complex numbers unit 4radicals. complex/imaginary numbers what is? why? there is no real number...
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![Page 1: COMPLEX NUMBERS Unit 4Radicals. Complex/Imaginary Numbers WHAT IS? WHY? There is no real number whose square is -25 so we have to use an imaginary number](https://reader036.vdocument.in/reader036/viewer/2022081420/5519d00355034649768b487c/html5/thumbnails/1.jpg)
COMPLEX NUMBERS
Unit 4Radicals
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Complex/Imaginary Numbers
WHAT IS?
25 25
WHY?
5
2552
There is no real number whose square is -25 so we have to use an imaginary number
125
15 i5
i 1“i” is an imaginary number. “i” is equal to the square root of -1
BASICALLY: any time you see a negative under a SQUARE ROOT an “i” gets pulled out.
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Simplifying Radicals with Imaginary Numbers
ALWAYS pull the “i” out first before multiplying together.
503281 yx
36
25i yxyi9
182i
23 2i
23
2251 32811 yx
3161
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Adding & Subtracting Complex Numbers
A complex number is a number with “i” in it. Complex numbers can be written in the form :
bia Real part
Imaginary part
To add or subtract complex numbers combine the real parts and combine the imaginary parts separately.
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Adding & Subtracting Complex Numbers
)34()65( ii )42()35( ii
i31 i77
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Multiplying Complex Numbers
You multiply complex numbers like you would binomials. (Double Distribute, Box, FOIL…etc)
)53)(32( ii )52)(6( ii
23012 ii )1(3012 i
i1230
i106 2159 ii )1(1516 i
i21
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Dividing Complex Numbers
Remember that we don’t want to leave a radical in the denominator.
To simplify a quotient, multiply by the conjugate of the denominator.
Conjugate – change only the middle sign
)43( i)42( i
)53(
CONJUGATE =
CONJUGATE =
CONJUGATE =
)43( i)42( i
)53(
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Rationalize the Denominator
i
i
41
32
Simplify
i
i
41
412
2
161
12382
i
iii
17
1110 i
i
i
5
27
i
i
5
52
2
25
1035
i
ii
25
3510
i
5
72
i
5
72 iImaginary # song
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i
1i
2i
3i
0i
1
i
1
12i
22i
33i
Since “i” raised to a power follows a pattern you can easily find the answer by dividing the exponent by 4 and using the remainder to simplify.
What about higher exponents? 4-7?
4 goes into 12, 3 times with a remainder of zero.
10 i
4 goes into 22, 5 times with a remainder of 2
12 i
4 goes into 33, 8 times with a remainder of 1
11 i