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j^2 = ?
j^2 = -1
form of complex numbers (and which is real and imaginary component)
x + yj
where x & y are real numbers
x is the real component
y is the imaginary component
when is a complex number real, when is it purely imaginary
when y = 0 the number is real.
when x = 0 the number is purely imaginary
what does the conjugate of z look like (z bar)
_
z = x - yj
1/z = 1/(x+yj) = ?
1/(x+yj) = (x-yj)/(x^2 + y^2)
good shorthand so you don’t have to do the math every time
z(bar)(bar) = ?
= z
when does z(bar) = z
z(bar) = z if and only if z is a REAL number
when does z(bar) = -z
z(bar) = -z if and only if z is purely imaginary
(z+w)(bar) = ?
(bar is over whole thing)
(z+w)(bar) = z(bar) + w(bar)
(zw)(bar) = ?
(zw)(bar) = z(bar) * w(bar)
(z^k)(bar) = ?
(z^k)(bar) = z(bar)^k
for k is a REAL number, k>= 0, (k =/= 0 if z=0)
z + z(bar) = ?
z + z(bar) = 2x = 2Re(z)
Re(z) = real component of z
z - z(bar) = ?
z - z(bar) = 2yj = 2Im(z)j
Im(z) = imaginary component of z
z * z(bar) = ?
z * z(bar) = x^2 + y^2
|z| = ?
|z| = sqrt(x^2 + y^2)
its just like pythagorean theorem
|z| = 0 if and only if (?)
|z| = 0 if and only if z = 0
|z(bar)| = ?
|z(bar)| = |z|
z * z(bar) = ?
z * z(bar) = |z|^2
|zw| = ?
|z| * |w|
What is the Triangle Inequality?
|z + w| <= |z| + |w|
z = x+ yj in polar form
z = r(cosA + jsinA)
where r = |z| = sqrt(x^2 + y^2) & theta is the angle between z & y=0
the value of the angle in polar form is called what
its called an Argument of z
how do you multiply complex numbers in polar form
(works in reverse for division)
you multiply the moduli(|z|, or r) and add the arguments(theta)
if z = r(cosA + jsinA)
and w = u(cosB = jsinB)
z*w = ru(cos(B+A) + jsin(B+A))
z^n = ?
z^n = r^n(cos(nA) + jsin(nA))
if n is an integer
or you can just do the foil in regular form and you get
z^2 = x^2 - y^2 + 2xyj