Introduction to Complex Numbers

Introduction to Complex Numbers Presented by: Roberto Savo Date: 19 August 2011 Document Number: Table of contents: 1.0 Introduction 3 2.0 The Square root of -1 3 3.0 Examples of roots with Graphs and Quadratic Equations 4 4.0 Addition and Subtraction of Complex Numbers 6 5.0 Multiplication of Complex Numbers 7 6.0 The Complex Conjugate 8 7.0 Vector Representation of a Complex Number 9 8.0 Vector Addition of a Complex Number 10 9.0 Polar Form of a Complex Number 12 10.0 The Exponential form of a Complex Number 14 11.0 Division of a Complex Number 16 12.0 Further Problems 17 1.0 Introduction The purpose of this course is to establish a good foundation of complex numbers 2.0 The Square root of -1 √25 = √ (5x 5) = 5 √64 = √ (8x 8) = 8 √ (-64) = √ [(-8) x (8)] = √ [(-1) x 8 x 8] = 8 √ (-1) Let i = √ (-1) √ (-64) = 8i 3.0 Examples of roots with Graphs and Quadratic Equations Y = (x+1)(x)(x-4) 3 Real Roots at X= -1, X=0 and X=4 Y = (x+1)(x)(x-4)+15 1 real root at x=-2.2. Where have the other roots gone? 4.0 Addition and Subtraction of Complex Numbers Z1 = A+iB and Z2 = C+iD Ztot = A+iB+C+iD Ztot = (A+C) + i(B+D) Exercise 4.1: Addition of Complex Numbers (3+i4) + (7+i6) = (8-i2) – (5+i9) = (6+i8) + (-6-i8) (3+i9) – (8-i5) + (-2-i3) = 5.0 Multiplication of Complex Numbers Exercise 5.1: Multiplication of Complex Numbers (2+i4)(6-i3) = (-8-i2)(-3+i7) = (3+i2)3 = i2 i3 i4 i5 i6 i7 6.0 The Complex Conjugate Z1 = A+iB Z*1 = A-iB ( Z1 )x(Z*1 )= (A+iB )x(A-iB) = A2+B2 Exercise 6.1: Complex Conjugate If Z = (9-i3), find Z* Calculate ZxZ* 7.0 Vector Representation of a Complex Number Imaginary Axis i Z1 = A+iB Z1 B A Real Axis θ = tan-1 (B/A) 8.0 Vector Addition of a Complex Number Z1 = A+iB and Z2 = C+iD Imaginary Axis i Z1 = A+iB Z2 = C+iD Z2 Z1 B C A Real Axis Imaginary Axis i Z2 Z1 = A+iB Z2 = C+iD D C B Z1 A Real Axis Imaginary Axis i Z2 Z1 = A+iB Ztot Z2 = C+iD D C B Z1 A Real Axis Exercise 8.1: Draw the complex numbers below on an Argand diagram: Z1 = 2+i5 Z2 = -3 - i4 Z3 = Z1+Z2 Z4 = Z1 - Z2 Z5 = Z1 x Z2 Z6 = Z1 x Z1* Z7 = Z2 x Z2* 9.0 Polar Form of a Complex Number Imaginary Axis i Z1 = A+iB Z1 B A Real Axis Imaginary Axis i Z1 = A+iB r B θ A Real Axis r = √ (A2+B2) θ = tan-1 (B/A) Exercise 9.1: Find the polar equivalents r(cosθ+isinθ) of the following complex numbers Z1 = 2+i3 Z2 = 6-i4 Z3 = 3i Z4 = -6-i5 Z5 = 3-i2 10.0 The Exponential form of a Complex Number ==================================================== Exercise 10.1: Find the exponential equivalents reiθ of the following complex numbers Z1 = 2+i3 Z2 = 6-i4 Z3 = 3i Z4 = -6-i5 Z5 = 3-i2 11.0 Division of a Complex Number Exercise 11.1: Calculate the results of the division and multiplication of the complex numbers shown below Z1 = 2+i3 Z2 = 6-i4 Z3 = 3i Z4 = -6-i5 Z5 = 3-i2 Find Z2 / Z1 = Z5 / Z4 = (Z5 x Z4) / (Z1 x Z2) (Z1 x Z4) / Z3 12.0 Further Problems 12.1 If a and b are real numbers, solve: 12.2 If a is real, show that is also real Solutions: 12.1 12.2. Hint: Expand the equations until you see a complex number Z and its conjugate Z*. Make use of the known fact that Z+Z* = a real number and Z-Z* = an imaginary number Page 18