CHAPTER 1 COMPLEX NUMBERS

Slides:



Advertisements
Similar presentations
PROGRAMME 2 COMPLEX NUMBERS 2.
Advertisements

Complex Numbers If we wish to work with , we need to extend the set of real numbers Definitions i is a number such that i2 = -1 C is the set of.
Copyright © 2009 Pearson Addison-Wesley Applications of Trigonometry.
Prepared by Dr. Taha MAhdy
De Moivres Theorem and nth Roots. The Complex Plane Trigonometric Form of Complex Numbers Multiplication and Division of Complex Numbers Powers of.
Complex numbers Definitions Conversions Arithmetic Hyperbolic Functions.
8 Applications of Trigonometry Copyright © 2009 Pearson Addison-Wesley.
8 Complex Numbers, Polar Equations, and Parametric Equations
INTRODUCTION OPERATIONS OF COMPLEX NUMBER THE COMPLEX PLANE THE MODULUS & ARGUMENT THE POLAR FORM.
Math 112 Elementary Functions
1 Week 1 Complex numbers: the basics 1. The definition of complex numbers and basic operations 2. Roots, exponential function, and logarithm 3. Multivalued.
Definition of Trigonometric Functions With trigonometric ratios of acute angles in triangles, we are limited to angles between 0 and 90 degrees. We now.
Copyright © 2011 Pearson, Inc. 6.6 De Moivre’s Theorem and nth Roots.
Copyright © 2011 Pearson, Inc. 6.6 De Moivre’s Theorem and nth Roots.
Sec. 6.6b. One reason for writing complex numbers in trigonometric form is the convenience for multiplying and dividing: T The product i i i involves.
Section 6.1 Notes Special Angles of the Unit Circle in degrees and radians.
Copyright © 2009 Pearson Addison-Wesley Complex Numbers, Polar Equations, and Parametric Equations.
Section 8.1 Complex Numbers.
LOCUS IN THE COMPLEX PLANE The locus defines the path of a complex number Recall from the start of the chapter that the modulus and the argument defines.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 8 Complex Numbers, Polar Equations, and Parametric Equations.
The Complex Plane; De Moivre’s Theorem. Polar Form.
Section 5.3 – The Complex Plane; De Moivre’s Theorem.
Lesson 78 – Polar Form of Complex Numbers HL2 Math - Santowski 11/16/15.
Copyright © 2009 Pearson Addison-Wesley De Moivre’s Theorem; Powers and Roots of Complex Numbers 8.4 Powers of Complex Numbers (De Moivre’s.
11.4 Roots of Complex Numbers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 1 Homework, Page 559 Plot all four points in the same complex plane.
Applications of Trigonometric Functions
CHAPTER 1 COMPLEX NUMBER. CHAPTER OUTLINE 1.1 INTRODUCTION AND DEFINITIONS 1.2 OPERATIONS OF COMPLEX NUMBERS 1.3 THE COMPLEX PLANE 1.4 THE MODULUS AND.
1Complex numbersV-01 Reference: Croft & Davision, Chapter 14, p Introduction Extended the set of real numbers to.
CHAPTER 1 COMPLEX NUMBERS
Chapter 40 De Moivre’s Theorem & simple applications 12/24/2017
Standard form Operations The Cartesian Plane Modulus and Arguments
Splash Screen.
Complex Numbers 12 Learning Outcomes
CHAPTER 1 COMPLEX NUMBERS
CHAPTER 1 COMPLEX NUMBERS
Solve this!.
Week 1 Complex numbers: the basics
Start Up Day 54 PLOT the complex number, z = -4 +4i
CHAPTER 1 COMPLEX NUMBERS
CHAPTER 1 COMPLEX NUMBER.
Polar Form and its Applications
Section 9-5 The Binomial Theorem.
HW # , , , Do Now Find the quotient of
Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Honors Precalculus: Do Now
8.3 Polar Form of Complex Numbers
Let Maths take you Further…
Topic Past Papers –Complex Numbers
Complex Algebra Review
De Moivre’s Theorem and nth Roots
Section 9.3 The Complex Plane
De Moivre’s Theorem and nth Roots
Chapter 2 Section 2.
Complex Numbers Arithmetic Operation Definition Complex Conjugate
7.6 Powers and Roots of Complex Numbers
Warm Up Simplify the fraction: a)
Complex Algebra Review
FP2 Complex numbers 3c.
Complex Algebra Review
10.5 Powers of Complex Numbers and De Moivre’s Theorem (de moi-yay)
Complex numbers nth roots.
Complex Numbers and i is the imaginary unit
De Moivre’s Theorem and nth Roots
( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )
The Complex Plane.
Complex Numbers and DeMoivre’s Theorem
The Complex Plane; DeMoivre's Theorem
6.5 Complex Numbers in Polar Form: DeMoivre’s Theorem
Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Presentation transcript:

CHAPTER 1 COMPLEX NUMBERS THE MODULUS AND ARGUMENT THE POLAR FORM THE EXPONENTIAL FORM DE MOIVRE’S THEOREM FINDING ROOTS

Argument of Complex Numbers Definition 1.7 The argument of the complex number z = a + bj is defined as 1st QUADRANT 2nd QUADRANT 4th QUADRANT 3rd QUADRANT

Argument of Complex Numbers Example 1.8 : Find the arguments of z:

THE POLAR FORM OF COMPLEX NUMBER (a,b) r Re(z) Im(z) Based on figure above:

The polar form is defined by: Example 1.9: Represent the following complex number in polar form:

Example 1.10 : Express the following in standard form of complex number:

Theorem 1: If z1 and z2 are 2 complex numbers in polar form where then,

Example 1.11 : If z1 = 2(cos40+jsin40) and z2 = 3(cos95+jsin95) . Find : If z1 = 6(cos60+jsin60) and z2 = 2(cos270+jsin270) . Find :

THE EXPONENTIAL FORM DEFINITION 1.8 The exponential form of a complex number can be defined as Where θ is measured in radians and

THE EXPONENTIAL FORM Example 1.15 Express the complex number in exponential form:

THE EXPONENTIAL FORM Theorem 2 If and , then:

THE EXPONENTIAL FORM Example 1.16 If and , find:

DE MOIVRE’S THEOREM Theorem 3 If is a complex number in polar form to any power of n, then De Moivre’s Theorem: Therefore :

DE MOIVRE’S THEOREM Example 1.17 If , calculate :

FINDING ROOTS Theorem 4 If then, the n root of z is: (θ in degrees) OR (θ in radians) Where k = 0,1,2,..n-1

FINDING ROOTS Example 1.18 If then, r =1 and : Let n =3, therefore k =0,1,2 When k =0: When k = 1:

FINDING ROOTS When k = 2: Sketch on the complex plane: y 1 y x nth roots of unity: Roots lie on the circle with radius 1

FINDING ROOTS Example 1.18 If then, and : Let n =4, therefore k =0,1,2,3 When k =0: When k = 1:

FINDING ROOTS Let n =4, therefore k =0,1,2,3 When k =2: When k = 3:

FINDING ROOTS Sketch on complex plane: y x