Complex Numbers Worksheet Page 2
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31.3
Operations on Complex Numbers
1.3.1
The Conjugate of a Complex Number
If z = a + bi is a complex number, then its conjugate, denoted by ¯ z is a
bi. For example,
z = 3 + 5i
¯ z = 3
5i
z = i
¯ z =
i
z = 3
¯ z = 3
Graphically, the conjugate of a complex number is it’s mirror image across the horizontal axis.
1.3.2
Addition/Subtraction, Multiplication/Division
To add (or subtract) two complex numbers, add (or subtract) the real parts and the imaginary parts separately:
(a + bi)
(c + di) = (a + c)
(b + d)i
To multiply, expand it as if you were multiplying polynomials:
2
(a + bi)(c + di) = ac + adi + bci + bdi
= (ac
bd) + (ad + bc)i
2
and simplify using i
=
1. Note what happens when you multiply a number by its conjugate:
2
2
2
2
2
2
z ¯ z = (a + bi)(a
bi) = a
abi + abi
b
i
= a
+ b
= z
z
Division by complex numbers z, w:
, is defined by translating it to real number division (rationalize the
w
denominator):
z
z ¯ w
z ¯ w
=
=
2
w
w ¯ w
w
Example:
1 + 2i
(1 + 2i)(3 + 5i)
7
11
=
=
+
i
3
5i
34
34
34
1.4
The Polar Form of Complex Numbers
1.4.1
Euler’s Formula
Any point on the unit circle can be written as (cos(θ), sin(θ)), which corresponds to the complex number
cos(θ) + i sin(θ). It is possible to show the following directly, but we’ll use it as a definition:
iθ
Definition (Euler’s Formula): e
= cos(θ) + i sin(θ).
1.4.2
Polar Form of a + bi:
The polar form is defined as:
iθ
2
2
z = re
where
r = z =
a
+ b
θ = arg(z)
To be sure that the polar form is unique, we restrict θ to be in the interval ( π, π]. You might think of arg(z)
as the four-quadrant inverse tangent- That is:
b
1
If (a, b) is in the first or fourth quadrant, then θ = tan
.
a
If a = 0 and b = 0, then θ is either π/2 (for b > 0) or
π/2.
1
b
If (a, b) is in the second quadrant, add π: θ = tan
+ π
a
b
1
If (a, b) is in the third quadrant, subtract π: θ = tan
π
a
The argument of zero is not defined.
Best way to remember these: Quickly plot a + bi to see if you need to add or subtract π.
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