Complexity

 

Just My Imagination (Running Away with Me)

But it was just my imagination
Runnin' away with me
It was just my imagination
Runnin' away with me

Reality is just one component of a complex number that requires an imagination.

A complex number, x, has a real component and an imaginary component, a+bi, where a is the coeffcint of the real component and b is the coeffcient of the imaginary component, i.  This can also be expressed as re^iθ, where r is the real component and θ is the angle to the imaginary axis.  It is thus proposed that x, the relationship to the absolute, is complex, and has both a real and an imaginary component.

For example, the Probability Density Function, PDF, of an exponential distribution, e.g. the formula for radioactive decay, is λe^-λx, where λ is the decay rate and x >0.  This has a Cumulative Distribution Function, CDF, which is the integral of the PDF, of 1-e^-λx.  This can be translated along the x-axis by an amount µ as PDF = λe^-λ(x-µ) which makes the CDF =1-e^-λ(x-µ).  The median is when the CDF=50%.  Thus 0.5=1-e^-λ(x-µ) requires that the median occurs when x-µ=-ln(0.5)/λ.  If µ=0, which means that x is a relationship to absolute zero, the median is -ln(0.5)/λ. But for an exponential distribution with a µ of zero, the median has previously been defined as 1/λ.  To resolve this apparent paradox, where the median must be both -ln(0.5)/λ and 1/λ, it is proposed that the median is a complex number and that 1/λ is only the real component of that complex number, -ln(0.5)/λ.  This means that -ln(0.5)/λ =(1/λ)e^iθ,  and that the angle θ, in radians, of the imaginary axis, is thus ln(-ln(0.5))=-0.36651i. Also, according to Euler’s Formula, iθ=ln(cos(θ)+isin(θ)), which means that (1/λ)e^-0.36651i= a*cos(θ)+i*b*sin(θ). This means that a decay rate, λ, defines the complex number, the angle of the imaginary axis, and the translation along the x-axis of the complex number x.