Complex Numbers Flashcards
What is the value of i?
i = √(-1)
What is z (usually)?
A complex number
What form does complex number take?
z = a + ib
a,b ∈ R
a = Re(z) b = Im(z)
What is the complex conjugate (z ̅) of z=a+ib?
z ̅ = a-ib
Re(z ̅) = Re(z)
Im(z ̅) = -Im(z)
What is the modulus |z| of z = a + ib?
|z| = √((a^2)+(b^2))
Think of it like the Pythagorean theorem with |z| being the hypotenuse and a,b being the other sides.
Scalar Multiplication:
For z = a+ib, what does 4z equal?
4z = 4a+4ib
For z = a + bi, what is the result of z * |z|?
z * |z| = a^2 + b^2
Complex Multiplication:
What is the result of z = u * v where u and v are complex numbers?
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
Complex Division:
What is the value of z = u/v where u and v are both complex numbers and v != 0?
z = u * (v ̅ / |v|^2)
How is a complex number represented as a matrix?
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
With a complex number in matrix form, how do you get the complex conjugate?
Perform a transposition of the matrix
With a complex number in matrix form, how do you get the modulus of the complex number?
Find the determinant of the matrix
How is a complex number represented as a 2-vector on an argand diagram?
z = a+bi
z = (a,b) (but vertical since its a vector)
With a complex number in vector form, how do you find the complex conjugate?
The reflection in the x-axis (the Re axis)
With a complex number in vector form, how do you find the modulus of the complex number?
The magnitude (size) of the vector
What flaws does vector form for complex numbers have?
Cannot do multiplication and division.
How is a complex number represented in Polar Form?
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
How is a complex number represented in Euler form?
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
What relation does the Euler form give us?
(cosθ + isinθ)^α = cos(αθ) + isin(αθ)
What downside does Euler’s form have which requires us to use principal values for arg z?
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
Complex Powers:
What is the result of u^v where u∈C and v∈Q+ (v is positive rational)?
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.
Can complex numbers be ordered?
No
For complex number v = a + ib, what is the value of w = 1/v?
w = (v ̅ / |v|^2)