COMPLEX NUMBERS Unit 4Radicals. Complex/Imaginary Numbers WHAT IS? WHY? There is no real number...
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Transcript of COMPLEX NUMBERS Unit 4Radicals. Complex/Imaginary Numbers WHAT IS? WHY? There is no real number...
COMPLEX NUMBERS
Unit 4Radicals
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.
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
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.
Adding & Subtracting Complex Numbers
)34()65( ii )42()35( ii
i31 i77
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
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(
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
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