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Imaginary
and complex numbers |
Multiplication and
division of
complex numbers |
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Polar
or trigonometric notation of complex numbers |
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Multiplication
and division of complex numbers in the polar form |
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| Multiplication and
division of
complex numbers |
| Multiplication: |
z1·
z2
=
(a + bi) · (c + di) = ac
+ bci + adi + bdi2
= (ac -
bd) +
(ad + bc)i |
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| Division: |
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| Examples: Given
are complex numbers, z1
= -3 +
2i and
z2
= 4 + 3i,
find z1 ·
z2 and
z1
/
z2.
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Solutions:
z1 · z2
= (-3 +
2i) · (4 + 3i) = -3 ·
4 + 2 · 4i +
(-3)
· 3i + 2 · 3 i2
= -18
-
i
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and |
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| Example:
For what real
number
a the real part of the complex number |
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equals
1. |
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Solution:
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| Example:
Evaluate the
expression |
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where
z = 1 -
i. |
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Solution:
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| Polar or trigonometric
notation of complex numbers |
| A point (x,
y)
of the complex plane that represents the complex number z
can also be specified by its distance r
from the origin and the angle j
between the line joining the point to the origin and the
positive x-axis. |
| Cartesian
coordinates expressed by polar coordinates: |
| x
= r cosj |
| y
= r sinj |
| plugged
into z
= x
+
yi
give |
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z
= r
(cos j
+
i sin j), |
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| where |
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| Thus,
obtained is the polar or trigonometric form of a complex number
where polar coordinates are r,
called the absolute value or modulus, and j,
that is called the argument, written j
= arg (z). |
| By using
Euler's formula eij
= cosj
+
i sinj,
a complex number can also be
written as |
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z
= r
eij |
which
is called the exponential form. |
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| To show
the equivalence between the algebraic and the trigonometric form of a complex number, |
| z
= r
eij
= r
(cosj
+
isinj) |
| express
the sine and the cosine functions in terms of the tangent |
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| and substitute
into above expression |
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| Example: Given
the complex number z
= 1
-
Ö3i,
express z
= x
+
yi
in the trigonometric form.
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| Solution:
The modulus |
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the argument |
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the trigonometric form is
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| Multiplication
and division of complex numbers in the polar form |
| If given
z1
= r1(cosj1
+ isinj1)
and z2
= r2(cosj2
+ isinj2)
then |
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z1
· z2 = r1
r2 · [cos(j1
+ j2)
+ isin(j1
+ j2)] |
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| and |
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College
algebra contents
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| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |
