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Kalkulus I
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RRules of Complex Arithmetic
1. i2 = -1
2 .Every complex number has the ``Standard
Form'' a + bi for some real a and b.
3.For real a and b,
4.
5.
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The point P is the image-point of
the complex number (a,b).
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Equal numbers
a + bi = c + di
<=>
a = c and b = d
Addition
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Conjugate
a + bi = conj(a + bi) = a - bi
Modulus of a complex number
We define modulus or absolute value of a + bi
as sqrt(a2 + b2) .
We write this modulus of a + bi as |a + bi|.
If p is the representation of a + bi in the
Gauss-plane, the distance from O to P is the
modulus of a + bi.
Ex: |3 + 4i| = 5
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Division in practice
a  bi (a  bi )(c  di )

c  di (c  di )(c  di )
(ac  bd )  i (bc  ad )
2
2
c d
ac  bd bc  ad

i
2
2
2
2
c d
c d
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Example
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Polar representation of complex numbers
a + ib = r (cos(t) + i sin(t))
The conjugate of r(cos(t) + i sin(t)) is r(cos(-t) + i sin(-t))
Natural power of a complex number
( r (cos(t) + i sin(t)) )n = rn .(cos(nt) + i sin(nt))
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Find all z so that z4 = -8(i-(3))
The 4th roots are
-8(i-
(3)) = 16.(cos(-/6) + i sin(-  /6))
z = 2.(cos(-  /24 + k.  /2) + i sin(-  /24 + k.  /2))
with k in Z
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