Complex Numbers

Question 1

What is the value of i?

Answer 1

i = √(-1)

Question 2

What is z (usually)?

Answer 2

A complex number

Question 3

What form does complex number take?

Answer 3

z = a + ib
a,b ∈ R

a = Re(z)
b = Im(z)

Question 4

What is the complex conjugate (z ̅) of z=a+ib?

Answer 4

z ̅ = a-ib

Re(z ̅) = Re(z)
Im(z ̅) = -Im(z)

Question 5

What is the modulus |z| of z = a + ib?

Answer 5

|z| = √((a^2)+(b^2))

Think of it like the Pythagorean theorem with |z| being the hypotenuse and a,b being the other sides.

Question 6

Scalar Multiplication:

For z = a+ib, what does 4z equal?

Answer 6

4z = 4a+4ib

Question 7

For z = a + bi, what is the result of z * |z|?

Answer 7

z * |z| = a^2 + b^2

Question 8

Complex Multiplication:

What is the result of z = u * v where u and v are complex numbers?

Answer 8

Re(z) = (Re(u) * Re(v)) - (Im(u) * Im(v))
Im(z) = (Re(u) * Im(v)) + (Im(u) * Re(v))

Try grid method multiplication to see how this works subbing in the value of (i^2) = -1

Question 9

Complex Division:

What is the value of z = u/v where u and v are both complex numbers and v != 0?

Answer 9

z = u * (v ̅ / |v|^2)

Question 10

How is a complex number represented as a matrix?

Answer 10

Matrix z = | a -b |
| b a |

I dunno how well that’ll show up when using the flashcards so maybe look at an image of the notes too

Question 11

With a complex number in matrix form, how do you get the complex conjugate?

Answer 11

Perform a transposition of the matrix

Question 12

With a complex number in matrix form, how do you get the modulus of the complex number?

Answer 12

Find the determinant of the matrix

Question 13

How is a complex number represented as a 2-vector on an argand diagram?

Answer 13

z = a+bi

z = (a,b) (but vertical since its a vector)

Question 14

With a complex number in vector form, how do you find the complex conjugate?

Answer 14

The reflection in the x-axis (the Re axis)

Question 15

With a complex number in vector form, how do you find the modulus of the complex number?

Answer 15

The magnitude (size) of the vector

Question 16

What flaws does vector form for complex numbers have?

Answer 16

Cannot do multiplication and division.

Question 17

How is a complex number represented in Polar Form?

Answer 17

z = (r,θ)

where:
r = |z|
θ = arg z = cos^-1(Re(z)/|z|) = sin^-1(Im(z)/|z|)
The angle is in radians and can be found from trigonometric calculations when viewing the complex number as a vector on an argand diagram

Question 18

How is a complex number represented in Euler form?

Answer 18

z = r * (e^iθ) = r(cosθ + isinθ)

where:
r = |z|
θ = arg z = cos^-1(Re(z)/|z|) = sin^-1(Im(z)/|z|)
The angle is in radians and can be found from trigonometric calculations when viewing the complex number as a vector on an argand diagram

Question 19

What relation does the Euler form give us?

Answer 19

(cosθ + isinθ)^α = cos(αθ) + isin(αθ)

Question 20

What downside does Euler’s form have which requires us to use principal values for arg z?

Answer 20

Because cos and sin are cyclical in nature, Each complex number has an infinite number of representations as arg z can take infinitely many values to represent the same thing.
To combat this we use the principal value where
0 <= arg z <= 2π

Question 21

Complex Powers:

What is the result of u^v where u∈C and v∈Q+ (v is positive rational)?

Answer 21

Representing u as u = |u|(e^(iarg u))) and v as v = 1/k:
u^(1/k) = (|u|^(1/k)) * e^((i*arg u)/k)

See slides/notes for better view of this as text like this doesn’t show superscripts.

Question 22

Can complex numbers be ordered?

Answer 22

No

Question 23

For complex number v = a + ib, what is the value of w = 1/v?

Answer 23

w = (v ̅ / |v|^2)