Download - Imaginary & Complex Numbers
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Imaginary & Complex Numbers
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Once upon a time…
1 no real solution
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-In the set of real numbers, negative numbers do not have square roots.
-Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions.
-These numbers were devised using an imaginary unit named i.
1i
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-The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1.
-The first four powers of i establish an important pattern and should be memorized.
Powers of i1 2 3 41 1i i i i i i
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Divide the exponent by 4No remainder: answer is 1.remainder of 1: answer is i.
remainder of 2: answer is –1.remainder of 3:answer is –i.
i4 1
i
i2 1
i i3
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Powers of i1.) Find i23
2.) Find i2006
3.) Find i37
4.) Find i828
i1i1
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Complex Number SystemReals
Rationals(fractions, decimals)
Integers(…, -1, -2, 0, 1, 2, …)
Whole(0, 1, 2, …)
Natural(1, 2, …)
Irrationals(no fractions)
pi, e
Imaginary
i, 2i, -3-7i, etc.
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1.) 5 1*5 1 5 5i
1*7 1 7 7i
1*99 1 99
3 11i
Simplify.3.)
2.) 7 4.)
3.) 995.)
i 3 3 11
-Express these numbers in terms of i.
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You try…6.7.
7
36
1608.
i 7
6i4 10i
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To multiply imaginary numbers or an imaginary number by a real number, it is important first to express the imaginary numbers in terms of i.
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94i
22 5i 2 52 21i
( 1) 21 21
Multiplying47 2i
2 5i
3 7
2 1 5i 2 5i i
i i3 7
9.
10.
11.
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a + biComplex Numbers
real imaginary
The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.
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7.) 7 9i i 16i
8.) ( 5 6 ) (2 11 )i i 3 5i
9.) (2 3 ) (4 2 )i i 2 3 4 2i i 2 i
Add or Subtract12.
13.
14.
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Multiplying & Dividing Complex Numbers
Part of 7.9 in your book
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REMEMBER: i² = -1
12 2i 12( 1) 12
2 27 i 49( 1) 49
Multiply3 4i i
27i
1)
2)
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You try…3)
4)
7 12i i
211i
284i )1(84 84
2211 i )1(121 121
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28 8i 21i 26i228 29 6i i
28 29 6( 1)i 28 29 6i 22 29i
ii 2734 5)
Multiply
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You try… ii 1032 6)
2103206 iii
i1716
210176 ii 110176 i
10176 i
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25 35i 35i 249i
25 49( 1)
25 4974
You try… ii 7575 7)
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Conjugate-The conjugate of a + bi is a – bi
-The conjugate of a – bi is a + bi
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Find the conjugate of each number…
3 4 i 3 4 i
4 7i 4 7i
5i 5i
6 6
8)
9)
10)
11)
iiBecause 06 as same theis 06
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11ii
2
14 41
ii
14 4
2i
7 2i
Divide…
5 91
ii
12)
5 5 9 91
2
2
i i ii i i
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3 53 5
ii
2
9 199 25
ii
9 19
34i
2 33 5ii
13)You try…
6 10 9 159 15 15 25
2
2
i i ii i i