Complex Numbers Yr 2

Question 1

What is De Moivre’s Theorem?

Answer 1

Question 2

Prove De Moivre’s Theorem by induction

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Question 3

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Question 4

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Question 5

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Question 6

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Question 7

What is the equation to the roots of z^n =1

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Question 8

How are the nth roots of unity shown on a complex number diagram?

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Question 9

What is the sum of the roots of unity?

Answer 9

Always = 0

Question 10

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Question 11

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Question 12

What is the equation of roots of any complex number?

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Question 13

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Question 14

Derive the equation for the nth roots of unity

Answer 14

Question 15

Derive the formula of equation for complex roots of any complex number

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Question 16

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Question 17

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Question 18

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Question 19

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Question 20

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Question 21

What is the form of e^i°?

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Question 22

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Question 23

What is Euler’s identity?

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Question 24

What are the two important identities for sin and cos ?

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Question 25

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Question 26

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Question 27

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Question 28

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Question 29

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Question 30

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Question 31

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Question 32

What are the nth roots of unity?

Answer 32

Complex numbers that when raised to the nth power, equal 1. Each root represents a point on the complex plane on the unit circle, creating a regular polygon with n vertices

Question 33

Derive the nth roots of unity formula

Answer 33

Question 34

General nth roots of unity formula

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Question 35

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Question 36

Derive the nth roots for any complex number formula

Answer 36

Question 37

What is the general formula for the nth roots of any complex number ?

Answer 37

Question 38

What is the nth roots for any complex number?

Answer 38

Similar to the nth roots of unity but instead of lying on a unit circle, they lie on a circle at with radius r^1/n and starting at angle theta/n, rather than at 1,0

Question 39

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Question 40

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Question 41

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Question 42

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Question 43

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Question 44

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Question 45

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Question 46

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Question 47

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Question 48

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Question 49

Complete the identities

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Question 50

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Question 51

Using the standard results fill out the following:

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Question 52

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Question 53

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Question 54

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Question 55

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Question 56

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Question 57

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Question 58

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Question 59

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Question 60

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Don’t forget the pi and the i !

Question 61

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Question 62

Part a

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Question 63

Part b

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Question 64

Exam practice

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Question 65

Part a

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Question 66

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Question 67

Part A

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Question 68

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