Complex Numbers

Question 1

Definition of a complex number

Answer 1

z = a + ib

Question 2

Definition of i

Answer 2

i^2 = -1

Question 3

Conjugate of z = a + ib

Answer 3

z* = a - ib

Question 4

If z = a + ib, what does zz* equal?

Answer 4

zz* = a^2 + b^2

Question 5

What does zz* also equal to?

Answer 5

|z|^2

Question 6

What does 1/z equal?

Answer 6

1/z = z*/(|z|^2)

Question 7

Representing a complex number in plane polar, what can r and θ be considered as?

Answer 7

r is the magnitude

θ is the argument

Question 8

Represent z = x + iy in plane polar form

Answer 8

z = r(cos(θ) + isin(θ))

Question 9

Represent 1/z in plane polar form

Answer 9

1/z = 1/r (cos(θ) - isin(θ))

Question 10

When two complex numbers are multiplied, what is the basic rule to work out the result?

Answer 10

Multiply moduli

Add angles

Question 11

De Moivre’s theorem

Answer 11

(cos(θ) + isin(θ))^n = cos(nθ) + isin(nθ)

Question 12

Euler’s formula

Answer 12

e^(iθ) = cos(θ) + isin(θ)

Question 13

Polar form of complex numbers using Euler’s formula

Answer 13

z = r(cos(θ) + isin(θ)) = re^(iθ)

Question 14

How can ln(z) be solved?

Answer 14

ln(z) = ln(re^(iθ)) = ln(r) + i(θ + 2nπ)

Question 15

What is z(1)^z(2) equal to?

Answer 15

z(1)^z(2) = e^(z(2)ln(z(1)))

Question 16

What is cos(iy) equal to?

Answer 16

cos(iy) = 1/2 (e^(-y) + e^(y)) = cosh(y)

Question 17

What is sin(iy) equal to?

Answer 17

1/2i (e^(-y) - e^(y)) = isinh(y)