Integrating Algebra and Geometry with Complex Numbers

Size: px
Start display at page:

Download "Integrating Algebra and Geometry with Complex Numbers"

Transcription

1 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 understanding of their use as algebraic tools. Complex numbers can be used to make connections among mathematical domains for students and teachers alike. Making these connections will enable teachers and students to see the usefulness and beauty of these numbers. The development of complex numbers should build appropriate understanding. One goal is to nourish understanding to the highest possible level by considering students' backgrounds and abilities, thus seeking to arouse curiosity and bring satisfaction. One way to do this is to relate the geometric meaning with the algebraic notion in every step of an exploration of complex numbers. This brief is a list of reminders, not a comprehensive overview, of complex numbers. In the first part, important contextual steps are given recalling algebraic notions and their geometric meanings. In the second part, the step- by- step progression is addressed by pointing to the geometric meaning followed by the algebraic notation. 1. The Imaginary Unit a. Algebraic perspective: i = 1, and the roots of x = 0 are +i and i. b. Geometric perspective: Where is (imaginary) number i with respect to the number line? Multiplication by 1 means a rotation of 180 on a number line (for example, (2)( 1) = 2). Multiplying a number with 1 twice returns the original to itself through a 360 rotation as seen in Figure 1. Figure 1 Thus, Figure 1 shows that multiplication of a number, a, by ( 1) has the meaning of rotation of 360 with respect to a number line; multiplication by 1

2 ( 1), a rotation of 180, and multiplication by 1 = 1 = i would then be a rotation of 90 as in Figure 2. Figure 2 2. Notion for a Complex Number a. Algebraic perspective: Adding a real number, a, and an imaginary number, bi, gives rise to the notion of the complex number a + bi, as a new entity often denoted by z. b. Geometric perspective: The new complex number, z, is seen in the complex plane as shown in Figure 3. Figure 3 3. Addition of Complex Numbers a. Algebraic perspective: (a + bi) + (c + di) = (a + c) + (b + d)i b. Geometric perspective: Addition of complex numbers Is pictured in Figure 4. 2

3 Figure 4 4. Notation When the complex plane is introduced, the horizontal axis is labeled as x (real numbers) and the vertical axis labeled as yi (where y is real). The set of complex numbers is denoted by C. For = a + bi, we write z C, a = Re(z) and b = Im(z). The component a is the real component and b the imaginary component of z. 5. Multiplication of Complex Numbers a. Algebraic perspective: (a + bi)(c + di) = ac + adi + bci + bdi 2 = (ac bd) + (ad + bc)i b. Geometric perspective: The multiplication of a complex number, a + bi, by a real number c, represents a scaling as in Figure 5, while the multiplication of a complex number, a + bi, by the imaginary number, i, represents a rotation of 90. Figure 5 Multiplication can be represented as a composition of scaling, rotation and addition. For a comprehensive geometric view of multiplication, refer to multiplication in polar form later in the brief. 3

4 6. Properties of Operations on Complex Numbers a. Algebraic perspective: Associativity, commutativity and distributivity properties follow from properties of operations on polynomials. b. Geometric perspective: Algebraic properties can be interpreted geometrically, for example commutativity as in Figure 6. Figure 6 7. Conjugates and Absolute Value (Modulus) of Complex Numbers a. Algebraic perspective: The conjugate of the complex number, z = a + bi is z = a bi. The modulus of a complex number z = a + bi is z = a + b = a + bi (a bi) = z (z) = z Though complex numbers cannot be ordered as real numbers can, they can be compared by their absolute values. b. Geometric perspective: The conjugate and modulus of a complex number, z, are seen in Figure 7. Figure 7 4

5 8. Complex Numbers as Vectors a. Algebraic perspective: The complex number, a + bi may be thought of as the vector, a, b and added as follows: a + bi + c + di = a + c + b + d i a, b + c, d = a + c, b + d b. Geometric perspective: Complex numbers treated as vectors are seen in Figure 8. Figure 8 c. To compare the meaning of the scalar (dot) product of collinear vectors and the product of associated complex numbers, think about absolute value (modulus). Multiplication of complex numbers is not equivalent to the scalar (or vector) product of vectors. 9. Division of Complex Numbers a. Algebraic perspective: Let z be the complex number c + di. Its multiplicative inverse 1/z is seen below: = = " = " = i = " " " Division of two complex numbers, w and z, can be treated as follows where w = a + bi and z = c + di: a + bi a + bi c di ab + bd + bc ad i = = c + di c + di c di c + d = (ab + bd) c + d (bc ad) + c + d i w w z = z z b. The geometric interpretation of complex number division can most easily be seen using the polar form given later. 5

6 10. Polar Form of a Complex Number a. Geometric perspective: A complex number z, is seen in polar form in Figure 9. Figure 9 b. Algebraic perspective: For = a + bi, we have z = r(cos φ + i sinφ), where r = a + b and tan (φ) =. c. Notation: Considering a complex number z = r(cos φ + i sin φ) in a polar form, we say that r is the absolute value (modulus) (see number 7 above) and φ is the argument of a complex number z. 11. Multiplication of Two Complex Numbers in Polar Form a. Geometric perspective: The multiplication of two complex numbers, w and z, is seen in Figure 10. Figure 10 6

7 b. Algebraic perspective: Let z = r(cos φ + i sin φ) and w = q(cos ρ + i sin ρ). Thus, zw = rq(cos (φ + ρ) + i sin(φ + ρ)). 12. Division of Two Complex Numbers in Polar Form a. Geometric perspective: A geometric look at the division of two complex numbers is seen in Figure 11. Figure 11 b. Algebraic perspective: The division of two complex numbers, w and z, is given in the following: = cos (ρ φ) + i sin(ρ φ ). 13. Power of a Complex Number a. Geometric perspective: Figure 12 shows different powers of the complex number, z. 7

8 Figure 12 b. Algebraic perspective: If z = r(cos φ + i sin φ), then z = r (cos (nφ) + i sin(nφ)). 14. Roots of a Complex Number in Polar Form a. Geometric perspective: Figure 13 depicts the roots of a complex number, z. Figure 13 8

9 b. Algebraic perspective: If z = r(cos φ + i sin φ), then ω 1 = z i sin. Hence, ω 1 3 = ω 2 3 = ω 3 3 =z. = r(cos + For all nth roots of z: r(cos " + i sin ", where k = 0, 1, 2,..., n- 1. All nth roots give rise to a regular n- gon inscribed in a circle of modulus r. 15. Isometries of a Plane through Complex Numbers Compositions of reflections, rotations and translations can be represented with complex numbers as seen in Table 1. Table 1 Geometric Algebraic Reflections, conjugation z z Rotations z u z where u C and u = 1 Translations z z + a where a C 16. e " Notation The usefulness and meaning of the notation e " = cos φ + i sin φ can be explored. In particular, formulas for cos (φ + ρ) and sin (φ + ρ) can easily be derived. 17. e " + 1 = 0 The formula e " + 1 = 0, which connects five most important constants in mathematics, its meaning and historical perspectives can be explored. 9

MAT01A1: Complex Numbers (Appendix H)

MAT01A1: Complex Numbers (Appendix H) MAT01A1: Complex Numbers (Appendix H) Dr Craig 14 February 2018 Announcements: e-quiz 1 is live. Deadline is Wed 21 Feb at 23h59. e-quiz 2 (App. A, D, E, H) opens tonight at 19h00. Deadline is Thu 22 Feb

More information

A Learning Progression for Complex Numbers

A 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 information

Quick Overview: Complex Numbers

Quick 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 information

Homework problem: Find all digits to the repeating decimal 1/19 = using a calculator.

Homework problem: Find all digits to the repeating decimal 1/19 = using a calculator. MAE 501 March 3, 2009 Homework problem: Find all digits to the repeating decimal 1/19 = 0.052631578947368421 using a calculator. Katie s way On calculator, we find the multiples of 1/19: 1/19 0.0526315789

More information

Discrete mathematics I - Complex numbers

Discrete mathematics I - Complex numbers Discrete mathematics I - Emil Vatai (based on hungarian slides by László Mérai) 1 January 31, 018 1 Financed from the financial support ELTE won from the Higher Education Restructuring

More information

MATHS (O) NOTES. SUBJECT: Maths LEVEL: Ordinary Level TEACHER: Jean Kelly. The Institute of Education Topics Covered: Complex Numbers

MATHS (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 information

The 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.

The 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 information

Section 5.5. Complex Eigenvalues

Section 5.5. Complex Eigenvalues Section 5.5 Complex Eigenvalues Motivation: Describe rotations Among transformations, rotations are very simple to describe geometrically. Where are the eigenvectors? A no nonzero vector x is collinear

More information

MAT01A1: Complex Numbers (Appendix H)

MAT01A1: 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: C-Ring 508 acraig@uj.ac.za Web: http://andrewcraigmaths.wordpress.com

More information

Overview of Complex Numbers

Overview 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 information

AH Complex Numbers.notebook October 12, 2016

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 information

P.6 Complex Numbers. -6, 5i, 25, -7i, 5 2 i + 2 3, i, 5-3i, i. DEFINITION Complex Number. Operations with Complex Numbers

P.6 Complex Numbers. -6, 5i, 25, -7i, 5 2 i + 2 3, i, 5-3i, 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 information

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.

Polynomial 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

Lesson 8: Complex Number Division

Lesson 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 information

Complex Numbers. Basic algebra. Definitions. part of the complex number a+ib. ffl Addition: Notation: We i write for 1; that is we

Complex 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 information

Complex Numbers. Copyright Cengage Learning. All rights reserved.

Complex 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 information

UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction

UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction Prerequisite Skills This lesson requires the use of the following skills: finding the product of two binomials simplifying powers of i adding two fractions with different denominators (for application

More information

Chapter 3: Complex Numbers

Chapter 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 information

Complex Numbers. Introduction

Complex 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 information

1.6 Lecture 2: Conjugation and inequalities

1.6 Lecture 2: Conjugation and inequalities 1.6. LECTURE 2: CONJUGATION AND INEQUALITIES 21 Lastly, we have the exceptional case αβ = 2δ and α 2 β 2 < 4γ. In this case, z = 1 [ ( α + i β ± )] β 2 2 α 2 + 4γ. As can be seen clearly from this exercise,

More information

Geometry in the Complex Plane

Geometry in the Complex Plane Geometry in the Complex Plane Hongyi Chen UNC Awards Banquet 016 All Geometry is Algebra Many geometry problems can be solved using a purely algebraic approach - by placing the geometric diagram on a coordinate

More information

Chapter 9: Complex Numbers

Chapter 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 information

Complex Numbers. April 10, 2015

Complex 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 information

Unit 3 Specialist Maths

Unit 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 information

Lecture 5. Complex Numbers and Euler s Formula

Lecture 5. Complex Numbers and Euler s Formula Lecture 5. Complex Numbers and Euler s Formula University of British Columbia, Vancouver Yue-Xian Li March 017 1 Main purpose: To introduce some basic knowledge of complex numbers to students so that they

More information

Section 3: Complex numbers

Section 3: Complex numbers Essentially: Section 3: Complex numbers C (set of complex numbers) up to different notation: the same as R 2 (euclidean plane), (i) Write the real 1 instead of the first elementary unit vector e 1 = (1,

More information

1 Review of complex numbers

1 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 information

Complex Numbers. Vicky Neale. Michaelmas Term Introduction 1. 2 What is a complex number? 2. 3 Arithmetic of complex numbers 3

Complex Numbers. Vicky Neale. Michaelmas Term Introduction 1. 2 What is a complex number? 2. 3 Arithmetic of complex numbers 3 Complex Numbers Vicky Neale Michaelmas Term 2018 Contents 1 Introduction 1 2 What is a complex number? 2 3 Arithmetic of complex numbers 3 4 The Argand diagram 4 5 Complex conjugation 5 6 Modulus 6 7 Argument

More information

Math Circles Intro to Complex Numbers Solutions Wednesday, March 21, Rich Dlin. Rich Dlin Math Circles / 27

Math Circles Intro to Complex Numbers Solutions Wednesday, March 21, Rich Dlin. Rich Dlin Math Circles / 27 Math Circles 2018 Intro to Complex Numbers Solutions Wednesday, March 21, 2018 Rich Dlin Rich Dlin Math Circles 2018 1 / 27 Today Today s Adventure Who is Rich Dlin? What do I need to know before we start?

More information

Complex Numbers. Emily Lawson

Complex Numbers. Emily Lawson Complex Numbers Emily Lawson 1418516 March 10, 2017 Contents 1 Introduction i 2 Definition of Complex Numbers i 2.1 Real and imaginary parts of a complex number................................. i 3 Conjugation

More information

Chapter 1. Complex Numbers. Dr. Pulak Sahoo

Chapter 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 E-mail : sahoopulak1@gmail.com 1 Module-1: Basic Ideas 1 Introduction

More information

Complex Numbers and Polar Coordinates

Complex 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 information

Know why the real and complex numbers are each a field, and that particular rings are not fields (e.g., integers, polynomial rings, matrix rings)

Know why the real and complex numbers are each a field, and that particular rings are not fields (e.g., integers, polynomial rings, matrix rings) COMPETENCY 1.0 ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Know why the real and complex numbers are each a field, and that particular rings are not fields (e.g., integers, polynomial rings, matrix rings)

More information

C. Complex Numbers. 1. Complex arithmetic.

C. 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 information

Number Systems III MA1S1. Tristan McLoughlin. December 4, 2013

Number Systems III MA1S1. Tristan McLoughlin. December 4, 2013 Number Systems III MA1S1 Tristan McLoughlin December 4, 2013 http://en.wikipedia.org/wiki/binary numeral system http://accu.org/index.php/articles/1558 http://www.binaryconvert.com http://en.wikipedia.org/wiki/ascii

More information

Chapter 6 Additional Topics in Trigonometry, Part II

Chapter 6 Additional Topics in Trigonometry, Part II Chapter 6 Additional Topics in Trigonometry, Part II Section 3 Section 4 Section 5 Vectors in the Plane Vectors and Dot Products Trigonometric Form of a Complex Number Vocabulary Directed line segment

More information

Zeros and Roots of a Polynomial Function. Return to Table of Contents

Zeros and Roots of a Polynomial Function. Return to Table of Contents Zeros and Roots of a Polynomial Function Return to Table of Contents 182 Real Zeros of Polynomial Functions For a function f(x) and a real number a, if f (a) = 0, the following statements are equivalent:

More information

Complex Numbers Introduction. Number Systems. Natural Numbers ℵ Integer Z Rational Q Real Complex C

Complex 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 information

PreCalculus 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 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 information

MATH Fundamental Concepts of Algebra

MATH Fundamental Concepts of Algebra MATH 4001 Fundamental Concepts of Algebra Instructor: Darci L. Kracht Kent State University April, 015 0 Introduction We will begin our study of mathematics this semester with the familiar notion of even

More information

Complex Numbers. Math 3410 (University of Lethbridge) Spring / 8

Complex Numbers. Math 3410 (University of Lethbridge) Spring / 8 Complex Numbers Consider two real numbers x, y. What is 2 + x? What is x + y? What is (2 + x)(3 + y)? What is (2x + 3y)(3x + 5y)? What is the inverse of 3 + x? What one fact do I know for sure about x

More information

Math Circles Complex Numbers, Lesson 2 Solutions Wednesday, March 28, Rich Dlin. Rich Dlin Math Circles / 24

Math Circles Complex Numbers, Lesson 2 Solutions Wednesday, March 28, Rich Dlin. Rich Dlin Math Circles / 24 Math Circles 018 Complex Numbers, Lesson Solutions Wednesday, March 8, 018 Rich Dlin Rich Dlin Math Circles 018 1 / 4 Warmup and Review Here are the key things we discussed last week: The numbers 1 and

More information

Re(z) = a, For example, 3 + 2i = = 13. The complex conjugate (or simply conjugate") of z = a + bi is the number z defined by

Re(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 information

MTH 362: Advanced Engineering Mathematics

MTH 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 information

Math 2 Variable Manipulation Part 3 COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number:

Math 2 Variable Manipulation Part 3 COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number: Math 2 Variable Manipulation Part 3 COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number: 1 Examples: 1 + i 39 + 3i 0.8 2.2i 2 + πi 2 + i/2 A Complex Number is just

More information

A Primer on Complex Numbers

A Primer on Complex Numbers ams 10/10A supplementary notes ucsc A Primer on Complex Numbers c 2013, Yonatan Katznelson 1. Imaginary and complex numbers. The real numbers are can be thought of as numbers that represent quantities

More information

Complex Numbers. James K. Peterson. September 19, Department of Biological Sciences and Department of Mathematical Sciences Clemson University

Complex Numbers. James K. Peterson. September 19, Department of Biological Sciences and Department of Mathematical Sciences Clemson University Complex Numbers James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 19, 2013 Outline 1 Complex Numbers 2 Complex Number Calculations

More information

Complex Numbers. Outline. James K. Peterson. September 19, Complex Numbers. Complex Number Calculations. Complex Functions

Complex Numbers. Outline. James K. Peterson. September 19, Complex Numbers. Complex Number Calculations. Complex Functions Complex Numbers James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 19, 2013 Outline Complex Numbers Complex Number Calculations Complex

More information

STAT 801: Mathematical Statistics. Moment Generating Functions. M X (t) = E(e tx ) M X (u) = E[e utx ]

STAT 801: Mathematical Statistics. Moment Generating Functions. M X (t) = E(e tx ) M X (u) = E[e utx ] Next Section Previous Section STAT 801: Mathematical Statistics Moment Generating Functions Definition: The moment generating function of a real valued X is M X (t) = E(e tx ) defined for those real t

More information

MATH 423 Linear Algebra II Lecture 3: Subspaces of vector spaces. Review of complex numbers. Vector space over a field.

MATH 423 Linear Algebra II Lecture 3: Subspaces of vector spaces. Review of complex numbers. Vector space over a field. MATH 423 Linear Algebra II Lecture 3: Subspaces of vector spaces. Review of complex numbers. Vector space over a field. Vector space A vector space is a set V equipped with two operations, addition V V

More information

In 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.

In 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 information

Complex Analysis Homework 1: Solutions

Complex 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 information

3 + 4i 2 + 3i. 3 4i Fig 1b

3 + 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 information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Sec. 6.1 Eigenvalues and Eigenvectors Linear transformations L : V V that go from a vector space to itself are often called linear operators. Many linear operators can be understood geometrically by identifying

More information

19.1 The Square Root of Negative One. What is the square root of negative one? That is, is there any x such that. x 2 = 1? (19.1)

19.1 The Square Root of Negative One. What is the square root of negative one? That is, is there any x such that. x 2 = 1? (19.1) 86 Chapter 9 Complex Numbers In order to work our way to the Mandelbrot set, we ll need to put Julia sets aside for a moment and focus on complex numbers. In the subsequent chapter we will consider dynamical

More information

Complex Numbers. 1 Introduction. 2 Imaginary Number. December 11, Multiplication of Imaginary Number

Complex 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

10.1 Complex Arithmetic Argand Diagrams and the Polar Form The Exponential Form of a Complex Number De Moivre s Theorem 29

10.1 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 information

With this expanded version of what we mean by a solution to an equation we can solve equations that previously had no solution.

With 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 information

or 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

or 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 information

Prentice Hall Mathematics, Algebra Correlated to: Achieve American Diploma Project Algebra II End-of-Course Exam Content Standards

Prentice Hall Mathematics, Algebra Correlated to: Achieve American Diploma Project Algebra II End-of-Course Exam Content Standards Core: Operations on Numbers and Expressions Priority: 15% Successful students will be able to perform operations with rational, real, and complex numbers, using both numeric and algebraic expressions,

More information

Vector Spaces. Chapter 1

Vector Spaces. Chapter 1 Chapter 1 Vector Spaces Linear algebra is the study of linear maps on finite-dimensional vector spaces. Eventually we will learn what all these terms mean. In this chapter we will define vector spaces

More information

MATH 135: COMPLEX NUMBERS

MATH 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 information

Inverses of Square Matrices

Inverses of Square Matrices Inverses of Square Matrices A. Havens Department of Mathematics University of Massachusetts, Amherst February 23-26, 2018 Outline 1 Defining Inverses Inverses for Products and Functions Defining Inverse

More information

B Elements of Complex Analysis

B 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

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : , MCN COMPLEX NUMBER C The complex number Complex number is denoted by ie = a + ib, where a is called as real part of (denoted by Re and b is called as imaginary part of (denoted by Im Here i =, also i =,

More information

Introduction to Complex Analysis

Introduction to Complex Analysis Introduction to Complex Analysis George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 413 George Voutsadakis (LSSU) Complex Analysis October 2014 1 / 67 Outline

More information

An introduction to complex numbers

An introduction to complex numbers An introduction to complex numbers The complex numbers Are the real numbers not sufficient? A complex number A representation of a complex number Equal complex numbers Sum of complex numbers Product of

More information

Complex Numbers. z = x+yi

Complex Numbers. z = x+yi Complex Numbers The field of complex numbers is the extension C R consisting of all expressions z = x+yi where x, y R and i = 1 We refer to x = Re(z) and y = Im(z) as the real part and the imaginary part

More information

18.03 LECTURE NOTES, SPRING 2014

18.03 LECTURE NOTES, SPRING 2014 18.03 LECTURE NOTES, SPRING 2014 BJORN POONEN 7. Complex numbers Complex numbers are expressions of the form x + yi, where x and y are real numbers, and i is a new symbol. Multiplication of complex numbers

More information

This leaflet describes how complex numbers are added, subtracted, multiplied and divided.

This 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 information

1 Complex Numbers. 1.1 Sums and Products

1 Complex Numbers. 1.1 Sums and Products 1 Complex Numbers 1.1 Sums Products Definition: The complex plane, denoted C is the set of all ordered pairs (x, y) with x, y R, where Re z = x is called the real part Imz = y is called the imaginary part.

More information

8. Complex Numbers. sums and products. basic algebraic properties. complex conjugates. exponential form. principal arguments. roots of complex numbers

8. 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 information

Math Precalculus Blueprint Assessed Quarter 1

Math Precalculus Blueprint Assessed Quarter 1 PO 11. Find approximate solutions for polynomial equations with or without graphing technology. MCWR-S3C2-06 Graphing polynomial functions. MCWR-S3C2-12 Theorems of polynomial functions. MCWR-S3C3-08 Polynomial

More information

Chapter 3: Polynomial and Rational Functions

Chapter 3: Polynomial and Rational Functions Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions A polynomial on degree n is a function of the form P(x) = a n x n + a n 1 x n 1 + + a 1 x 1 + a 0, where n is a nonnegative integer

More information

Math 1302 Notes 2. How many solutions? What type of solution in the real number system? What kind of equation is it?

Math 1302 Notes 2. How many solutions? What type of solution in the real number system? What kind of equation is it? Math 1302 Notes 2 We know that x 2 + 4 = 0 has How many solutions? What type of solution in the real number system? What kind of equation is it? What happens if we enlarge our current system? Remember

More information

What is the purpose of this document? What is in the document? How do I send Feedback?

What is the purpose of this document? What is in the document? How do I send Feedback? This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Number

More information

- 1 - Items related to expected use of technology appear in bold italics.

- 1 - Items related to expected use of technology appear in bold italics. - 1 - Items related to expected use of technology appear in bold italics. Operating with Geometric and Cartesian Vectors Determining Intersections of Lines and Planes in Three- Space Similar content as

More information

College Trigonometry

College 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 information

Sometimes can find power series expansion of M X and read off the moments of X from the coefficients of t k /k!.

Sometimes can find power series expansion of M X and read off the moments of X from the coefficients of t k /k!. Moment Generating Functions Defn: The moment generating function of a real valued X is M X (t) = E(e tx ) defined for those real t for which the expected value is finite. Defn: The moment generating function

More information

Mathematics Specialist Units 3 & 4 Program 2018

Mathematics 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 information

The Complex Numbers c ). (1.1)

The Complex Numbers c ). (1.1) The Complex Numbers In this chapter, we will study the basic properties of the field of complex numbers. We will begin with a brief historic sketch of how the study of complex numbers came to be and then

More information

Lecture 1 Complex Numbers. 1 The field of complex numbers. 1.1 Arithmetic operations. 1.2 Field structure of C. MATH-GA Complex Variables

Lecture 1 Complex Numbers. 1 The field of complex numbers. 1.1 Arithmetic operations. 1.2 Field structure of C. MATH-GA Complex Variables Lecture Complex Numbers MATH-GA 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 information

Los Angeles Unified School District Secondary Mathematics Branch

Los Angeles Unified School District Secondary Mathematics Branch Math Analysis AB or Trigonometry/Math Analysis AB (Grade 10, 11 or 12) Prerequisite: Algebra 2AB 310601 Math Analysis A 310602 Math Analysis B 310505 Trigonometry/Math Analysis A 310506 Trigonometry/Math

More information

Lesson 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 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 information

Chennai Mathematical Institute B.Sc Physics Mathematical methods Lecture 1: Introduction to complex algebra

Chennai Mathematical Institute B.Sc Physics Mathematical methods Lecture 1: Introduction to complex algebra Chennai Mathematical Institute B.Sc Physics Mathematical methods Lecture 1: Introduction to complex algebra A Thyagaraja January, 2009 AT p.1/12 1. Real numbers The set of all real numbers (hereafter denoted

More information

Algebra II Standards of Learning Curriculum Guide

Algebra II Standards of Learning Curriculum Guide Expressions Operations AII.1 identify field properties, axioms of equality inequality, properties of order that are valid for the set of real numbers its subsets, complex numbers, matrices. be able to:

More information

THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS

THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS The real number SySTeM C O M P E T E N C Y 1 THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS This competency section reviews some of the fundamental

More information

0.0.1 Moment Generating Functions

0.0.1 Moment Generating Functions 0.0.1 Moment Generating Functions There are many uses of generating functions in mathematics. We often study the properties of a sequence a n of numbers by creating the function a n s n n0 In statistics

More information

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5 Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x - 15 2. x 2-9x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There

More information

Berkeley Math Circle, May

Berkeley Math Circle, May Berkeley Math Circle, May 1-7 2000 COMPLEX NUMBERS IN GEOMETRY ZVEZDELINA STANKOVA FRENKEL, MILLS COLLEGE 1. Let O be a point in the plane of ABC. Points A 1, B 1, C 1 are the images of A, B, C under symmetry

More information

Mathematical Focus 1 Complex numbers adhere to certain arithmetic properties for which they and their complex conjugates are defined.

Mathematical 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 information

Some commonly encountered sets and their notations

Some 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 information

THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS

THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS THE REAL NUMBER SYSTEM C O M P E T E N C Y 1 THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS This competency section reviews some of the fundamental

More information

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers

Chapter 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 information

+ py 1v + py 1 v + qy 1 v = 0

+ py 1v + py 1 v + qy 1 v = 0 September 25, 2012 8-1 8. Reduction of Order and more on complex roots Reduction of Order: Suppose we are given a general homogeneous second order d.e. L(y) = y + p(t)y + q(t)y = 0. (1) We know that, in

More information

MATHia Unit MATHia Workspace Overview TEKS

MATHia Unit MATHia Workspace Overview TEKS 1 Function Overview Searching for Patterns Exploring and Analyzing Patterns Comparing Familiar Function Representations Students watch a video about a well-known mathematician creating an expression for

More information

Section 5.5. Complex Eigenvalues

Section 5.5. Complex Eigenvalues Section 5.5 Complex Eigenvalues A Matrix with No Eigenvectors Consider the matrix for the linear transformation for rotation by π/4 in the plane. The matrix is: A = 1 ( ) 1 1. 2 1 1 This matrix has no

More information

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS SPECIALIST MATHEMATICS (Year 11 and 12) UNIT A A1: Combinatorics Permutations: problems involving permutations use the multiplication principle and factorial notation permutations and restrictions with

More information

CHAPTER 1 COMPLEX NUMBER

CHAPTER 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 information

MTH4101 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) 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 information