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