Imagination Matters

 

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 µ_c=µ_r+0*i and σ_c²=s_r²*π²/3+0²*i. While on all surfaces it is perfectly acceptable to treat µ_c as equal to µ_r 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%.