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Entry task- Solve two different ways
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4.8Complex Numbers
Target: I can identify and perform operations with
complex numbers
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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.
1i
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Imaginary numbers:
i 1
i2 1
i is not a variable it is a symbol for a specific
number
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With your a/b partner determine the values for the cycle of i
i
-i
1
i
1
-1 -1 -1
i
-i
1
1
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Definition of Imaginary Numbers
Any number in form a+bi, where a and b are
real numbers and i is imaginary unit.
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Definition of Pure imaginary numbers:
Any positive real number b,
where i is the imaginary unit and bi is called the pure
imaginary number.
b2 b2 1 bi
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Simplify the expression.
1. 81 81 1 9i
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4. 8i 3i 24i2 241
Remember i2 1
Simplify each expression.
24
5. 5 20 i 5i 20Remember that 1 i
i2 100 110Remember i
2 1
10
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When adding or subtracting complex numbers, combine like terms.
Ex: 8 3i 2 5i 8 2 3i 5i
10 2i
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8 7i 1211i
8 12 7i 11i
418i
Simplify.
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Simplify.
9 12 6i 2i
3 8i
9 6i 122i
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Multiplying complex numbers.
To multiply complex numbers, you use the
same procedure as multiplying polynomials.
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Simplify.
8 5i 2 3i
16 24i 10i 15i2F O I L
16 14i 15 31 14i
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Simplify.
3018i 10i 6i2F O I L
3028i 6 2428i
62i 5 3i
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-Express these numbers in terms of i.
1.) 5 1*5 1 5 5i
2.) 7 1*7 1 7 7i
3.) 99 1*99 1 99
9*11i
3 11i
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Conjugates
In order to simplify a fractional complex number, use a conjugate.
What is a conjugate?
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a b c d and a b c d
are said to be conjugates of each other.
Ex: 3 2i 5 and 3 2i 5
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Lets do an example:
Ex: 8i
1 3i
8i
1 3i1 3i
1 3iRationalize using the conjugate
Next
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8i 24i2
1 98i 24
10
4i 12
5Reduce the fraction
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Lets do another example
Ex: 4 i
2i4 i
2i
i
i
4i i2
2i2
Next
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4i i2
2i2 4i 1
2
Try these problems.
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1. 3
2 5i
2. 3 - i2 - i
1. 2 5i
9
2. 7 i
5
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MULTIPLYING COMPLEX NUMBERS
2
1. 4( 2 3 )
2. ( )( 3 )
3. (2 )(4 3 )
4. (3 2 )(3 2 )
5. (3 2 )
Multiply
i
i i
i i
i i
i
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1. 4( 2 3 ) 8 12i i
ANSWERS
2. ( )( 3 )i i 23 3( 1) 3i
3. (2 )(4 3 )i i 28 6 4 3i i i 28 2 3i i
(-1)
11 2i
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4. (3 2 )(3 2 )i i (-1)
1325. (3 2 ) (3 2 )(3 2 )i i i
29 6 6 4i i i =9 12 4( 1)i
5 12i
249 i
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Use the quadratic formula to solve the following:
22. 3 2 5 0x x a=3, b= -2, c=5
22 ( 2) 4(3)(5)x
2(3)
6
4
14
2 2 14i
6
1 14i
3
6042 x6
562 x
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Let’s Review
You need to be able to:– 1) Recognize what i, i2, i3 ect. is equal to (slide 5)– 2) Simplify Complex numbers– 3) Combine like terms (add or subtract)– 4) Multiply (FOIL) complex numbers– 5) Divide (multiply by complex conjugates)
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Assignment Pg.253 -254
Homework – #9-43 odds, skip 13,15,17
Challenge - 70