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Pre-Calculus Lesson 4: The Wondrous World of Imaginary and Complex NumbersDefinitions, operations, and calculator shortcuts
![Page 2: Pre-Calculus Lesson 4: The Wondrous World of Imaginary and Complex Numbers Definitions, operations, and calculator shortcuts](https://reader035.vdocument.in/reader035/viewer/2022062314/56649ea25503460f94ba5cf3/html5/thumbnails/2.jpg)
Definition
The lowercase letter “i” is equal to the square root of negative one.
1i
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Examples
9 64216x
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Examples
9 64216x
i3
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Examples
9 64216x
i3 i8
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Examples
9 64216x
i3 i8 xi4
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Powers of i
i2 = -1
i3 = -i
i4 = 1
http://supermanjaviolivares.iespana.es/poster1.jpg
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Powers of i
i2 = -1
i3 = -i
i4 = 1
Higher powers of i?
http://supermanjaviolivares.iespana.es/poster1.jpg
![Page 9: Pre-Calculus Lesson 4: The Wondrous World of Imaginary and Complex Numbers Definitions, operations, and calculator shortcuts](https://reader035.vdocument.in/reader035/viewer/2022062314/56649ea25503460f94ba5cf3/html5/thumbnails/9.jpg)
Let’s exploit 74i
14 i
![Page 10: Pre-Calculus Lesson 4: The Wondrous World of Imaginary and Complex Numbers Definitions, operations, and calculator shortcuts](https://reader035.vdocument.in/reader035/viewer/2022062314/56649ea25503460f94ba5cf3/html5/thumbnails/10.jpg)
Let’s exploit 74i 2184 ii
14 i
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Let’s exploit 74i 2184 ii
2181 i
14 i
![Page 12: Pre-Calculus Lesson 4: The Wondrous World of Imaginary and Complex Numbers Definitions, operations, and calculator shortcuts](https://reader035.vdocument.in/reader035/viewer/2022062314/56649ea25503460f94ba5cf3/html5/thumbnails/12.jpg)
Let’s exploit 74i 2184 ii
2181 i21 i
14 i
![Page 13: Pre-Calculus Lesson 4: The Wondrous World of Imaginary and Complex Numbers Definitions, operations, and calculator shortcuts](https://reader035.vdocument.in/reader035/viewer/2022062314/56649ea25503460f94ba5cf3/html5/thumbnails/13.jpg)
Let’s exploit 74i 2184 ii
2181 i21 i
2i
14 i
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Let’s exploit 74i 2184 ii
2181 i21 i
2i 1
14 i
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Complex NumbersWhat is a complex number?
A number with a real and an imaginary component
Standard form:a + bi
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Adding and Subtractingii 23 ii 97
ii 3265 ii 2752
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Adding and Subtractingii 23 ii 97
ii 3265 ii 2752
i5
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Adding and Subtractingii 23 ii 97
ii 3265 ii 2752
i5 i2
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Adding and Subtractingii 23 ii 97
ii 3265 ii 2752
i5 i2
i93
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Adding and Subtractingii 23 ii 97
ii 3265 ii 2752
i5 i2
i93 i39
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FOIL-ing Complex Numbers Remember: i2=-1 Example:
ii 3265
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FOIL-ing Complex Numbers Remember: i2=-1 Example:
ii 3265 218121510 iii
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FOIL-ing Complex Numbers Remember: i2=-1 Example:
ii 3265 218121510 iii
218310 ii
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FOIL-ing Complex Numbers Remember: i2=-1 Example:
ii 3265 218121510 iii
218310 ii 18310 i
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FOIL-ing Complex Numbers Remember: i2=-1 Example:
ii 3265 218121510 iii
218310 ii 18310 ii328
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Conjugates of Complex Numbers
The conjugate of a+bi is a-bi.
The conjugate of a-bi is a+bi.
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Dividing Complex NumbersWe cannot leave i in the
denominator of a fraction. (Why not?)
Multiply the numerator and denominator of the fraction by the conjugate of the denominator.
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Sample Problem
i
i
32
65
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Sample Problem i
i
32
65
ii
32
32
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Sample Problem i
i
32
65
ii
32
32
2
2
9664
18121510
iii
iii
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Sample Problem i
i
32
65
ii
32
32
2
2
9664
18121510
iii
iii
94
182710
i
![Page 32: Pre-Calculus Lesson 4: The Wondrous World of Imaginary and Complex Numbers Definitions, operations, and calculator shortcuts](https://reader035.vdocument.in/reader035/viewer/2022062314/56649ea25503460f94ba5cf3/html5/thumbnails/32.jpg)
Sample Problem i
i
32
65
ii
32
32
2
2
9664
18121510
iii
iii
94
182710
i
13
278 i
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Sample Problem i
i
32
65
ii
32
32
2
2
9664
18121510
iii
iii
94
182710
i
13
278 i i
13
27
13
8
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Graphing Calculator
Press MODE.Select a+bi mode.
The i key is the decimal key.
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Sample Problem
i
i
3
54
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Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator.
i
i
3
54
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Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
i
i
3
54
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Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
Press ENTER.
i
i
3
54
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Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
Press ENTER. 1.7-1.1i
i
i
3
54
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Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
Press ENTER. 1.7-1.1i
Prefer fractions? Press MathFrac.
i
i
3
54
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Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
Press ENTER. 1.7-1.1i
Prefer fractions? Press MathFrac. 17/10-11/10i
i
i
3
54
![Page 42: Pre-Calculus Lesson 4: The Wondrous World of Imaginary and Complex Numbers Definitions, operations, and calculator shortcuts](https://reader035.vdocument.in/reader035/viewer/2022062314/56649ea25503460f94ba5cf3/html5/thumbnails/42.jpg)
Sample Problem
Type into the calculator with parenthesis around the numerator and the denominator. (4-5i)/(3-i)
Press ENTER. 1.7-1.1i
Prefer fractions? Press MathFrac. 17/10-11/10i
i
i
3
54
IMPORTANT NOTE:
You must WRITE the answer as
i
10
11
10
17
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Practice Questions1. (3 + 2i) + (4 – 5i)2. (5 – 6i) – (3 – 2i)3. (2 + 4i)(3 – 5i)4. (4 + i)2 5. 4 + 3i
2 – 3i