5.4 Complex Numbers (p. 272). Imaginary Unit Until now, you have always been told that you cant take...
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Transcript of 5.4 Complex Numbers (p. 272). Imaginary Unit Until now, you have always been told that you cant take...
5.45.4Complex Complex NumbersNumbers
(p. 272)(p. 272)
Imaginary UnitImaginary Unit• 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!
• The imaginary unit is ¡.• ¡= • It is used to write the square root of
a negative number.
1
Property of the square root Property of the square root of negative numbersof negative numbers
• If r is a positive real number, then
r ri
Examples:
3 3i 4 4i i2
then,1- If i
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
4
23
.etcii - which use So, 3
ii 23
ExamplesExamples2)3( 1. i
22 )3(i)3*3(1
)3(13
26103 Solve 2. 2 x
363 2 x122 x
122 x
12ix 32ix
Complex NumbersComplex Numbers• A complex number has a real part &
an imaginary part.• Standard form is:
bia
Real part Imaginary part
Example: 5+4iExample: 5+4i
The Complex planeThe Complex plane
Imaginary Axis
Real Axis
Graphing in the complex planeGraphing in the complex plane
i34 .
i52 .i22 .
i34
.
Adding and SubtractingAdding and Subtracting(add or subtract the real parts, then (add or subtract the real parts, then add or subtract the imaginary parts)add or subtract the imaginary parts)
Ex: )33()21( ii )32()31( ii
i52
Ex: )73()32( ii )73()32( ii
i41
Ex: )32()3(2 iii )32()23( iii
i21
MultiplyingMultiplyingTreat the i’s like variables, then Treat the i’s like variables, then change any that are not to the change any that are not to the
first powerfirst power
Ex: )3( ii 23 ii
)1(3 i
i31
Ex: )26)(32( ii 2618412 iii
)1(62212 i62212 i
i226
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
Absolute Value of a Absolute Value of a Complex NumberComplex 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
ExamplesExamples1. i52
22 )5()2(
254 29
2. i622 )6()0(
360
366
Which of these 2 complex numbers is closest to the origin? -2+5i
Assignment