Download 4.8 Day 1 Complex Numbers.notebook

4.8 Day 1 Complex Numbers.notebook
November 12, 2015
4.8 Complex Numbers
Normally, you can't take the square root of a negative
number in the real number system, but you can in the
imaginary number system
What is the
?
Now you can simplify problems that were previously not possible to
simplify further.
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4.8 Day 1 Complex Numbers.notebook
November 12, 2015
The reason for the name "imaginary" numbers
is that when these numbers
were first proposed several hundred years ago,
people could not "imagine" such a number.
It is said that the term "imaginary" was coined by René Descartes in the seventeenth century and
was meant to be a derogatory reference since, obviously, such numbers did not exist. Today, we
find the imaginary unit being used in mathematics and science. Electrical engineers use the imaginary
unit (which they represent as j ) in the study of electricity.
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4.8 Day 1 Complex Numbers.notebook
November 12, 2015
Imaginary Numbers are
Useful...
AC (Alternating Current)
Electricity changes between positive and negative in a sine wave.
If you combine two AC currents they may not match properly,
and it can be very hard to figure out the new current.
But using imaginary numbers and real numbers together makes
it a lot easier to do the calculations.
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4.8 Day 1 Complex Numbers.notebook
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Imaginary numbers occur when a quadratic
equation has no roots in the set of real
numbers.
y
5
4
3
Look ,
no real roots!
2
1
x
­5
­4
­3
­2
­1
0
1
2
3
4
5
­1
­2
­3
­4
­5
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4.8 Day 1 Complex Numbers.notebook
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Simplify each number using the imaginary number, i.
1.
2.
3.
4.
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4.8 Day 1 Complex Numbers.notebook
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Remember that:
Find i on your calculator.
Simplify each number using the imaginary number, i.
1.
2.
3.
4.
To graph in complex number plane, the point (a,b) represents
the complex number a + bi.
The real part is graphed on the horizontal axis
and the imaginary part on the vertical axis.
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4.8 Day 1 Complex Numbers.notebook
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To add or subtract complex numbers, combine the real parts and imaginary
parts separately-combine the like terms.
You can use your
calculator too!
1. (7 - 2i) + (- 3 + i)
3. (5 - 3i) - (- 2 + 4i)
2. (- 3 + 9i) + (4 + 9i)
4. (1 + 5i) - (3 - 2i)
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4.8 Day 1 Complex Numbers.notebook
Hint: Any
November 12, 2015
becomes -1.
What is the product?
5. (3i)(- 5 + 2i)
6. (7i)(3i)
To multiply complex numbers a + bi and c + di, use FOIL or the
box method.
7. (4 + 3i)(- 1 - 2i)
8. (2 - 3i)(4 + 5i)
4 + 3i
-1
- 2i
2 - 3i
4
+ 5i
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4.8 Day 1 Complex Numbers.notebook
November 12, 2015
Multiply (a + bi) by its complex conjugate (a - bi). The
product of complex conjugates is a real number.
For quotients, we don't want to leave an i in the denominator.
What is each quotient?
Complex conjugate
9.
to make it...
-just change the sign
of the imaginary
term!
Try with the calculator!
10.
11.
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4.8 Day 1 Complex Numbers.notebook
November 12, 2015
The Unit Imaginary Number, i, has an interesting property.
It "cycles" through 4 different values each time you multiply by i.
For later on....
The circle of i
i
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4.8 Day 1 Complex Numbers.notebook
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HW p.253; 8­12 all, 19­31 odd
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4.8 Day 1 Complex Numbers.notebook
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HW p.253; 8­12 all, 19­31 odd
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