Complex Numbers 1

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Complex Numbers 1
a bi
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We can add, subtract, multiply or divide complex
numbers. After performing these operations if
weve simplified everything correctly we should
always again get a complex number (although the
real or imaginary parts may be zero). Below is
an example of each to refresh your memory.
Combine real parts and combine imaginary parts
(3 2i) (5 4i)
8 6i
ADDING
Be sure to distribute the negative through before
combining real parts and imaginary parts
(3 2i) - (5 4i)
SUBTRACTING
-2 2i
3 2i - 5 4i
FOIL and then combine like terms. Remember i 2
-1
(3 2i) (5 4i)
MULTIPLYING
15 12i 10i8i2
Notice when Im done simplifying that I only have
two terms, a real term and an imaginary one. If
I have more than that, I need to simplify more.
15 22i 8(-1) 7 22i
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DIVIDING
FOIL
Combine like terms
Recall that to divide complex numbers, you
multiply the top and bottom of the fraction by
the conjugate of the bottom.
This means the same complex number, but with
opposite sign on the imaginary term
Well put the 41 under each term so we can see
the real part and the imaginary part
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Lets solve a couple of equations that have
complex solutions to refresh our memories of how
it works.
Square root and dont forget the ?
?
-25
-25
The negative under the square root becomes i
Use the quadratic formula
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Powers of i
We could continue but notice that they repeat
every group of 4. For every i 4 it will 1
To simplify higher powers of i then, we'll group
all the i 4ths and see what is left.
4 will go into 33 8 times with 1 left.
4 will go into 83 20 times with 3 left.
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This "discriminates" or tells us what type of
solutions we'll have.
If we have a quadratic equation and are
considering solutions from the complex number
system, using the quadratic formula, one of three
things can happen.
1. The "stuff" under the square root can be
positive and we'd get two unequal real solutions
2. The "stuff" under the square root can be zero
and we'd get one solution (called a repeated or
double root because it would factor into two
equal factors, each giving us the same solution).
3. The "stuff" under the square root can be
negative and we'd get two complex solutions that
are conjugates of each other.
The "stuff" under the square root is called the
discriminant.
The Discriminant
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Acknowledgement I wish to thank Shawna Haider
from Salt Lake Community College, Utah USA for
her hard work in creating this PowerPoint. www.sl
cc.edu Shawna has kindly given permission for
this resource to be downloaded from
www.mathxtc.com and for it to be modified to suit
the Western Australian Mathematics Curriculum.
Stephen Corcoran Head of Mathematics St
Stephens School Carramar www.ststephens.wa.edu.
au