities that have both n u m e r i c a l a n d d i r e c t i o n a l properties Mathematical operations of vectors in this chapter A d d i t i o
More informationChapter P. Prerequisites. Slide P- 1. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide P- 1 Chapter P Prerequisites 1 P.1 Real Numbers Quick Review 1. List the positive integers between -4 and 4.. List all negative integers greater than -4. 3. Use a calculator to evaluate the expression
More informationComplex numbers. Learning objectives
CHAPTER Complex numbers Learning objectives After studying this chapter, you should be able to: understand what is meant by a complex number find complex roots of quadratic equations understand the term
More informationGrade Demonstrate mastery of the multiplication tables for numbers between 1 and 10 and of the corresponding division facts.
Unit 1 Number Theory 1 a B Find the prime factorization of numbers (Lesson 1.9) 5.1.6 Describe and identify prime and composite numbers. ISTEP+ T1 Pt 1 #11-14 1b BD Rename numbers written in exponential
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 informationAH 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 informationGrade 3 Unit Standards ASSESSMENT #1
ASSESSMENT #1 3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties
More informationAlgebra II Notes Quadratic Functions Unit Complex Numbers. Math Background
Complex Numbers Math Background Previously, you Studied the real number system and its sets of numbers Applied the commutative, associative and distributive properties to real numbers Used the order of
More informationMath 120, Sample Final Fall 2015
Math 10, Sample Final Fall 015 Disclaimer: This sample final is intended to help students prepare for the final exam The final exam will be similar in structure and type of problems, however the actual
More informationLinear Equations & Inequalities Definitions
Linear Equations & Inequalities Definitions Constants - a term that is only a number Example: 3; -6; -10.5 Coefficients - the number in front of a term Example: -3x 2, -3 is the coefficient Variable -
More informationComplex Numbers and Polar Coordinates
Chapter 5 Complex Numbers and Polar Coordinates One of the goals of algebra is to find solutions to polynomial equations. You have probably done this many times in the past, solving equations like x 1
More informationSection 1.3 Review of Complex Numbers
1 Section 1. Review of Complex Numbers Objective 1: Imaginary and Complex Numbers In Science and Engineering, such quantities like the 5 occur all the time. So, we need to develop a number system that
More informationIntroduction to Complex Numbers Complex Numbers
Introduction to SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/ Retell, Activating Prior Knowledge, Create Representations The equation x 2 + 1 = 0 has special historical and mathematical significance.
More informationVectors. Introduction
Chapter 3 Vectors Vectors Vector quantities Physical quantities that have both numerical and directional properties Mathematical operations of vectors in this chapter Addition Subtraction Introduction
More informationK K.OA.2 1.OA.2 2.OA.1 3.OA.3 4.OA.3 5.NF.2 6.NS.1 7.NS.3 8.EE.8c
K.OA.2 1.OA.2 2.OA.1 3.OA.3 4.OA.3 5.NF.2 6.NS.1 7.NS.3 8.EE.8c Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to Solve word problems that
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 non-negative e.g. (4) 6 and ( 4) 6. Therefore, square root of 6 is ± 4. What about the square
More informationCHAPTER 6. Section Two angles are supplementary. 2. Two angles are complementary if the sum of their measures is 90 radians
SECTION 6-5 CHAPTER 6 Section 6. Two angles are complementary if the sum of their measures is 90 radians. Two angles are supplementary if the sum of their measures is 80 ( radians).. A central angle of
More information4.2 Graphs of Rational Functions
4.2. Graphs of Rational Functions www.ck12.org 4.2 Graphs of Rational Functions Learning Objectives Compare graphs of inverse variation equations. Graph rational functions. Solve real-world problems using
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS
EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS CONTENTS 3 Be able to understand how to manipulate trigonometric expressions and apply
More informationSection 1.1: Patterns in Division
Section 1.1: Patterns in Division Dividing by 2 All even numbers are divisible by 2. E.g., all numbers ending in 0,2,4,6 or 8. Dividing by 4 1. Are the last two digits in your number divisible by 4? 2.
More information1.5 F15 O Brien. 1.5: Linear Equations and Inequalities
1.5: Linear Equations and Inequalities I. Basic Terminology A. An equation is a statement that two expressions are equal. B. To solve an equation means to find all of the values of the variable that make
More informationCHAPTER 45 COMPLEX NUMBERS
CHAPTER 45 COMPLEX NUMBERS EXERCISE 87 Page 50. Solve the quadratic equation: x + 5 0 Since x + 5 0 then x 5 x 5 ( )(5) 5 j 5 from which, x ± j5. Solve the quadratic equation: x x + 0 Since x x + 0 then
More informationMathematics Grade 3. grade 3 21
Mathematics Grade 3 In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within
More informationThe Plane of Complex Numbers
The Plane of Complex Numbers In this chapter we ll introduce the complex numbers as a plane of numbers. Each complex number will be identified by a number on a real axis and a number on an imaginary axis.
More informationChapter 9: Complex Numbers
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
More informationMATH Spring 2010 Topics per Section
MATH 101 - Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line
More informationProf. Anyes Taffard. Physics 120/220. Foundations Circuit elements Resistors: series & parallel Ohm s law Kirchhoff s laws Complex numbers
Prof. Anyes Taffard Physics 120/220 Foundations Circuit elements Resistors: series & parallel Ohm s law Kirchhoff s laws Complex numbers Foundations Units: ü Q: charge [Coulomb] ü V: voltage = potential
More information3 What You Should Know About Complex Numbers
3 What You Should Know About Complex Numbers Life is complex it has a real part, and an imaginary part Andrew Koenig. Complex numbers are an extension of the more familiar world of real numbers that make
More informationDay 1: Introduction to Vectors + Vector Arithmetic
Day 1: Introduction to Vectors + Vector Arithmetic A is a quantity that has magnitude but no direction. You can have signed scalar quantities as well. A is a quantity that has both magnitude and direction.
More informationComplex Numbers. Essential Question What are the subsets of the set of complex numbers? Integers. Whole Numbers. Natural Numbers
3.4 Complex Numbers Essential Question What are the subsets of the set of complex numbers? In your study of mathematics, you have probably worked with only real numbers, which can be represented graphically
More informationCircuit Analysis-III. Circuit Analysis-II Lecture # 3 Friday 06 th April, 18
Circuit Analysis-III Sinusoids Example #1 ü Find the amplitude, phase, period and frequency of the sinusoid: v (t ) =12cos(50t +10 ) Signal Conversion ü From sine to cosine and vice versa. ü sin (A ± B)
More informationMatrices and Vectors
Matrices and Vectors James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 11, 2013 Outline 1 Matrices and Vectors 2 Vector Details 3 Matrix
More informationMICHIGAN STANDARDS MAP for a Basic Grade-Level Program. Grade Eight Mathematics (Algebra I)
MICHIGAN STANDARDS MAP for a Basic Grade-Level Program Grade Eight Mathematics (Algebra I) L1.1.1 Language ALGEBRA I Primary Citations Supporting Citations Know the different properties that hold 1.07
More informationCoordinate Systems. Chapter 3. Cartesian Coordinate System. Polar Coordinate System
Chapter 3 Vectors Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels instructions
More informationFunction Operations and Composition of Functions. Unit 1 Lesson 6
Function Operations and Composition of Functions Unit 1 Lesson 6 Students will be able to: Combine standard function types using arithmetic operations Compose functions Key Vocabulary: Function operation
More informationLesson #33 Solving Incomplete Quadratics
Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique
More informationCHAPTER 1 POLYNOMIALS
1 CHAPTER 1 POLYNOMIALS 1.1 Removing Nested Symbols of Grouping Simplify. 1. 4x + 3( x ) + 4( x + 1). ( ) 3x + 4 5 x 3 + x 3. 3 5( y 4) + 6 y ( y + 3) 4. 3 n ( n + 5) 4 ( n + 8) 5. ( x + 5) x + 3( x 6)
More informationHonors Pre-Calculus Summer Work
Honors Pre-Calculus Summer Work Attached you will find a variety of review work based on the prerequisites needed for the Honors Pre-Calculus curriculum. The problems assigned should be the minimum you
More informationHandout 11: AC circuit. AC generator
Handout : AC circuit AC generator Figure compares the voltage across the directcurrent (DC) generator and that across the alternatingcurrent (AC) generator For DC generator, the voltage is constant For
More informationChapter 7 Quadratic Equations
Chapter 7 Quadratic Equations We have worked with trinomials of the form ax 2 + bx + c. Now we are going to work with equations of this form ax 2 + bx + c = 0 quadratic equations. When we write a quadratic
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 informationSolutions For the problem setup, please refer to the student worksheet that follows this section.
1 TASK 3.3.3: SHOCKING NUMBERS Solutions For the problem setup, please refer to the student worksheet that follows this section. 1. The impedance in part 1 of an AC circuit is Z 1 = 2 + j8 ohms, and the
More informationLESSON 9.1 ROOTS AND RADICALS
LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical
More informationAll work must be shown or no credit will be awarded. Box all answers!! Order of Operations
Steps: All work must be shown or no credit will be awarded. Box all answers!! Order of Operations 1. Do operations that occur within grouping symbols. If there is more than one set of symbols, work from
More informationElementary Algebra - Problem Drill 01: Introduction to Elementary Algebra
Elementary Algebra - Problem Drill 01: Introduction to Elementary Algebra No. 1 of 10 Instructions: (1) Read the problem and answer choices carefully (2) Work the problems on paper as 1. Which of the following
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 information