Polar Coordinates And Complex Numbers Worksheets With Answers - Math 611b Page 42
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50F
I
H G
K J
b
−
θ
+
=
=
1
The argument, 2 , or amplitude of a complex number z = a + bi is
Arg a
(
bi
)
tan
a
This is the angle the vector makes with the positive side of the horizontal axis.
From trigonometry we know that:
a
θ
θ
=
=
cos
or
a
r
cos
r
b
θ
θ
=
=
sin
or
b
r
sin
and
.
r
Thus z = a + bi = rcos 2 + irsin 2 = r(cos 2 + isin 2 ) = rcis 2
Multiplication of complex numbers
The modulus of the product of two complex numbers is the product of their modulii.
The argument of the product of two complex numbers is the sum of their arguments.
θ
θ
θ θ
+
θ
θ
θ
θ
⋅
=
=
⋅
=
+
i
i
i
(
)
r e
r e
r r e
r cis
r cis
r r cis
(
)
1
2
1
2
1
2
1 2
1
1
2
2
1 2
1
2
Ex. 4cis 45° @ 5cis 15° = 20cis 60°
Division of complex numbers
The modulus of the quotient of two complex numbers is the quotient of their modulii.
The argument of the quotient of two complex numbers is the difference of their modulii.
θ
r cis
r
θ
θ
=
−
1
1
1
cis
(
)
θ
1
2
r cis
r
2
2
2
o
8
cis
75
=
o
4
cis
45
Ex.
o
2
cis
30
42
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