Quadratic

 

Zero, My Hero

What's so wonderful about a zero? It's nothing, isn't it?
Sure, it represents nothing alone
But place a zero after one, and you've got yourself a ten
See how important that is?
When you run out of digits, you can start all over again
See how convenient that is?
That's why with only ten digits including zero
You can count as high as you could ever go
Forever, towards infinity
No one ever gets there, but you could try 

Zero is even more important than that! 

In a previous blog post  https://dbeagan.blogspot.com/2024/10/absolute-zero.html I suggested that zero as the coefficient of the imaginary part of a complex number on a hyperbolic surface is important.

The solution  of quadratic equation ax²+bx+c is typically given as x=(-b±√(b²-4ac))/(2a) if x is a real number or if x is a complex number on a Euclidean, flat, surface.  However if x is the real part of a complex number on a non-Euclidean hyperbolic surface then the quadratic equation is really ax²+bx+c+0²i and its solution is x=ln(cosh((-b±√(b²-4a*c))/(2a))±sinh((-b±√(b²-4a*c))/(2a))). Then if b²‑4a*c is negative, and its square root is imaginary, since the solution of cosh(ki) is always real but the solution of sinh(ki) is always imaginary, the solution of x will be imaginary. This is true for the period of 5/6, 83.3%, of the solutions where the traditional solution of the quadratic equation is the approximation as well as  the period of the remaining 16.7%, 1/6, where the approximation no longer applies.

The approximation means that in this case the real part will itself contradictorily be a complex number, a+b*i+0²i. For the remaining 16.7%, 1/6, the approximation will be the negative of that complex number, while the hyperbolic solution will always give the correct complex number.

For other solutions where the imaginary coefficient should always be zero, the hyperbolic solution always applies. Thus the relativistic dilation is γ=ln(cosh(√(1-v^2/c²))±sinh(√(1-v^2/c²))); Pythagoras’ Theorem is c=ln(cosh(√(a²⁺b²))±sinh(√(a²⁺b²))); in statistics the coefficient of EVERY moment about the mean is 0 not just odd moments; the real radius, r,  of a complex number, x+y*i in Cartesian coordinates translated to polar coordinates, re^iθ, is  r=ln(cosh( (√(x²⁺y²))±sinh(√(x²⁺y²))), etc.