Continued from page 1
I posed riddle, how do you create a diagram of Gamma function when it spirals in one place and not in other?
I solved that riddle with a surprising answer. NATURE IN THIS CONTEXT REJECTS THE IMAGINARY. There is no solution to Gamma Function if you use complex maths. There is no complex space.
That is to say, square root of -1 has two answers, but Nature insists that you must take them BOTH AT THE SAME TIME.
This is quite different to roots of real numbers. The roots of 4 are 2 *OR* -2.
The roots of -1 are i1 *AND* -i1. Take BOTH or NONE, in case of Gamma Function. Otherwise, riddle is insoluble.
But i1 makes spiral go left whilst -i1 makes spiral go right. Making spiral go equally left and right will STOP it from spiralling. So both is equivalent of none. Nature has REJECTED imaginary.
At this point I wrote (http://www.wehner.org/euler/solution.htm ) that "If it can be shown for other functions unrelated to Gamma function, and having a negation function, that Nature rejects imaginary axis in this way, then it can be said that Nature rejects complex mathematics as a human product. "
I had not expected that Nature would then reveal to me a function that rejects imaginary axis without having a negation function. It is INVERSE function to exponential. That is to say, it is LOGARITHM.
From first page above, you can visualise Euler's equation for exponential (exp(i X) = cos(X) + i sin(X)) as a spiral. You can see that exp(i Pi) gives -1. So does exp (i 3Pi). So does exp (i 5Pi). So do ALL odd imaginary multiples of Pi. So do all negative values of those odd imaginary multiples.
So log of -1 is ALL of those positive and negative odd imaginary multiples of Pi. It does not really matter whether they are AND or OR. It only matters that they are there.
I then considered logarithm of zero. This is known to be MINUS INFINITY.
But what of MINUS ZERO? This may seem like a silly question, but it is not. We have considered log of any negative number to be log of positive equivalent plus one or more odd imaginary multiples of Pi. So log of minus zero will be minus infinity just like log of zero - but with a DIFFERENCE.
For all negative numbers, logarithm is as for positive but TOGETHER WITH THE SERRIED RANKS OF ODD IMAGINARY PIs. As numbers rise, they get closer and closer towards zero, and real part of logarithm goes downwards towards minus infinity.
Then they hit zero. Those serried ranks of imaginary numbers STOP. That's it! They VANISH.
One would call this a SINGULARITY, but it is more than that. There are INFINITE odd imaginary multiples of Pi, and they ALL stop together at point X=0.
So Nature again is REFUSING imaginary.
Natural laws tend to flow smoothly throughout number continuum. For an entire family of numbers - all odd imaginary multiples of Pi - and therefore by inference a whole SET - set of all imaginary - to stop is quite unnatural.
So imaginary numbers are imaginings of Humankind. Nature does not imagine.
Charles Douglas Wehner
Born in 1944, Charles Wehner is an electronics engineer, technical author and former factory manager.