Download PC 2.4 Complex Numbers

Chapter 2
Polynomial and Rational Functions
Warm Up 2.4
 From 1980 to 2002, the number of quarterly
periodicals P published in the U.S. can be
modeled by
P  0.138t 4  6.24t 3  86.8t 2  239t  1450
 Where t is the number of years since 1980.
Describe the end behavior of the graph.
b. Graph the model on the domain 0 ≤ t ≤ 22.
c. Use the model to predict the number of
quarterly periodicals in the year 2010.
a.
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2.4 Complex Numbers
Objectives:
Use the imaginary unit i to write complex numbers.
Add, subtract, and multiply complex numbers.
Use complex conjugates to write the quotient of
two complex numbers in standard form.
Plot complex numbers in the complex plane.
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The Imaginary Unit, i
If we find the square root of a number, are there any
restrictions?
Yes – the term inside the radical must be ≥ 0.
What happens if the term inside the radical is negative?
For example, what if x2 + 1 = 0?
This equation has no real solution.
The Imaginary Unit i was created for this purpose.
That is, i   1 and i 2  1
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Complex Numbers
A complex number consists of a real part and
an imaginary part.
Standard form of a complex number is a ± bi,
where a is the real part and bi is the imaginary
part.
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Adding and Subtracting
Complex Numbers
Add or subtract the real parts and add or subtract the
imaginary parts.
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Examples
Perform the following operations. Simplify if possible.
1.
3  i   2  3i  
2. 3   2  3i  
3.
3  2i   4  i  
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Multiplying Complex Numbers
FOIL or distribute and simplify.
Note: Be sure numbers are written in
complex form before you begin.
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Examples
Perform the following operations. Simplify if possible.
1.
 4   16 
2.
2  i 4  3i  
3. 4i  1  5i  
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Imaginary Units of Higher Power
How can we easily find i raised to any integer power?
i  i
1
i  1
2
i 3  i 2  i1  1 i  i
i  i  i  1 1  1
4
2
2
What is the next one?
What is the pattern?
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Complex Conjugates
The product of two complex numbers is usually
another complex number.
However, the product of two complex numbers can be
a real number if the two numbers are conjugates.
Numbers of the form a + bi and a – bi are complex
conjugates.
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Example
Multiply 3 – 5i by its complex conjugate.
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Complex Number Quotients
When we divide a complex number by another
complex number, we can write it as a quotient.
We want to make sure that the denominator is real,
not imaginary or complex.
To do so, we must multiply both the numerator and
the denominator of the quotient by the complex
conjugate of the denominator.
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Example
Write the quotient in standard form.
2  3i
4  2i
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Homework 2.4
Worksheet 2.4
#5, 9, 11, 17 – 35 odd, 37, 45 – 61 odd
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