Absolute Zero

 

Key's in the Conch Shell

It's the third thatched roof on the right
Right beside crystal blue water
First wave of the day
Almost got away but my sail board caught her.

Let’s talk about that first wave.

There is a tendency to dismiss zero. But there are two kinds of zero, a relative zero and an absolute zero. An absolute zero CAN be dismissed, but a relative zero CAN NOT be dismissed. What is meant by a relative zero? Think of temperature. 0º Celsius ( 32º Fahrenheit for you non-STEM Americans) is the freezing point of water. But it is 273.15º on the absolute Kelvin temperature scale. The reason being that when measuring temperature on the Celsius scale, negative numbers are allowed. -40º Celsius means that it is cold, not that there is negative temperature. On an absolute scale there is no temperature less than zero. So there is a world of difference between a relative zero, e.g. 0º Celsius, and an absolute zero, e.g. 0º Kelvin. While it is convenient to act like ‑500º Celsius has a reality, in fact it is less than would be allowed on the Kelvin scale and thus is NOT real.

Why does this matter. A complex number is a+b*i. If b=0 is an absolute zero then it can be ignored and only the real coefficient, a, can be a concern. However if b=0 is a relative zero then it should NOT be ignored. This has a bearing on all forms of mathematics. This is because there are different solutions for linear regression, statistics, calculus, the quadratic equation, Pythagoras Theorem, relativistic dilation, etc. when that zero coefficient of an imaginary number is dropped or retained.

Pythagoras’s Theorem is c=√(a²+b²). This is true because c²=a²+b²+0²i where there is an absolute zero coefficient of the imaginary component of a complex number. But this needs imaginary solutions when a²+b² is negative. However if that zero is a relative zero and the surface is hyperbolic then the solution is c=ln(cosh(√((a²+b²))±sinh(√((a²+b²))) and no imagainary solutions are required. The hyperbolic solution is approximately √(a²+b²) most of the times that matter to us, 83.3%, 5/6, of the time. A hyperbolic solution when x is less than 16.7%, 1/6, away from the absolute will look like a parabola any way.  It is only as it nears the absolute that the approximation breaks down and it looks like a hyperbolic curved line that never approaches the absolute.

If the absolute is a wave, then it may be approximately true as you get further away from the start of that wave, even if that wave will infinitely repeat. However the approximation may break down in the first wave, before it repeats, where you are closest to the absolute. It is true that the solution to ln(cosh(x)±sinh(x)) is equal to both x and -x. However that is true only if x is larger than 1x10^-15 (at least according to Excel Microsoft Office 365 Apps version 2409)    Less than this value, the sharp discontinuity required by the approximation at x=0 appears to smooth out and the solutions are no longer equal. That first wave is different.