Module 4 Lesson 2 Notes

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Module 4 Lesson 2 Complex Numbers

• Remediation Notes

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In the set of real numbers, negative numbers do
not have square roots.  For example, x2 -1
has no real solutions because the square of any
real number is never negative.
To fix this problem, a new kind of number, called
an imaginary number was invented so that negative
numbers would have a square root.
Imaginary Unit i
The imaginary unit can be used to find the square
root of any negative number.
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Follow the pattern to find the value an in.
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Solutions

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The powers of i are cyclic and repeat in patterns
of 4 i,-1,-i, 1. To calculate any high power
of i, you can convert it to a lower power by
dividing the exponent by 4 and matching the
remainder to one of the powers of i in the table
below.

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Example
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Complex Numbers

• A complex number has a real part an imaginary
  part.
• Standard form is

Real part
Imaginary part
Example 3 2i
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Properties

• Imaginary unit, i, is a variable. Therefore,
  properties for simplifying also apply.
• To add or subtract complex numbers, combine like
  terms. That is combine the real parts and
  combine the imaginary parts.
• To multiply complex numbers, you can use FOIL.
  Remember i2 -1.
• To divide complex numbers we must multiply the
  denominator by its conjugate.

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Adding Subtracting
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Multiplying