Complex Numbers: Product, Quotient and Inverse
Martti E. Pesonen, last revised 30.11.2006
1. Product of Complex Numbers
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Beside you see two complex numbers as plane vectors
z
and
w
and their product is
zw
.
Problems
1. When
z
= (-2,1) and
w
= (4,1), what is
zw
?
2. When
z
=
i
, what must
w
be, so that the product
zw
= (2,-3)?
3. If
z
is kept constant, when is
zw
real?
4. Keep the point
z
fixed and move
w
. Which geometric property stays invariant?
2. Division of Complex Numbers
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Beside you see two complex numbers as plane vectors
z
and
w
and their quotient is
z/w
.
Problems
1. When
z
= (-8,4) and
w
= (3,1), what is
z/w
?
2. Solve the equation (4-3i)/
w
= 1 - 2i.
3. When
z
is purely imaginary, for what kind of numbers
w
are
w
and
z/w
on a same straight line through the origin?
4. When
w
=
i
, what can you say about
z
and
z/w
?
3. Inversion of Complex Numbers
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Beside you see a complex number as a plane vector
z
and its inverse 1/
z
.
Also the conjugate conj(1/
z
) can be switches visible.
Problems
1. What is the inverse of 3 + i ?
2. Where is the inverse of the conjugate of
z
when
z
is
inside
the unit disk
D
?
3. What is the inverse of a
circle
like?
4. What does the inverse of an origin-centered square look like?