It Don’t Matter
to Me
It don’t matter to me
If you really feel that
You need sometime to be free
Time to go out searching for yourself
Hoping to find
Time to go to find.
But some things do matter
Imagination matters. If the answer is a complex number,
and the coefficient of the imaginary portion of that complex number is zero, it
does NOT mean that it is no longer a complex number, and the imaginary axis can
be dropped. It only means that the coefficient of the imaginary axis is zero.
A case in point is the formula for the variance of a logistics,
sech squared, distribution. As in most random distributions, it is defined by
two parameters, a location, µ, and a range, a function of the variance,
s. Its Probability Density function, PDF, for a given value of x is
1/(4*s)*sech2((x-µ)/(2*s))
Its Cumulative Distribution Function, CDF, the integral of
the PDF, is a scaled version of the hyperbolic tangent
½ + ½*tanh(x-µ)/(2*s))
In this function the mean/median is also µ and the variance,
σ2, whose square root is the Standard Deviation is
σ2=s2*π2/3.
When the parameters
are real numbers, then they can also be expressed as complex numbers as µc=µr+0*i
and σc2=sr2*π2/3+02*i.
While on all surfaces it is perfectly acceptable to treat µc
as equal to µr which effectively seems to ignore
the imaginary coefficient. However, the variance expressed as a complex number is also the formula for a triangle, and the surface on which the triangle resides matters. If that surface is flat or a very large sphere, then it is true that there
is a single solution and that solution is σ=s*π/√3. But on a hyperbolic
surface, there are two solutions (one for each sheet of a two sheeted hyperboloid formed
by rotating that surface around an imaginary axis), σ=ln(0 ± 2*cosh(s*π/√3)). This is true for any non-zero value of s.
For example for s =1/2, where s could be interpreted as the mean
of the number of absolutes, the Standard
Deviation, σ, square root of the variance, σ2, on a flat
surface, also know as Euclidean surface, is σ=0.9069. But on a hyperbolic surface σ=1.0579. In this case by considering imagination, the Standard Deviation
has increased by 16%.