1. CARTESIAN COMPLEX NUMBERS - Weebly

1 UNIVERSITI KUALA LUMPUR COMPLEX number E2 1. CARTESIAN COMPLEX NUMBERS INTRODUCTION Try to solve this quadratic equation : 0522=++xx By using quadratic formula : the discriminant , 16)5)(1(4)2(422 = = = acb the solution : )1(216)2( =x but it is not possible to evaluate 1 however if an operator j is defined as then the solution may be expressed as : 12 =j 21242)1(216)2(jjx = = = 21j+ and are known as COMPLEX NUMBERS .

2 21j Both solutions are of the form : = x1 2j COMPLEX number Real part Imaginarypart z = ajb this form is known as the CARTESIAN COMPLEX NUMBERS ( ALGEBRAIC FORM ) E2 - 1 - MATHEMATICS UNIT UNIVERSITI KUALA LUMPUR COMPLEX number E2 EXAMPLES EXAMPLE 1 : Solve the quadratic equation , 042=+x 240422jxxx = ==+ EXAMPLE 2 : Solve the quadratic equation , 2 x 2 + 3 x + 5 = 0 2343463436344093)2(2)5)(2(4332jxjxxxx = = = = = E2 - 2 - MATHEMATICS UNIT UNIVERSITI KUALA LUMPUR COMPLEX number E2 POWERS OF j j 0 0)1( 1 j 1 1 j j 2 )1)(1( -1 j 3 j2j = (-1)j -j j 4 j2j2 = (-1)(-1) 1 j 5 j4j = (1)j j In general we can bring the power to the nearest multiplication of 4.

3 J 4 p + 0 = 1 j 4 p + 1 = j j 4 p + 2 = -1 j 4 p + 3 = - j where p Z DOMAIN The domain of the COMPLEX number is C where R is an element of C R C R C E2 - 3 - MATHEMATICS UNIT UNIVERSITI KUALA LUMPUR COMPLEX number E2 THE ARGAND DIAGRAM A COMPLEX number may be represented graphically on rectangular or CARTESIAN axes.

4 The horizontal ( or x ) axis is used to represent the real axis and the vertical ( or y ) axis is used to represent the imaginary axis . Such a diagram is called an ARGAND DIAGRAM . EXAMPLES : Represent Argand points A = 3 + j2 , B = -2 + j4 , C = -3 j3 , D = 2 j2 Real Axis Imaginary Axis D C B A 2 -2 -3 -3 -2 4 2 3 E2 - 4 - MATHEMATICS UNIT UNIVERSITI KUALA LUMPUR COMPLEX number E2 2. ADDITION AND SUBTRACTION - ALGEBRAIC FORM Two COMPLEX NUMBERS are added / subtracted by adding / subtracting separately the two real parts and two imaginary parts.

5 Given two COMPLEX number Z = a + j b and W = c + j d IDENTITY If two COMPLEX NUMBERS are equal , then their real parts are equal and their imaginary parts are equal . Hence , two COMPLEX NUMBERS are identical , Z = W if : a = c and b = d EXAMPLE : Solve the COMPLEX equations ; (a) 36)(2jjyx =+ SOLUTION 3622jyjx =+ Therefore [ Re ] 32662===xxx [ Im ] 2332 = =yy E2 - 5 - MATHEMATICS UNIT UNIVERSITI KUALA LUMPUR COMPLEX number E2 (b) ()()jbajj+= +3221 SOLUTION jbajjbajjbajj+= += ++ += +74)34()62()32)(21( Therefore : [ Re ] : a = 4 and [ Im ].

6 B = - 7 ADDITION & SUBTRACTION The sum of two COMPLEX number , Z + W )()()()(dbjcawzjdcjbawz+++=++++=+ EXAMPLE Given : and 32jz+=41jw = jwzjwz =+ ++ +=+1)]4(3[)]1(2[ The difference of two COMPLEX number , Z - W )()()()(dbjcawzjdcjbawz + = + += EXAMPLE Given : and 32jz+=41jw = 73)]4(3[)]1(2[jwzjwz+= + = E2 - 6 - MATHEMATICS UNIT UNIVERSITI KUALA LUMPUR COMPLEX number E2 The addition and subtraction of COMPLEX NUMBERS may be achieved graphically in the Argand diagram.

7 Represent Example 1 and Example 2 in the Argand diagram . IMAGINARYAXIS3 2 -1-4w z Addition jwz =+1 REAL AXIS Subtraction 73jwz+= IMAGINARY AXIS 73zz w-4 -1 32 REAL AXIS E2 - 7 - MATHEMATICS UNIT UNIVERSITI KUALA LUMPUR COMPLEX number E2 SCALAR MULTIPLICATION If Z = a + j b and k R , where k is a scalar ; then k Z , jkbkakzjbakkz+=+=)( EXAMPLE 3 : Given Z1 = 2 + j4 and Z2 = 3 - j Determine : (a) 4Z1 = 168)42(4jj+=+ (b) 5 Z2 = 515)3(5jj = E2 - 8 - MATHEMATICS UNIT UNIVERSITI KUALA LUMPUR COMPLEX number E2 3.

8 MULTIPLICATION AND DIVISION - ALGEBRAIC FORM MULTIPLICATION Multiplication of COMPLEX NUMBERS is achieved by assuming all quantities involved are real and using j 2 = -1 to simplify : Given two COMPLEX NUMBERS : Z = a + jb and W = c + jd The product of two COMPLEX number , Z . W ))((jdcjbawz++= by using F O I L method )()(2bcadjbdacwzbdjbcjadacwzbdjjbcjadacw zjjbdjbcjadacwz++ = ++= +++= +++= Z . W = )()(bcadjbdac++ EXAMPLE : multiply the following COMPLEX number (a) ( 3 + j2 )( 4 - j5 ) = 1222)5)(2()4)(12()5)(3()4)(3(jjjjj = ++ + (b) ( -2 + 5j )( -5 + 2j ) = 29)5)(5()2)(5()2)(2()5)(2(jjjjjj =++ + E2 - 9 - MATHEMATICS UNIT UNIVERSITI KUALA LUMPUR COMPLEX number E2 COMPLEX CONJUGATE The COMPLEX conjugate of a COMPLEX number is obtained by changing the sign of the imaginary part.

9 Hence the COMPLEX conjugate of : Z = a + j b is Z= a - j b W = c - j d is W= c + j d EXAMPLE : Let Z = 2 + j5 1. The COMPLEX conjugate of Z , is 52jz = 2. Calculate ZZ. : 2252+=zz= 49+ = 13 CONCLUSION : The product of the COMPLEX number and its conjugate ZZ. is always a real number . EXAMPLE : Let Q = 1 + j2 and R = 3 + j4 1. Calculate RQ+ Solution 64)43()21(jjjRQ+=+++=+ Therefore 64jRQ =+ or 644321jRQjRjQ =+ = = E2 - 10 - MATHEMATICS UNIT UNIVERSITI KUALA LUMPUR COMPLEX number E2 2.

10 Calculate QR Solution 105)64()83()43)(21(jQRjQRjjQR = + = = or 105105)64()83()43)(21(jQRjQRjQRjjQR =+ =++ =++= 3. Calculate 2Q Solution 43441)21(22222jQjjQjQ =+ = = or 4343441)21(222222jQjQjjQjQ =+ =++=+= From the previous examples , we can conclude that the : PROPERTIES OF the COMPLEX CONJUGATES ()nnzzwzzwwzwz= =+=+ E2 - 11 - MATHEMATICS UNIT UNIVERSITI KUALA LUMPUR COMPLEX number E2 The geometric interpretation of the COMPLEX conjugate ( shown below ) Z is the reflection of Z in the real axis.