Complex Numbers

Question 1

What is De Moivre’s Theorem?

Answer 1

zⁿ = rⁿ (cos(nx) + isin(nx))

Question 2

How many roots does the equation zⁿ have?

Answer 2

‘n’ number of roots

Question 3

What is the formula for the roots of unity?

Answer 3

z = cos(2 * pi * k / n) + isin (2 * pi * k / n)

Where k = 0, 1, 2, 3 … (n-1)

Question 4

What do the roots of unity form on the argand diagram?

Answer 4

When roots are plotted on a diagram, they form a regular polygon with ‘n’ sides

Question 5

What is the first root of unity equal to?

Answer 5

1

Question 6

What is the sum of all the roots of unity?

Answer 6

1

Question 7

What are the steps to finding the power of a complex number?

Answer 7

Put the complex number in modulus argument form
Find the n^th root of the modulus
Divide argument by ‘n’
Start with r(cos(x) + isin(x)) and keep adding 2pi/n to the argument

Question 8

How do you express a a trigonometric identity with a multiple angle to an identity with a normal power?

Answer 8

Using De Moivre’s theorem, you can convert expressions such as cos(5x):

cos(5x) + isin(5x) = (cos(x) + isin(x))⁵

Question 9

What trig identity involving cos turns powers to multiple angles?

Answer 9

zⁿ + z^-n = 2cos(x)

Question 10

What trig identity involving sin turns powers to multiple angles?

Answer 10

zⁿ - z^-n = 2isin(x)

Question 11

How do you convert a power of a function to angle multiple?

Answer 11

Use sine or cosine identity
Raise it to the right power
Binomially expand
Factor out same powers
Use cosine identity to convert to multiples of angles

Question 12

What is the exponential equation involving sine and cosine?

Answer 12

e^{i * (x)} = cos(x) + isin(x)

Question 13

What Is the exponential form of a complex number?

Answer 13

zⁿ = rⁿe^{i * n * (x)}

Question 14

What is the equation involving cosine and the exponential function?

Answer 14

2cos(x) = e^{i * (x)} + e^{-i * (x)}

Question 15

What is the equation involving sine and the exponential function?

Answer 15

2isin(x) = e^{i * (x)} - e^{-i * (x)}

Question 16

What is -1 + i in modulus argument and exponential form?

Answer 16

2^1/2(cos(3pi/4) + isin(3pi/4))

2^1/2e^{3pi/4 * i}

Question 17

How do you find the roots of a complex number in exponential form?

Answer 17

Taking the root of a magnitude
Divide the by the power
Find the roots by adding 2kpi/n to the argument

Question 18

What is the formula for the sum of a series of complex numbers?

Answer 18

w(zⁿ - 1) / z - 1

‘w’ is the original complex number
‘z’ is the common ratio
‘n’ is the upper limit + 1

Question 19

What is the formula for the sum of a series of complex numbers to infinity?

Answer 19

w / 1 - z

‘w’ is the original complex number
‘z’ is the common ratio

Question 20

How do you simplify expressions like ‘e^{i * x} - 1’ ?

Answer 20

e^{i * x/2}(e^{i * x/2} - e^{-i * x/2})
e^{i * x/2}(2isin(x/2))