Section 4.4 The Derivative in Graphing and Applications- “Absolute Maxima and Minima”
Complex Numbers 22 11 Definitions Graphing 33 Absolute Values.
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Transcript of Complex Numbers 22 11 Definitions Graphing 33 Absolute Values.
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Complex Numbers
2
1Definitions
Graphing
3Absolute Values
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2
Imaginary Number (i)
Defined as:
Powers of i
1i
1i
12 i
ii 3
14 i
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Complex Numbers
A complex number has a real part & an imaginary part.
Standard form is:
bia
Real part Imaginary part
Example: 5+4i
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4
Definitions
Pure imaginary number Monomial containing i
Complex Number An imaginary number combined with a real
number Always separate real and imaginary parts
ii
5
3
5
2
5
32
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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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Absolute Value of a Complex Number
The distance the complex number is from the origin on the complex plane.
If you have a complex number the absolute value can be found using:) ( bia
22 ba
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Examples
1. i52
22 )5()2(
254 29
2. i622 )6()0(
360
366
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9
Simplifying Monomials
Simplify a Power of i Steps
Separate i into a power of 2 or 4 taken to another power
Use power of i rules to simplify i into -1 or 1 Take -1 or 1 to the power indicated Recombine any leftover parts
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Operations
Simplify a Power of iSimplify
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Simplifying Monomials Example Square Roots of Negative NumbersSimplify
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Addition & Subtraction
Add and Subtract Complex Numbers Treat i like a variableSimplify
ii 4523
ii 4523
i22
ii 3146
ii 3146
i7
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Ex: )33()21( ii
ii 3231 i52
Ex: )73()32( ii )73()32( ii
i41
Ex: )32()3(2 iii iii 3223
i21
Addition & Subtraction Examples
)7332 ii
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Multiplying Complex Numbers Multiply Pure Imaginary Numbers Steps
Multiply real parts Multiply imaginary parts Use rules of i to simplify imaginary parts
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Monomial Multiplication Example
Multiply Pure Imaginary NumbersSimplify
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Multiplication Example
Multiply Complex NumbersSimplify ji 5731
)57(3)57(1 iii 2152157 iii 2152157 iii
)1(152157 ii152157 ii
i1622
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Solving ax2+b=0
Equation With Imaginary SolutionsSolve
Note: ± is placed in the answer because both 4 and -4 squared equal 16
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Multiply the numerator and denominator by the complex conjugate of the complex number in the denominator.
7 + 2i3 – 5i The complex conjugate
of 3 – 5i is 3 + 5i.
Multiplying Complex Numbers
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Dividing Complex Numbers
Divide Complex Numbers No imaginary numbers in the
denominator! i is a radical
Remember to use conjugates if the denominator is a binomial
Simplify
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i
i
i
iEx
21
21*
21
113 :
)21)(21(
)21)(113(
ii
ii
2
2
4221
221163
iii
iii
)1(41
)1(2253
i
41
2253
i
5
525 i
5
5
5
25 i
i 5
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Division Example
Simplify
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7 + 2i3 – 5i
21 + 35i + 6i + 10i2
9 + 15i – 15i – 25i221 + 41i – 10
9 + 25
(3 + 5i)(3 + 5i)
11 + 41i 34
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Try These.
1. (3 + 5i) – (11 – 9i)
2. (5 – 6i)(2 + 7i)
3. 2 – 3i 5 + 8i
4. (19 – i) + (4 + 15i)
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Try These.
1. (3 + 5i) – (11 – 9i) -8 + 14i
2. (5 – 6i)(2 + 7i) 52 + 23i
3. 2 – 3i –14 – 31i 5 + 8i 89
4. (19 – i) + (4 + 15i) 23 + 14i