3. COMPLEX NUMBER
Combination of real
and imaginary number
Real
number
Examples :
4 – 3i
Imaginary
number
-8i + 2
Imaginary
number
Real
number
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4. i2 = -1
i
=
-1
Simplify the expressions below :
Replace
with -1
a. i9 = i2(4) i
= (-1)4 i
=i
b. 8i58 + 6i97 = 8i2(29) + 6i2(47) i
= 8(-1)29 + 6(-1)47 i
= -8 – 6i
c.
-81 +
-29 =
81(-1) +
= 9i + 5.4 i
= 14.4 i
29 (-1)
Replace
with i
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5. MODULUS & ARGUMENT
MODULUS, R
ARGUMENT, Arg
x
tan
2
1
y
y
2
BASED ON QUADRANT
QUADRANT II
ARG = 180 -
QUADRANT I
ARG =
x
ARG
Example
QUADRANT III
ARG = 180 +
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QUADRANT IV
ARG = 360 -
6. Example :
1. Find Modulus and Argument for 5 – 3i
Solution :
Modulus , R =
x
2
y
2
2
= 5 (-3)
= 5.83
Argand
Diagram
2
5
Argument , Arg= tan
1
= tan
1
y
x
3
Ignore the
–ve value
-3
5
= 30.96
Based on Quadrant IV = 360 – 30.96
= 329.04
Quadrant IV
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7.
ARGAND DIAGRAM
Sketch Argand Diagram for -3 – 4i
SOLUTION :
- 3 – 4i = -3 as a real number plot at x-axis
= -4 as a imaginary number plot at y-axis
Calculate the Modulus and Argument
Modulus, R =
( 3)
2
(-4)
2
Argument = tan
=5
ARG is Based on Quadrant III
1
4
3
= 53.13° ( refer to the Quadrant )
-3
= 180° +
= 180 + 53.13
= 233.13
-4
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9.
CARTESIAN FORM
z = a + bi
Example :
i) -5 + 3i
ii) 4 + 6i
iii) -7 – 7i
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10.
POLAR FORM
z = Modulus
= R Arg
Argument
Example :
i) 4.5 45°
ii) 6 123.6°
iii) 7.8 330°
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11.
TRIGONOMETRIC
FORM
z = Modulus ( Cos Argument + Sin Argument i )
= R ( Cos Arg + Sin Arg i )
Example :
i) 4.5 ( Cos 45° + Sin 45 i )
ii) 6 ( Cos 123.6° + Sin 123.6° i )
iii) 7.8 ( Cos 330° + Sin 330° i )
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12.
EXPONENTIAL
FORM
z = Modulus.e Arg i
= Re Arg i
Arg - should
be in radian
Example :
i) 4.5e 0.75 i
ii) 6e 1.2 i
iii) 7.8e 0.43 i
Exercise
BNSA/JMSK
13. 1. Given z = 5 + 12i , express z in :
i) Polar Form
ii) Trigonometric Form
iii) Exponential Form
Solution :
i) Polar Form
ii) Trigonometric Form
iii) Exponential Form
R Arg
R ( Cos Arg + Sin Arg i )
Re Arg i
So, we find the Modulus
and Argument
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14. Modulus , R = 5
= 13
2
(12)
Argument , Arg = tan
2
1
12
5
= 67.38
Based on Quadrant I, Arg =
So, Arg = 67.38
i) Polar Form = 13 67.38°
ii) Trigonometric Form = 13 ( Cos 67.38° + Sin 67.38 i )
iii) Exponential Form = 13e 1.18 i
Arg should be in Radian , so we must convert the Degree to Radian
by using 67 . 38
180
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15. 2. Express 25 ( Cos 137 + Sin 137 i ) into
Cartesian and Polar Form.
Solution :
ii) Polar Form = R Arg
i) Cartesian Form = a + bi
So, Polar Form = 25 137
a = 25 x Cos 137
= -18.28
b = 25 x Sin 137
= 17.05
So, Cartesian Form = -18.28 + 17.05i
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17. ADDITION
• Should be in Cartesian Form only.
• Real number1 + Real number2
• Imaginary number1 + Imaginary number2
Imaginary
number
Example
1. ( 3 + 4i ) + ( -5 – 2i ) = ( 3 + -5 ) + (4i + -2i)
= -2 + 2i
Real
number
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18. SUBTRACTION
• Should be in Cartesian Form only.
• Real number1 - Real number2
• Imaginary number1 - Imaginary number2
Imaginary
number
Example
1. ( -6 + 3i ) - ( 3– 2i ) = ( -6 - 3) + (3i - -2i)
= -9 + 5i
Real
number
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19. MULTIPLICATION
• Can be in Cartesian or Polar Form only
• In Cartesian Form = ( a + bi)(c + di)
• In Polar Form = Modulus1 x Modulus2 Arg1 + Arg2
( 3 + 4i )X (5 + 2i ) = 15 + 6i + 20i + 4i2
= 15 + 26i + 4(-1)
= 11 + 26i
Cartesian Form
6 123.6 x 3.5 35
= 6 x 3.5 123.6 + 35
= 21 158.6
Polar Form
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20. DIVISION
• Can be in Cartesian or Polar Form only
• In Cartesian Form = ( a + bi) (c + di) X Conjugate ( to get real number)
• In Polar Form = Modulus1 Modulus2 Arg1 - Arg2
Conjugate
2
3i
2
3i
4
4i
4
4i
= 4 4i
4
4i
=
=
=
8
8 i 12 i 12 i
2
2
16
16 i 16 i 16 i
8
15 125.5
2 45
= 15 2 125.5 - 45
= 7.5 80.5
8 i 12 i 12 ( 1)
16
4
16 ( 1)
20 i
32
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21.
COMPLEX NUMBER EQUALITY
Comparison of Real number and Imaginary
number for the both side (right and left side )
Example
1. Find the value of x and y for the equation,
3x + 2yi = ( 3 + i)(4 – 5i)
Real
number
Solution :
3x + 2yi = ( 3 + i)(4 – 5i)
3x + 2yi =12 -15i + 4i – 5i2
3x + 2yi =17 -11i
So, compare with both side :
3x =17
2yi = -11i
x = 5.67
y = -5.5
Imaginary
number
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