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WARM-UP1. 4.
2. 5.
3.
25 5
3 3
5 27 3 10x x x
4 63 24x y
2
2x y
3
7 2
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COMPLEX NUMBERS
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IMAGINARY NUMBERSImaginary Numbers are numbers can be
written as a real number times “i.”
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IMAGINARY NUMBERSImaginary Numbers are numbers that can be
written as a real number times “i.”
Steps:1. View the radical without the negative and
simplify it2. After simplified form, attach an “i” onto the real
number3. Simplify further, if needed
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EXAMPLE 1SimplifyAfter simplified form, attach an “i” onto the real
number
4
4i 2i2i2i
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EXAMPLE 1
x = 0
Where does it cross the x-axis?
+2i
Simplify
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EXAMPLE 2
2i 2i
4
Simplify
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EXAMPLE 3
11i 11i
11
Simplify
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EXAMPLE 4
2 5i2 5i
20
Simplify
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YOUR TURN
5 3i 5 3i
75
Rules with Imaginaries:i1 = ii2 = –1
Answers CAN have an i but CAN NOT have two i’s
Two i’s make –1
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IMAGINARY NUMBERS
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WHY IS I2 EQUAL TO –1?
1i
1i 2 2
2 1i
How do we cancel the radicals?
Multiply
What is another way of writing this problem?
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EXAMPLE 5
3 3
2
3i
3 3i i
2i 2
31 3
Multiply
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EXAMPLE 6
16 16
24i
Multiply
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YOUR TURN
50 50
2
5 2i
Multiply
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EXAMPLE 7
6 6
6 6
6 6 6 6 36
6 6 6 6i i
22 6i
2
6 6
Multiply
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EXAMPLE 8
15 15
3 5
Multiply
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EXAMPLE 9
4i4i
8 2
Multiply
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YOUR TURN
28x 28x
34 16x x
Rationalize
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REVIEW1
2 6
1
2 6 1 6
2 6 6
6
2 6
6
12
6
12
Rationalize
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EXAMPLE 101
3i
23
i
i 1 3
i
3
i
3
i
1
3
i
i i
No i ’s AND radicals in the denominator
Rationalize
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EXAMPLE 111
6
1 6
6 6
i
i i
2
6
6
i
i
6
6
i
6
6
i
6
1 6
i
No i ’s AND radicals in the denominator
Rationalize
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EXAMPLE 1249
10
7 10
10
i
7 10
10
i
Rationalize
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YOUR TURN40
8
5i 5i
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COMPLEX NUMBERSComplex Numbers
a + b i
Conjugate is the complex number’s opposite sign Example: 2 + 3i ‘s conjugate is 2 – 3i
Remember: NO IMAGINARY NUMBERS in the denominator
SIMPLIFY RADICALS FIRST then OPERATE
Real Number
Imaginary Number
Add
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REVIEW 2 5 3 4x x
5 x5 x
Add
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EXAMPLE 13 2 5 3 4i i
5 i5 i
Subtract
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EXAMPLE 14 7 3 5 2i i
2 5i2 5i
7 3 5 2i i
Subtract
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YOUR TURN 2 4 2i i
3i 3i
MultiplyFOIL (First Outer Inner Last)
24 2 2x x x
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REVIEW 22 x
2 2x x
44 2x4 2 2x x 2 4 4x x
FOIL (First Outer Inner Last)Multiply
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EXAMPLE 15 22 i
24 2 2i i i 2 2i i
44 2i4 2 2i i 24 4i i
4 4 1i 3 4i3 4i
Multiply
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EXAMPLE 16 2 3 6i i
15 16i15 16i
212 2 18 3( )i i i
12 16 3( 1)i
12 16 3i
Multiply
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YOUR TURN 9 2 9 2i i
8585
Rationalize
What is the problem with the bottom?
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REVIEW1
4 3
1
4 3
4 3
4 3
We try to get rid of the radicals on the
bottom so the bottom will be even
With the CONJUGATE
4 3
16 3
4 3
13
4 3
13 13
or
Rationalize
What is the problem with the bottom?
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EXAMPLE 171
4 3i
1
4 3i
4 3
4 3
i
i
We try to get rid of the radicals on the
bottom so the bottom will be even
With the CONJUGATE
2
4 3
16 3 3 3
i
i i i
4 3
19
4 3
19 19
i
or
i
Rationalize
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EXAMPLE 181
2 3i
2 3
13
i
Rationalize
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EXAMPLE 192
1
i
i
1 3
2
i
Rationalize
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YOUR TURN3 2
3 2
i
i
5 12
13
i
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IMAGINARY POWERSSteps:1. Look at the exponent of the
imaginary number and divide by 4 2. View only the remainder or decimal3. Convert the problem with the number
REMEMBER: No i’s in the denominator
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IMAGINARY NUMBERS
Imaginary Imaginary ExponentExponent
RemainderRemainder DecimalDecimal Imaginary Imaginary NumberNumber
i1 R1 0.25 i
i2 R2 0.50 –1
i3 R3 0.75 –i
i4 R0 0.00 1
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IMAGINARY NUMBERS40 i 1
11 i i
22 i -1
33 i i
0 0.0R
1 0.25R
2 0.50R
3 0.75R
Simplify
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EXAMPLE 207i
7i 7 4 1 3
1.75
r
Imaginary Imaginary ExponentExponent
RemainderRemainder DecimalDecimal Imaginary Imaginary NumberNumber
i1 R1 0.25 i
i2 R2 0.50 –1
i3 R3 0.75 –i
i4 R0 0.00 1
i
Simplify
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EXAMPLE 2125i
i
24i i
25
46 1r
1r i
Simplify
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YOUR TURN589,276,538i
1
Simplify
i25 = ii36 = 1
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EXAMPLE 2225 36i i
i
Simplify
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YOUR TURN21 30i i
1 i
Simplify
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EXAMPLE 24 7
2i
128i
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