sossego said:
Does 4 √i have a value of (+, -, ±, unsigned) or not?"
As you probably know, any complex number
z (except zero) has exactly
n (all distinct) complex
n-th roots. In the complex plane these lie on a circle centred at the origin with radius
n√|
z| (i.e. the
n-th root of the absolute value of
z, which is obviously a positive real number), spaced at angles that are 2π/
n apart, with the argument (angle in polar representation) of the principal root being the argument of
z itself divided by
n. So the four quartic roots of
i are e^
iφ = cos(φ) +
i sin(φ) where φ is in the set {π/8, 5π/8, 9π/8,13π/8}. Note that none of those quartic roots are either real or purely imaginary, they really are properly complex (i.e. both real and imaginary parts are non-zero).
However, this only applies when
n is a finite natural number (and for convenience let's disregard 0 and 1). Once you start throwing infinity into the mix things get screwed up royally. The modulus of the roots converges to 1 (except when
z equals zero) but the arguments of the roots diverge. They could essentially be anything (i.e. the entire circle), but raising such a root to infinite power means multiplying the argument by ∞, which means the argument becomes ∞, which is indeterminate modulo 2π :q When
z is real and positive, the
principal root will always be likewise (the argument remains zero after all), but beyond that it escalates.
sossego said:
Hmmm, I was trying to find a formula to describe a qubit.
That sounds interesting, so feel free to elaborate. You might want to have a look at
Bloch spheres if you hadn't already.
P.S. I can't blame Freddie (
@phoenix) for thinking of certain squishy lady parts when you wrote (ω) :e
