Zero

 

Zorro

Out of the night, when the full moon is bright,
Comes a horseman known as Zorro
This bold renegade
Carves a 'Z' with his blade,
A 'Z' that stands for "Zorro"

What about a 'Z' that stand for “Zero”

Is zero

1) a mid-point between two absolutes, -∞ and ∞;

2) a coordinate transformation, a relative, positional notation, or

3) the absence of an absolute.

If there are negative numbers, then zero can be the mid-point between two absolutes. If there is only one absolute, then there is an absolute zero (which is an absence of that absolute), and negative numbers are not allowed. Relative, positional, zeros exist when there is an absolute zero  (as in Temperature measured in degrees Kelvin) and the zero is merely a coordinate transformation to a different scale (e. g. 273.15˚ Kelvin is 0˚ Celsius). Zero degrees Celsius is merely a convenient reference point, i.e., the freezing point of water. -40˚ Celsius does not mean that there is NEGATIVE temperature, just that with respect to the temperature in Celsius at which water freezes, the temperature is 40 degrees below that temperature.

Thus there are really only two kinds of zero:

·        mid-point zero, Case (1)  and

·        absolute zero, Case (3).

If negative numbers are allowed, then the zero is a mid-point and there are two absolutes. If there is only one absolute, then there are no negative numbers. Case (2), a postitional zero, is thus merely a subset of Case (3), where negative numbers exist with respect to a reference point, but there is no -∞.

Vectors have both an amplitude, radius, and an angle. The radius expressed as a scalar can NEVER be negative. If you turn a vector by 180˚,  then you still have a radius of 1, not a radius of -1.

Are there mathematical functions which also do not allow negative numbers? Absolutely. An exponential distribution has a Probability Density Function, PDF, of λe^-λx and a Cumulative Distribution Function, CDF, of 1- e^-λx, both with x≥0. Logarithms are undefined for x<0.

Let x be the position in space. The dimension perpendicular to the x-axis might be the value of the PDF and CDF, but for convenience let’s refer to this axis as time. While the PDF and the CDF of an exponential distribution will not ordinarily be zero, in converting to Minkowski space, the time axis is with respect to a fixed point, i.e. now; negative numbers are defined as the past; and positive numbers are defined as the future. For example, a time of 2024 CE only means that -2024 is 2024 years before the Common Era, or 2024 BCE. Alternatively that measurement can begin at the Big Bang. Negative numbers could then be the time before the Big Bang, and positive numbers could be the time since the Big Bang. Reality is the set of choices that have been made. Thus negative numbers could be the set of choices before that reality and positive numbers could be the set of choices that will be made after that reality.

Thus there are three dimensions in Minkowski space:

·        space, which does NOT allow negative numbers;

·        time, which allows negative numbers; and

·        unreality, imaginary/imagination, which allows negative numbers. 

A complex number in three polar dimensions can be expressed as re^iθ .  Transfomed to cylindrical coordinates where the cylindrical volume is formed by rotating the real surface (r, space-time) about the imaginary, i, axis, this is r*cos(θ)+r*sin(θ)*i. Thus Euler’s Formula, e^ix=cos(x)+sin(x)*i, is the special case where r =1 and is on the real surface which is rotated about the imaginary axis. If that surface is flat, Euclidean, then x=θ and r=√((r*cos(x))²+(r*sin(x))²). Since cos²+sin²=1, this becomes r=r. If the surface that is being rotated is spherical, then then as the radius of the spherical surface,  R, approaches infinity, its  limit is also r=√((r*cos(x))²+(r*sin(x))^2. But if the surface being rotated is hyperbolic, then it is     

  r=ln(cosh(r*cos(x))*cosh(r*sin(x))±√((cosh(r*cos(x))*cosh(r*sin(x)))²-1)).

The origin (0,0,0) has two solutions for the coefficient of the imaginary axis. Sin(0) and sin(π) are both 0. But cos(0)=1, while cos(π)=-1. And cosh(0)=1. This mean that a rotation of a hyperbolic surface that passes through the imaginary plane at the origin also has two solutions

r=ln(cosh(r*cos(0))*cosh(r*sin(0)) ± √((cosh(r*cos(0))*cosh(r*sin(0)))²-1)) 

and

r=ln(cosh(r*cos(π))*cosh(r*sin(π)) ± √((cosh(r*cos(π))*cosh(r*sin(π)))²-1)).

Using the hyperbolic identity that cosh²-sinh²=1, and thus sinh=√(cosh²-1), and the values for 0 and π , these can be simplified to r=ln(cosh(r) ± sinh(r)) and r=ln(cosh(-r) ± sinh(-r)). Because cosh is an odd function, that is cosh(x)=cosh(-x), these both can be expressed as r=ln(0 ± 2*cosh(r)). In Minkowski space it is conventional to describe a light cone and an inverted light cone whose peaks intersect at the origin. But this assumes that the surfaces being rotated are flat, Euclidean. If the surface being rotated is hyperbolic, then the light cones become two sheets of a hyperboloid where the two sheets intersect at the origin. There is one solution in each sheet of the two sheeted hyperboloid. The radius appears negative in one sheet and positive in the other sheet, and there is a rotation of π when passing through the origin between those two sheets. But from  the perspective of each sheet, its radius is positive, and radius of the other sheet is negative, and its rotation of the imaginary axis is 0 and the rotation of the imaginary axis in the other sheet is sheet is π  from its imaginary axis. Thus while there is one solution in each sheet, it may be percieved differently, relatively, in each sheet.

This is all because zero as the coefficient of the imaginary axis is a positional, not an absolute zero. This zero does not signify nothing and can thus can be ignored.  It signifies a relative postion that can NOT be ignored.