Download 6.5-complex numbers

6.5-Complex Numbers
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In the quadratic equation, the x-intercepts are
found.
What if there are no x-intercepts (zeros)
Consider the graph:
y  .5 x  3x  6.5
2
y  .5( x  3)  2
2
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The quadratic formula yields:
(3)  (3)  4(.5)(6.5)
x
2(.5)
2
6.5-Complex Numbers
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Solving for x yields:
3  9  13 3  4
x

1
1
When the b2  4ac (the discriminant) is
negative, there are no real solutions.
 If the solutions are not real, they must be
imaginary. This is where complex numbers
are used.
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i  1...so... 4  4 1  2i
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A complex number has a real part and an
imaginary part.
3  2i  3  2i..and..3  2i
6.5-Complex Numbers
When a quadratic has no real zeros, the
complex answers will be a conjugate pair in
the form
a  bi  a  bi..and..a  bi
 Another name for a conjugate pair is complex
conjugate.
 Roots (or solutions) of polynomials can be real
or complex ,or there may be some of both.
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 If
the polynomial has real coefficients, any complex
roots will always come in conjugate pairs.
2i..and  2i
3  4i..and..3  4i
6.5-Complex Numbers
Complex Arithmetic:
 Addition/Subtraction: (3  2i )  (4  5i )  7  3i
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(3  2i)  (1  4i)  2  6i (2  6i )  (2  6i)  4
 Adding
conjugate pairs will always result in a real solution
i  ( 1)  1
2
(3  2i)(5  4i)  15  12i  10i  8i  15  2i  8  23  2i
2
(7  3i)(7  3i)  49  21i  21i  9i  49  9  58
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Multiplication:
 Multiplying
2
2
conjugate pairs will always result in a
real solution because the inner/outer terms
2
i
cancel and the  1
6.5-Complex Numbers
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Division of Complex Numbers:
form must have no i (or √ ) in the
denominator (it must be a rational number).
 To simplify, multiply top and bottom of fraction
by the complex conjugate.
 Simplest
3  2i (4  i) (3  2i)(4  i) 12  3i  8i  2i 2

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2
4  i (4  i) (4  i)(4  i)
16  4i  4i  i
12  5i  2(1) 14  5i


16  (1)
17
The denominator
is rationalized
6.5-Complex Numbers
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Complex numbers
can be graphed on a
modified coordinate
system.
A
C
 x-axis
represents real
 y-axis represents Real
imaginary
axis (x)
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Point A (2+3i)
Point B (3-4i)
Point C (-1+2i)
Point D (0-3i)
D
B
Imaginary axis (y)