Section 2.4 Complex Numbers

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Section 2.4 Complex Numbers
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What you should learn

• How to use the imaginary unit i to write complex
  numbers
• How to add, subtract, and multiply complex
  numbers
• How to use complex conjugates to write the
  quotient of two complex numbers in standard form
• How to find complex solutions to quadratic
  equations

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Real Number System

• 1, 2, 3, 4,
• How many natural numbers are there?

Natural
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Real Number System

• 0, 1, 2, 3, 4,
• How many whole numbers are there?

Natural
Whole
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Real Number System
Natural

• ...-3, -2, -1, 0, 1, 2, 3,
• How many integers numbers are there?

Whole
Integers
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Real Number System

• Fractions
• How many rational numbers are there?

Natural
Whole
Integers
Rational
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Real Number System
Natural

• How many irrational numbers are there?

Whole
Integers
Irrational
Rational
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Real Number System
Natural

• Each set is a subset of the Real Number System.
• The union of all these sets forms the real number
  system.
• The number line is our model for the real number
  system.

Whole
Integers
Irrational
Rational
Real Numbers
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Definition of Square Root

• If a2 n then a is a square root of n.
• 42 (4)(4) 16
• ? 4 is a square root of 16
• (-4)2 (-4)(-4) 16
• ? -4 is a square root of 16

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What square root of -16?

• Whatever it is it is not on the real number line.

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Definition of i
The number i is such that
Imaginary Unit
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Complex Numbers
Imaginary
REAL
Complex
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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 then the number a bi a is a real
  number.
• If b ? 0, then the number a bi is called an
  imaginary number.
• A number of the form bi, where b ? 0 is called a
  pure imaginary number.

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Examples
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If you square a radical you get the radicand
2
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Whenever you have i2 the next turn you will have
-1 and no i.
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Equality of Complex numbers

• If a bi c di, then a c and b d.

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Is a negative times a negative always positive?
Trick question. This is not a negative times a
negative.
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Example
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Example
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Example
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Example
Cancel the i factor
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Add
Collect like terms.
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Subtract
First distribute the negative sign.
Now collect like terms.
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Multiplication
F
O
I
L
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Simplify each expression. Express your answer in
form.
F-O-I-L
Recall i2-1
Combine like terms.
Combine like terms.
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Write in the form
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Multiply by the conjugate factor.
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Powers of i
Anything other than 0 raised to the 0 is 1.
Anything raised to the 1 is itself.
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Simplify as much as possible.
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Use the Quadratic Formula
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Homework Section 2.4Puzzle