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Imaginary
and Complex Numbers |
Polar or trigonometric
notation of complex numbers |
Exponentiation
and root extraction of complex numbers in the polar form |
Powers and roots of
complex numbers, use of de Moivre’s formulas |
Exponentiation
and root extraction of complex numbers examples
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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 cos j |
y
= r sin j |
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
= cos
j
+
i sin j,
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
e
ij
= r
(cos j
+
i sin j) |
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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Exponentiation
and root extraction of complex numbers in the polar form - de
Moivre's formula |
We use the polar form
for exponentiation and root extraction of complex numbers that
are known as de Moivre's formulas. |
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zn
= rn · [cos
(nj)
+ i sin (nj)] |
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and |
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Exponentiation
and root extraction of complex numbers examples
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Example: |
Compute |
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Solution: |
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or
in the polar form, |
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and |
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since
exponentiation with integer exponent |
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then |
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Example: |
Compute |
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Solution:
As square root of a
complex number is a complex number, then |
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and, two
complex numbers are equal if their real parts are equal and
their imaginary parts are equal, that is |
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Example: |
Calculate |
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using de Moivre's
formula. |
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These complex numbers satisfy the equation z3
= -8
and by the Fundamental theorem of algebra, since this equation
is of degree 3, there must be 3 roots. |
Thus, for
example to check the root zk=2
we cube this solution, |
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then |
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Example: |
Calculate |
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r
= 64 and
j =
p |
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thus, |
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These complex numbers satisfy the equation z6
= -64
and by the Fundamental theorem of algebra, since this equation
is of degree 6, there must be 6 roots. |
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Example: |
Calculate |
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Example: |
Calculate |
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Pre-calculus
contents A
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