The Hyperbolic function

Here they are.  The first thing we do is a change of base to the twos binary. Then we both add and subtract each of the functions above.  Here is what we get:

cosh + sinh =  2^N, whener N is the digit index in the twos system

and
cosh - sinh = 1/(2^N) where N is the fractional digits in the twos binary.


The two functions are actually separated by one digit, and I will get into that.  But the cosh is the wave, the sinh the null.  When the cosh and sinch define a group, no problem, the digit system works within the group and the fraction and whole part fit right into the spots they belong in the complete integer. Again, the proton holds the fractional part. When using the natural logarithm, we are using the qubit system, and we can directly draw the probability functions that make the structure look real.