The Complex Plane.

Slides:



Advertisements
Similar presentations
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.
Advertisements

Complex Numbers in Polar Form; DeMoivre’s Theorem 6.5
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.
Copyright © 2011 Pearson, Inc. 6.6 Day 1 De Moivres Theorem and nth Roots Goal: Represent complex numbers in the complex plane and write them in trigonometric.
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
6.5 Complex Numbers in Polar Form. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Objectives: Plot complex number in the complex plane. Find the.
Definition of Trigonometric Functions With trigonometric ratios of acute angles in triangles, we are limited to angles between 0 and 90 degrees. We now.
Mathematics.
Complex Numbers. Complex number is a number in the form z = a+bi, where a and b are real numbers and i is imaginary. Here a is the real part and b is.
Complex Numbers in Polar Form; DeMoivre’s Theorem
Applied Symbolic Computation1 Applied Symbolic Computation (CS 300) Review of Complex Numbers Jeremy R. Johnson.
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.
Roots of Complex Numbers Sec. 6.6c HW: p odd.
Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Copyright © 2009 Pearson Addison-Wesley Complex Numbers, Polar Equations, and Parametric Equations.
Section 6.5 Complex Numbers in Polar Form. Overview Recall that a complex number is written in the form a + bi, where a and b are real numbers and While.
Section 8.1 Complex Numbers.
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.
Copyright © 2009 Pearson Addison-Wesley De Moivre’s Theorem; Powers and Roots of Complex Numbers 8.4 Powers of Complex Numbers (De Moivre’s.
What does i2 equal? 1 -1 i Don’t know.
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.
The Geometry of Complex Numbers Section 9.1. Remember this?
Welcome to Week 6 College Trigonometry.
Standard form Operations The Cartesian Plane Modulus and Arguments
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 NUMBER.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Polar Form and its Applications
CHAPTER 1 COMPLEX NUMBERS
Roots of Complex Numbers
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
ARGAND DIAGRAM Learning Outcomes:
Complex Algebra Review
De Moivre’s Theorem and nth Roots
Section 9.3 The Complex Plane
De Moivre’s Theorem and nth Roots
Applied Symbolic Computation (CS 300) Review of Complex Numbers
Complex Numbers Arithmetic Operation Definition Complex Conjugate
Applied Symbolic Computation (CS 567) Review of Complex Numbers
7.6 Powers and Roots of Complex Numbers
Warm Up Simplify the fraction: a)
Complex Algebra Review
Complex Algebra Review
Trigonometric Form Section 6.5 Precalculus PreAP/Dual, Revised ©2016
Complex Numbers and i is the imaginary unit
De Moivre’s Theorem and nth Roots
Complex Numbers MAΘ
Vector-Valued Functions
Complex Numbers and DeMoivre’s Theorem
The Complex Plane; DeMoivre's Theorem
6.5 Complex Numbers in Polar Form: DeMoivre’s Theorem
Presentation transcript:

The Complex Plane

The Complex Plane The complex number z = a + bi can plotted as a point with coordinates z(a,b). Re (z) x – axis Im (z) y – axis Im(z) Re(z) O(0,0) z(a,b) a b

The Complex Plane Definition 1.6 (Modulus of Complex Numbers) The modulus of z is defined by Im(z) Re(z) O(0,0) z(a,b) a b r

The Complex Plane Example 1.7 : Find the modulus of z:

The Complex Plane The Properties of Modulus

Argument of Complex Numbers Definition 1.7 The argument of the complex number z = a + bi 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+isin40) and z2 = 3(cos95+isin95) . Find : If z1 = 6(cos60+isin60) and z2 = 2(cos270+isin270) . 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

Thank You Prepared By Smt. Sasmita kumari swain Lecturer in Mathematics Khemundi College