Imaginary

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Imaginary Complex Numbers
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Once upon a time
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-In the set of real numbers, negative numbers do
not have square roots.
-Imaginary numbers were invented so that negative
numbers would have square roots and certain
equations would have solutions.
-These numbers were devised using an imaginary
unit named i.
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-The imaginary numbers consist of all numbers bi,
where b is a real number and i is the imaginary
unit, with the property that i² -1.
-The first four powers of i establish an
important pattern and should be memorized.
Powers of i
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Divide the exponent by 4 No remainder answer is
1. remainder of 1 answer is i. remainder of 2
answer is 1. remainder of 3answer is i.
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Powers of i
1.) Find i23
2.) Find i2006
3.) Find i37
4.) Find i828
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Complex Number System
Reals
Imaginary i, 2i, -3-7i, etc.
Rationals (fractions, decimals)
Integers (, -1, -2, 0, 1, 2, )
Irrationals (no fractions) pi, e
Whole (0, 1, 2, )
Natural (1, 2, )
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Simplify.
-Express these numbers in terms of i.
3.)
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You try
6.
7.
8.
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To multiply imaginary numbers or an imaginary
number by a real number, it is important first to
express the imaginary numbers in terms of i.
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Multiplying
9.
10.
11.
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Complex Numbers
a bi
imaginary
real
The complex numbers consist of all sums a bi,
where a and b are real numbers and i is the
imaginary unit. The real part is a, and the
imaginary part is bi.
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Add or Subtract
12.
13.
14.
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Multiplying Dividing Complex Numbers

• Part of 7.9 in your book

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Multiply
REMEMBER i² -1
1)
2)
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You try
3)
4)
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Multiply
5)
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You try
6)
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You try
7)
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Conjugate
-The conjugate of a bi is a bi
-The conjugate of a bi is a bi
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Find the conjugate of each number
8)
9)
10)
11)
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Divide
12)
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You try
13)