5.4 Complex Numbers Until now, you have always been told that you can’t take the square root of a...
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![Page 1: 5.4 Complex Numbers Until now, you have always been told that you can’t take the square root of a negative number. If you use imaginary units, you can!](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e405503460f94b3101b/html5/thumbnails/1.jpg)
5.4 Complex Numbers5.4 Complex Numbers
Until now, you have always been told that you can’t take the square root of a negative number.
If you use imaginary units, you can!
OBJ: To solve quadratic equations with complex solutions & perform operations with complex numbers
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Imaginary UnitImaginary Unit
• The imaginary unit is ¡
• ¡= • It is used to write the square root of
a negative number
1
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Property of the square root of Property of the square root of negative #snegative #s
If r is a positive real number, then:
r riEx 1:
3 4
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then,1- If i12 iii 3
14 iii 5
16 iii 7
18 i
*For larger exponents, divide the exponent by 4, then use the remainder
as your exponent instead.
Ex 2: ?23 i3 ofremainder a with 5
4
23
.etc
ii - which use So, 3
ii 23
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Ex 3:Ex 3:2)3(i
22 )3(i)3*3(1
)3(13
26103 Solve 2 x
363 2 x122 x
122 x
12ix 32ix
a. b.
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Complex NumbersComplex NumbersA complex number has a real part & an
imaginary part.
bia
Real part Imaginary part
Like: 5+4iLike: 5+4i
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Adding and SubtractingAdding and Subtracting(add or subtract the ”matching” parts)(add or subtract the ”matching” parts)
Ex 4: )33()21( ii )32()31( ii
i52
Ex 5: )73()32( ii )73()32( ii
i41
Ex: 6 )32()3(2 iii )32()23( iii
i21
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HomeworkHomework
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MultiplyingMultiplying((Treat the i’s like variables, then simplify any Treat the i’s like variables, then simplify any that are not to the first power)that are not to the first power)
Ex 7: )3( ii 23 ii
)1(3 i
i31
Ex 8: )26)(32( ii 2618412 iii
)1(62212 i62212 i
i226
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i
i
i
i
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
Ex 9:
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DividingDividing(Multiply the numerator and the
denominator by the “conjugate”)
Ex 10:
i
i
21
35
i
i
21
21
2
2
4221
63105
iii
iii
5
131 i
ior5
13
5
1
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Absolute Value of a Complex #Absolute Value of a Complex #
If you have a complex number the absolute value can be found using:
) ( bia
22 ba
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Ex 11Ex 11::a. i52
22 )5()2(
254 29
b. i622 )6()0(
360
366