Complex Numbers Practice
HSSP Audio and Speaker-building
Review the rules for complex numbers:
Definition:
i=
i2 =
Addition:
(a + bi) + (c + di) =
Multiplication:
(a + bi)(c + di) =
a+bi
c+di
Division:
=
=
|z|2 =
Magnitude (z = a + bi) :
Phase:
z=
�
√
Mar. 3, 2007
−1
(1)
−1
(2)
(a + c) + (b + d)i
(ac − bd) + (bc + ad)i
a + bi c − di
c + di c − di
(ac + bd) + (bc − ad)i
c2 + d2
a2 + b2
b
tan−1
a
(3)
(4)
Manipulating numbers numerically
Take these two numbers (they are called a conjugate pair):
√
z1 = 1 + i 3
√
z2 = 1 − i 3
• Firstly, what are the magnitudes and phase angles?
|z1 | =
�
|z2 | =
z1 =
�
z2 =
• How about their sum?
z1 + z2 =
• And the product?
Use either the individual components, or the magnitude and phase:
|z1 z2 | = |z1 ||z2 |, � (z1 z2 ) = � z1 + � z2
z1 z2 =
Algebraic expressions
• If A is a constant, find:
1−
1
=
1 + iA
• If ω, R, L and C are constants, and:
z1 = R + iωL
and
Find the following:
z1 z2
=
z1 + z 2
1
z2 =
1
iωC
(5)
(6)
(7)
(8)
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Audio and Speaker Electronics
Spring 2007
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