This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CHAPTER 3 COMPLEX NUMBER 3.1 INTRODUCTION Lets a quadratic equation be in the form of 2 = + + c bz az ( 1 ) The usual method to find the roots from this equation is by using formula a ac b b z 2 4 2  = If 4 2 < ac b , then the solution of z is in term of square root of negative number y x z 1 = If 1 is written as i then the solution become iy x z + = and iy x z = ( 2 ) Equation (2) are known as Complex Number , composed of a real part x and an imaginary part iy . Complex number has its conjugate , the only difference between them is the sign of the imaginary part, that is iy x z = is the conjugate of iy x z + = , and vise versa. We denoted the conjugate as z . 3.2 OPERATIONS Operation of addition, subtraction, multiplication and division are also can be performed on complex number as usual together with the rule that 1 2 = i . Example 1: Let i z 5 2 1 + = and i z 2 1 2 + = , find (a) 2 1 z z + (b) 2 1 z z (c) 2 1 z z (d) 2 1 z z Documents PDF Complete Click Here & Upgrade Expanded Features Unlimited Pages Solution (a) ( 29 ( 29 i i i z z 7 1 2 1 5 2 2 1 + = + + + = + (b) ( 29 ( 29 i i i z z 3 3 2 1 5 2 2 1 + = + + = (c) ( 29 ( 29 i i i i i i i z z = = + + = + + = 12 10 2 10 5 4 2 2 1 5 2 2 2 1 (d) ( 29 ( 29 ( 29 ( 29 i i i i i i i i i i i z z 5 9 5 8 5 9 8 4 1 10 9 2 2 1 2 1 2 1 5 2 2 1 5 2 2 2 2 1 = = = + + = + + = In example 1 (d), for division of complex number, we convert the complex number in the denominator into real number, by multiplying both numerator and denominator by the conjugate of the denominator. 3.3 THE ARGAND DIAGRAM We will present complex number graphically on Cartesian coordinate system and Polar coordinate system. 3.3.1 CARTESIAN FORM Let iy x z + = 1 , the graph using Cartesian coordinate system is as figure below. iy iy x z + = 1 iy x x In Cartesian coordinate system, xaxis represented real part while yaxis represented imaginary parts of complex number. The addition and subtraction also can be shown graphically. Documents PDF Complete Click Here & Upgrade Expanded Features Unlimited Pages Let 1 1 1 iy x z + = and 2 2 2 iy x z + = , the graph of 2 1 z z + is as figure below. iy ( 29 2 1 2 1 2 1 y y i x x z z + + + = + 1 z ( 29 2 1 y y i + 2 z x 2 1 x x + 3.3.2 POLAR OR TRIGONOMETRIC FORM Consider the following figure iy ) , ( r r iy x x ) , ( r is the polar coordinate for z , r is the modulus of z and is written as z and is called the argument of z. If iy x z + = , from graph we conclude = cos r x = sin r y Thus...
View
Full
Document
This note was uploaded on 12/01/2010 for the course SUKAN md taught by Professor Saipon during the Spring '10 term at Albany College of Pharmacy and Health Sciences.
 Spring '10
 saipon

Click to edit the document details