Complex_3

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Mathematics – Complex - Exercises - Solutions 
Click on the exercise number or the ‘Video’ button to start a video. 
 
 
01 Complex number, negative: 
 
 
 
 
 Solution: 
 The negative of a complex number z = x +iy is given by -z = -(x+iy) = -x-iy 
 
 [-2 4 ] 2 4 ci i− + = − 
 
 
 
 
 
02 Complex number, conjugate: 
 
 
 
 
 
 Solution: 
 The conjugate of a complex number x + iy is given by x- iy. 
 
 -2 4 -2 - 4 bi i+ = 
 
 
 
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_01.mp4
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_02.mp4
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_01.mp4
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_02.mp4
 03 Complex equation: 
 
 
 
 Solution: 
 Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal 
 if the real parts are equal and the imaginary parts are equal, that is: 
 x1 = x2 and y1 = y2 . 
 
2(2 ) (6 ) 2( 2)
4 2 6 2 4
(4 2 ) (6 ) 4 2
4 2 4 6 2
2 6 d
x y x i xi
x y i xi xi
x y x i xi
x y x x
x y
+ + − = −
+ + − = −
+ + − = − +

+ = −  − =

=  = −
 
 
 
 
 
 
 
 
 
 
 
 
 
 
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_03.mp4
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_03.mp4
04 Complex number in polar form: 
 
 
 Solution: 
 2 2
1
 (cos sin ) 
 
 tan
iz x iy r i re
r x y
y
x
 
 −
= + = + =
= +
 
=  
 
 
 
 
1 1 1
( 3 ) 3 cos30 sin 30
2 2 2
 cos30 ( sin 30 )
 cos(360 30 ) sin(360 30 )
 cos330 sin 330 
o o
o o
o o o o
o o
i i i
i
i
i
− = − = −
= + −
= − + −
= +
2 2
1
 d
1 1 1
( 3 ) 3
2 2 2
1 1 3 1
3 1
2 2 4 4
1
2tan 330
1
3
2
1 1 1
( 3 ) 3 cos330 sin 330 
2 2 2
o
o o
or
i i
r
i i i
 −
− = −
   
= + − = + =   
   
 
− 
= = 
 
 
− = − = +
 
 
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_04.mp4
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_04.mp4
05 Complex number in rectangular form: 
 
 
 
 
 Solution: 
 
 2 2
1
 (cos sin ) 
 
 tan
iz x iy r i re
r x y
y
x
 
 −
= + = + =
= +
 
=  
 
 
 
 
 
 
 
5 5 5 5 1 1 3 2 3 2
3 cos sin 3cos sin 3 ( 2) 3( 2) a
4 4 4 4 2 2 2 2
i i i i
    
+ = + =  − +  − = − − 
 
 
 
 
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_05.mp4
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_05.mp4
06 Complex number simplify: 
 
 
 Solution: 
 1
2 2
1 1 (1 6 ) 1 6 1 6 1 6
( 1 6 ) b
1 6 (1 6 )(1 6 ) 1 (6 ) 7 7 7
i i i
i i
i i i i
−  − − −
− − = − = − = − = − = − +
+ + − −
 
 
 
 
07 Complex number, multiplication: 
 
 
 
 
 
 Solution: 
 For complex numbers we have: 
 
 
(cos sin )
( ) (cos sin )
i
n n n n in
z x iy r i re
z x iy r n i n r e


 
 
= + = + =
= + = + =
 
 
 
 
( ) ( )
3
3 33 3
3 2 2 4 4 4(1 ) 1 1 2 2 2
3
135 b
4
i i i
o
i e e e
  


 
+ = + = = 
 
= =
 
 
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_06.mp4
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_07.mp4
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_06.mp4
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_07.mp4
08 Complex number, multiplication: 
 
 
 
 
 
 
 
 Solution: 
 For complex numbers we have: 
 
 
(cos sin )
( ) (cos sin )
i
n n n n in
z x iy r i re
z x iy r n i n r e


 
 
= + = + =
= + = + =
 
 
 
 
( ) ( )
3
3 33 3
3 2 2 4 4 4(1 ) 1 1 2 2 2
2 2 c
i i i
i e e e
r
   
+ = + = = 
 
=
 
 
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_08.mp4
https://grimstad.uia.no/perhh/phh/fag/forkurs/mathematics/o/dig/video/complex_08.mp4