daily check:
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Daily Check:. Perform the indicated operation . Find the area and perimeter of the box . 3. Perimeter = ____ 4. Area = ____. 2x-3. 2x+1. Homework Review. CCGPS Analytic Geometry Day 32 (9-20-13). UNIT QUESTION: In what ways can algebraic methods be used in problem solving? - PowerPoint PPT PresentationTRANSCRIPT
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Daily Check:
Perform the indicated operation.
2 21. (3 4 7) (5 4 10)x x x x
2. (3 2)( 5)x x
Find the area and perimeter of the box.
3. Perimeter = ____
4. Area = ____2x+1
2x-3
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Homework Review
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CCGPS Analytic GeometryDay 32 (9-20-13)
UNIT QUESTION: In what ways can algebraic methods be used in problem solving?Standard: MCC9-12.N.RN.1-3, N.CN.1-3, A.APR.1
Today’s Question:How do we take the square root of negative numbers?Standard: MCC9-12..N.CN.1-3
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2
2
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i1i
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• You can't take the square root of a negative number, right?
• When we were young and still in Math I, no numbers that, when multiplied by themselves, gave us a negative answer.
• Squaring a negative number always gives you a positive. (-1)² = 1. (-2)² = 4 (-3)² = 9
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So here’s what the math people did: They used the letter “i” to represent the square root of (-1). “i” stands for “imaginary”
1i
So, does
1really exist?
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Examples of how we use
1i
16 16 1
4 i 4i
81 81 1
9 i 9i
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Examples of how we use
1i
45 45 1
3 3 5 1
3 5 1
3 5 i
3 5i
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2 2 2 5 5 1
200 200 1
2 5 2 1
10 2 i
10 2i
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1.3Powers of i and
Complex Operations
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Since -1, theni
12 i
ii 3
14 i
ii 5
16 i
ii 7
18 i
*For larger exponents, divide the exponent by 4, then use the remainder
as your exponent instead.
Example: ?23 i3 ofremainder a with 5
423
.etcii - which use So, 3
ii 23
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!Try These131. i272. i
543. i
724. i
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$25,000 Pyramid
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$25,000 Pyramid
2i-1 -i 1 -1
i -1 -i
-i 1
i
3i 4i 14i
45i 54i 63i
71i 92i
141i
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$25,000 Pyramid
2i-1 -i 1 -i
-i -1 i
-1 -i
i
3i 4i 23i
35i 66i 77i
82i 95i
173i
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Complex Numbers
A complex number has a real part & an imaginary part.
Standard form is:
bia
Real part Imaginary partExample: 5+4i
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The Complex Plane
Imaginary Axis
Real Axis
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Graphing in the complex plane
i34 .
i52 .i22 .
i34
.
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Adding and SubtractingAdd or subtract the real parts, and then, add or subtract the imaginary parts.
Ex: (3 2 ) (7 6 )i i (3 7) (2 6 )i i
10 8i Ex: (6 5 ) (1 2 )i i
(6 5 ) ( 1 2 )i i (6 1) (5 2 )i i
5 3i
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1) (9-4i)-(-2+3i)
11 7i
Your Turn!
2) 9-(10+2i)-5i
1 7i
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4 3 4 33) (11i 2 ) (2 6 )i i i
9 8i
Your Turn!
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MultiplyingTreat the i’s like variables, then change any that are not to the first power
Ex: )3( ii 23 ii
)1(3 i
i31
Ex: )26)(32( ii 2618412 iii
)1(62212 i62212 i
i226
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1) ( 3 )(8 5 )i i
29 7i
Your Turn!
2) (4 3 )(4 3 )i i
25
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3) 2 (1 4 )i i
8 2i
Your Turn!
4) (3 2 )( 5 9 )i i
33 17i
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Conjugates: Two complex numbers of the form a + bi anda – bi are complex conjugates. The product is always a real number
Ex: (2 4 )(2 4 )i i 24 8 8 16i i i
4 16( 1)
20
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Conjugates: Two complex numbers of the form a + bi anda – bi are complex conjugates. The product is always a real number
Ex: (2 4 )(2 4 )i i 24 8 8 16i i i
4 16( 1)
20
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Dividing Complex Numbers
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Conjugates: Two complex numbers of the form a + bi anda – bi are complex conjugates. The product is always a real number
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Dividing Complex Numbers
Multiply the numerator and denominator by the conjugate of the denominator.
Simplify completely.
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5 2 3 81) *3 8 3 8
i ii i
(5 2 )(3 8 )(3 8 )(3 8 )
i ii i
2
2
15 40 6 169 24 24 64
i i ii i i
15 46 16( 1)9 64( 1)i
15 46 169 64i
1 4673
i
1 4673 73
i
Writing in Standard Form
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52) 1 i
5 52 2
i
Your Turn!
8 33) 1 2
ii
2 195 5
i
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6 34) 2ii
3 32
i
Your Turn!
5 65) 3ii
523
i
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Assignment
Complex Numbers Practice WS