Consider the quadratic equation x2 1 = 0.

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Complex Numbers

• Consider the quadratic equation x2 1 0.
• Solving for x , gives x2 1

We make the following definition
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Complex Numbers
Note that squaring both sides yields therefore a
nd so and

And so on
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• Real numbers and imaginary numbers are subsets of
  the set of complex numbers.

Complex Numbers
Imaginary Numbers
Real Numbers
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Definition of a Complex Number

• If a and b are real numbers, the number a bi is
  a complex number, and it is said to be written in
  standard form.
• If b 0, the number a bi a is a real number.

If a 0, the number a bi is called an
imaginary number.
Write the complex number in standard form
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Addition and Subtraction of Complex Numbers

• If a bi and c di are two complex numbers
  written in standard form, their sum and
  difference are defined as follows.

Sum
Difference
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• Perform the subtraction and write the answer in
  standard form.
• ( 3 2i ) ( 6 13i )
• 3 2i 6 13i
• 3 11i

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Multiplying Complex Numbers

• Multiplying complex numbers is similar to
  multiplying polynomials and combining like terms.
  
• Perform the operation and write the result in
  standard form. ( 6 2i )( 2 3i )
• F O I L
• 12 18i 4i 6i2
• 12 22i 6 ( -1 )
• 6 22i

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• Consider ( 3 2i )( 3 2i )
• 9 6i 6i 4i2
• 9 4( -1 )
• 9 4
• 13
• This is a real number. The product of two
  complex numbers can be a real number.

This concept can be used to divide complex
numbers.
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Complex Conjugates and Division

• Complex conjugates-a pair of complex numbers of
  the form a bi and a bi where a and b
  are real numbers.
• ( a bi )( a bi )
• a 2 abi abi b 2 i 2
• a 2 b 2( -1 )
• a 2 b 2
• The product of a complex conjugate pair is a
  positive real number.

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• To find the quotient of two complex numbers
  multiply the numerator and denominator by the
  conjugate of the denominator.

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• Perform the operation and write the result in
  standard form.

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• Perform the operation and write the result in
  standard form.

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Expressing Complex Numbers in Polar Form

• Now, any Complex Number can be expressed as
• X Y i
• That number can be plotted as on ordered pair in
  
• rectangular form like so

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Expressing Complex Numbers in Polar Form

• Remember these relationships between polar and
• rectangular form

So any complex number, X Yi, can be written
in polar form
Here is the shorthand way of writing polar form
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Expressing Complex Numbers in Polar Form

• Rewrite the following complex number in polar
  form
• 4 - 2i

Rewrite the following complex number
in rectangular form
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Expressing Complex Numbers in Polar Form

Express the following complex number
in rectangular form
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Expressing Complex Numbers in Polar Form

Express the following complex number in polar
form 5i
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Products and Quotients of Complex Numbers in
Polar Form

The product of two complex numbers,
and Can be obtained by using the following
formula
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Products and Quotients of Complex Numbers in
Polar Form

The quotient of two complex numbers,
and Can be obtained by using the following
formula
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Products and Quotients of Complex Numbers in
Polar Form

Find the product of 5cis30 and 2cis120
Next, write that product in rectangular form
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Products and Quotients of Complex Numbers in
Polar Form

Find the quotient of 36cis300 divided by 4cis120
Next, write that quotient in rectangular form
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Products and Quotients of Complex Numbers in
Polar Form

Find the result of Leave your answer in polar
form.
Based on how you answered this problem, what
generalization can we make about raising a
complex number in polar form to a given power?

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De Moivres Theorem
De Moivre's Theorem is the theorem which shows us
how to take complex numbers to any power easily.

De Moivre's Theorem Let r(cos Fisin F) be a
complex number and n be any real number.
Then r(cos Fisin Fn rn(cos nFisin nF)
What is this saying?
The resulting r value will be r to the nth power
and the resulting angle will be n times the
original angle.

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De Moivres Theorem
Try a sample problem What is 3(cos
45isin45)5?

To do this take 3 to the 5th power, then multiply
45 times 5 and plug back into trigonometric form.
35 243 and 45 5 225 so the result is 243(cos
225isin 225)
Remember to save space you can write it in
compact form. 243(cos 225isin 225)243cis
225
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De Moivres Theorem

Find the result of
Because of the power involved, it would easier to
change this complex number into polar form and
then use De Moivres Theorem.

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De Moivres Theorem
De Moivre's Theorem also works not only for
integer values of powers, but also rational
values (so we can determine roots of complex
numbers).

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De Moivres Theorem

Simplify the following

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De Moivres Theorem
Every complex number has p distinct pth
complex roots (2 square roots, 3 cube roots, etc.)

To find the p distinct pth roots of a complex
number, we use the following form of De Moivres
Theorem
where n is all integer values between 0 and
p-1.
Why the 360? Well, if we were to graph the
complex roots on a polar graph, we would see that
the p roots would be evenly spaced about 360
degrees (360/p would tell us how far apart the
roots would be).
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De Moivres Theorem

Find the 4 distinct 4th roots of -3 - 3i

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De Moivres Theorem

Solve the following equation for all
complex number solutions (roots)