Complex Numbers

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Complex Numbers

• Lesson 3.3

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The Imaginary Number i

• By definition
• Consider powers if i

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Using i

• Now we can handle quantities that occasionally
  show up in mathematical solutions
• What about

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Complex Numbers

• Combine real numbers with imaginary numbers
• a bi
• Examples

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Try It Out

• Write these complex numbers in standard form a
  bi

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Operations on Complex Numbers

• Complex numbers can be combined with
• addition
• subtraction
• multiplication
• division
• Consider

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Operations on Complex Numbers

• Division technique
• Multiply numerator and denominator by the
  conjugate of the denominator

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Complex Numbers on the Calculator

• Possible result
• Reset modeComplex formatto Rectangular
• Now calculator does desired result

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Complex Numbers on the Calculator

• Operations with complex on calculator

Make sure to use the correct character for i. Use
2nd-i
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Warning

• Consider
• It is tempting to combine them
• The multiplicative property of radicals only
  works for positive values under the radical sign
• Instead use imaginary numbers

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Try It Out

• Use the correct principles to simplify the
  following

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The Discriminant
Return of the

• Consider the expression under the radical in the
  quadratic formula
• This is known as the discriminant
• What happens when it is
• Positive and a perfect square?
• Positive and not a perfect square?
• Zero
• Negative ?

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Example

• Consider the solution to
• Note the graph
• No intersectionswith x-axis
• Using the solve andcsolvefunctions

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Fundamental Theorem of Algebra

• A polynomial f(x) of degree n 1 has at least
  one complex zero
• Remember that complex includes reals
• Number of Zeros theorem
• A polynomial of degree n has at most n distinct
  zeros
• Explain how theorems apply to these graphs

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Conjugate Zeroes Theorem

• Given a polynomial with real coefficients
• If a bi is a zero, then a bi will also be a
  zero

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Assignment

• Lesson 3.3
• Page 211
• Exercises 1 78 EOO