Week 4 Complex Numbers Flashcards
what are natural numbers
counting numbers so positive integers and zero
what are rational numbers
numbers that can be represented by a fraction
what are real numbers
all rational and irrational numbers
what are complex numbers
they are imaginary numbers that involve i
what is the value of i
sqrt(-1)
what is the general complex number equation and name the two parts
z = x + iy
x is the real part
iy is the imaginary part
what are the three possible solutions for a quadratic equation and name the conditions for each
b^2-4ac > 0 => 2 real roots
b^2-4ac = 0 => 1 repeated root
b^2-4ac < 0 => 2 complex roots
what happens when a quadratic has 2 complex solutions
the two solutions are a complex conjugate pair
what is the complex conjugate pair equation for a general complex number
z* = x - iy
how do we obtain the square of the modulus of a complex number
multiplying the complex number by its complex conjugate
I z I^2 = z*z
what can we say if two complex numbers are equal
if z1 = z2
then
x1 = x2
y1 = y2
what is the general equation for adding or subtracting complex numbers z1 ± z2
z1 ± z2 = (x1 ± x2) + i(y1 ± y2)
what can be said if a complex number is equal to 0
x = y = 0
what can I z I be thought of as
it can be considered to be the length of a complex number since it is always a real number
what is the result of the product of two complex numbers
z1z2
z1z2 = (x1x2 - y1y2) + i(x1y2 + x2y1)
what is the procedure for dividing a complex number
multiply both the numerator and denominator by the complex conjugate
describe the argand diagram
cartesian diagram where the two axes are Re(z) and Im(z)
what are the two components for the polar argand diagram
the two components are theta and I z I
what is taking the complex conjugate equivalent to in polar terms
it is equivalent to changing the sign of theta
what is the exponential form of a complex number
I z Ie^iθ
what is Euler’s formula
e^iθ = cosθ + isinθ
what does De Moivres Theorem (DMT) allow us to do
gives us a way of finding powers and roots of complex numbers that has a trigonometric association
what is the DMT formula
(cosθ + isinθ)^n = cos(nθ) + isin(nθ)
why are complex numbers important in physics
they are useful for expressing oscillating functions
