complex numbers 10.7 1.write imaginary numbers using i. 2.perform arithmetic operations with complex...
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Complex Numbers10.710.7
1. Write imaginary numbers using i.2. Perform arithmetic operations with complex numbers.3. Raise i to powers.
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Imaginary unit:
Imaginary number: A number that can be expressed in the form bi, where b is a real number and i is the imaginary unit.
i3 i9 i52
1i12 i
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16
21
32
1 16 4i 4i
1 21 21i
1 32 16 2i 4 2i
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i
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Complex number: A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.
i34 i57 i54
32
Examples:
realim
aginary
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Complex Numbers: a + bComplex Numbers: a + bii
b = 0: Real numbers
a = 0: Imaginary numbers
real imaginary
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Add Complex NumbersAdd Complex Numbers
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Add the real parts – add the imaginary parts
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Subtract Complex NumbersSubtract Complex Numbers
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i9
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Slide 10- 9Copyright © 2011 Pearson Education, Inc.
Simplify. (4 + 7i) – (2 + i)
a) 2 + 7i2
b) 2 + 8i
c) 6 + 6i
d) 6 + 8i
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Slide 10- 10Copyright © 2011 Pearson Education, Inc.
Simplify. (4 + 7i) – (2 + i)
a) 2 + 7i2
b) 2 + 8i
c) 6 + 6i
d) 6 + 8i
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Multiply Complex NumbersMultiply Complex Numbers
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228i
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Multiply Complex NumbersMultiply Complex Numbers
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i
25635 ii
15635 i
5635 i
i3556
standard a + bi form
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Multiply Complex NumbersMultiply Complex Numbers
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12320 i
2320 i
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Multiply Complex NumbersMultiply Complex Numbers
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193025 i
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Rewrite & Foil
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Slide 10- 15Copyright © 2011 Pearson Education, Inc.
Multiply. (4 + 7i)(2 + i)
a) 15 + 10i
b) 1 + 10i
c) 15 + 18i
d) 15 + 18i
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Slide 10- 16Copyright © 2011 Pearson Education, Inc.
Multiply. (4 + 7i)(2 + i)
a) 15 + 10i
b) 1 + 10i
c) 15 + 18i
d) 15 + 18i
![Page 17: Complex Numbers 10.7 1.Write imaginary numbers using i. 2.Perform arithmetic operations with complex numbers. 3.Raise i to powers](https://reader035.vdocuments.us/reader035/viewer/2022062519/5697c01e1a28abf838cd0f8c/html5/thumbnails/17.jpg)
Divide Complex NumbersDivide Complex Numbers
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76
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Binomial denominator conjugate
Divide Complex NumbersDivide Complex Numbers
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3730 standard a + bi form
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Slide 10- 19Copyright © 2011 Pearson Education, Inc.
Write in standard form.
a)
b)
c)
d)
4
2 3
i
i
5 14
13 13
i
5 14
13 13
i
11 14
13 13
i
11 14
13 13
i
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Slide 10- 20Copyright © 2011 Pearson Education, Inc.
Write in standard form.
a)
b)
c)
d)
4
2 3
i
i
5 14
13 13
i
5 14
13 13
i
11 14
13 13
i
11 14
13 13
i
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Powers of i:Powers of i:1
12
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iiiii 123
12 i
111
1
111
1
448
347
246
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iii
iiiii
iii
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41i ii 104 ii 1
15i 334 ii ii 1
i
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iiiii 123
12 i
Powers of i:Powers of i:
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432 iiii
Simplify:
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