Complex Numbers Page 2
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1
2
3Appendix B
Complex Numbers
P. Danziger
Example 3
Put 1
i in polar form.
2
2
tan θ =
1, in fourth quadrant so θ =
. r =
1
+ 1
=
2. So
4
7
1
i =
2e
=
2e
.
4
4
Put 2e
in rectangular form.
3
1
3i
2e
= 2
+
=
3 + i.
3
2
2
2.2
Operations with Complex numbers
Let z = x + iy = re and w = u + iv = qe then we have the following operations:
The imaginary part of z, Im(z) = y.
The real part of z, Re(z) = x.
The Complex Conjugate of z, z = x
iy = re
.
Note: Complex conjugation basically means turn every occurence of an i to a
i.
2
2
The modulus of z, z =
zz =
x
+ y
= r.
1
The argument of z, arg(z) = tan
y/x = θ.
2
2
2
Note: zz = z
, so z = z
/z, so z/ z
= 1/z this is used to do division.
Example 4
Let z =
2 + i and w = 1
i then:
1. Re(z) =
2, Im(z) = 1, Re(w) = 1 and Im(w) =
1.
(
1
) .
arctan
1
2
2
2. z =
( 2)
+ 1
=
5, arg(z) = arctan
so z =
5e
2
2
1
2
2
3. w =
1
+ ( 1)
=
2, arg(w) = arctan
=
so w =
2e
.
4
1
4
5
4. z =
2
i =
5e
and w = 1 + i =
2e
.
6
4
Addition z + w = (x + u) + i(y + v) (Includes Subtraction).
( + )
Multiplication zw = (x + iy)(u + vi) = (xu
yv) + i(xv + yu) = qre
.
z
zw
Division
=
.
2
w
w
Example 5
Let z =
2 + i and w = 1
i then:
1. z + w = ( 2 + 1) + (1
1)i =
1.
2
2. zw = ( 2 + i)(1
i) =
2 + 2i + i
i
=
2 + 1 + 3i =
1 + 3i.
1
1
1
2
2
3. z/w = zw/ w
=
( 2 + i)(1 + i) =
( 2
2i + i + i
) =
( 3
i).
2
2
2
2
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