complex numbers

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Complex Numbers

• N-CN.1 Know there is a complex number i such that
  i2 1, and every complex has the form a bi
  with a and b real.
• N-CN.2 Use the relation i2 1 and the
  commutative, associative, and distributive
  properties to add, subtract, and multiply complex
  numbers.

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First a little review
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Real Number System

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

Natural
Infinite
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Real Number System

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

Natural
Whole
Infinite
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Real Number System
Natural

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

Whole
Integers
Infinite
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• Fractions
• How many rational numbers are there?

Natural
Whole
Infinite
Integers
Rational
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Real Number System
Natural

• How many irrational numbers are there?

Whole
Integers
Infinite
Rational
Irrational
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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

The square root of a number is a value that, when
multiplied by itself, gives the number.
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What square root of -16?

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

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• You may remember being told that you can't take
  the square root of a negative number, because no
  numbers when squared resulted in a negative
  product.
•   Squaring a negative number always gives you a
  positive.  

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• however, if we had a number for the square root
  of negative one, we could take the square root
  of a negative numbers.
• So mathematicians came up
  with the number i

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Definition of i
Imaginary Unit
The number i is such that
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Definition of pure imaginary numbers
i is not a variable it is a symbol for a
specific number, the square root of negative one.
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Imaginary Numbers
A number that when squared gives a negative
result.
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Complex Numbers

• A complex number is a combination of a
• real number and an imaginary number.
• Standard form is

Real part
Imaginary part
Example 54i
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Complex Numbers
Imaginary
REAL
Complex
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If you square a radical you get the radicand
2
2
Whenever you have i2 , it can be simplified to
-1 because i2 -1
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Examples
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Is a negative times a negative always positive?
Simplify negative radicals before multiplying
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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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Equality of Complex numbers

• If a bi c di, then a c and bi di.