Garlonicon: great reply.  Thank you

To you and kTimesG:

First, kTimesG, thoughtful moniker.

Above all: With Bitcoin, and other such concepts, rules, required, and allowed operations are important.  Following the rules, regardless of origin and their cause, is what makes bitcoin work.

And recognizing my low knowledge of mathematics, I still have questions.

Looking at the equations, they all use addition.  Addition cannot be directly applied to a point on the curve.  The equation of y^2 = x^3 + 7 makes it impossible.  However, addition can be applied to the constituent parts of the points.  In my thoughts, that differentiation between the points and the constituent parts of the points is critical.  I recognize it may be wrong, but it seems correct.  At least right now.

Addition is nothing more than a short cut to counting.  I have learned that 4 + 4 is eight.  But it is a shortcut for counting.  To make my point,  4 plus 1 is 5, plus 1 is 6, etc.

What is subtraction?  Once at 6, I can subtract 2, then again, and get back to 4.  Subtraction is nothing more than discovering what I can add 2 to in order to get 6.  If addition is allowed, then subtraction is definitely valid.

What is multiplication?  It is nothing more than addition repeated a specified number of times.  I can add 2 to itself, then do it again, and again, and get 6.  Multiplication is nothing more than a shortcut for repeated addition. 

What is division.  Divide 6 by 2 to get 3.  It is nothing more than a shortcut to determine how many times to add 2 to itself to get 6.  Same for multiplication.  If we can add, then we can divide.

If we have addition, then it does follow to say that, by the definition of the words and the operations, there is subtraction, multiplication and division.  But to be redundant, they cannot be applied directly to points on the curve, but to their constituent parts to produce the result desired.

Re: There is no concept of "non-integer" in modular arithmetic. Actually, you shouldn't even think at integers, you can think as modular arithmetic as a set of ordered emoji (or fruits) as its elements. Division is not prohibited - it's simply non-existent, …

As described above, I have some difficulties with that.  If we can add, then we can subtract.  If we can add more than once, then we can multiply.  If we can subtract and multiply then we can divide.

But, and this is huge, and a bit redundant, if the rules of Bitcoin state that we must replace divide by modular inverse, then we must do that.  Otherwise we will never succeed at the task.

That said, I do recognize that you guys know far more than I.  I took Calculus 1 a second time to get an A.  For Calculus II, that semester it was the only course I took, I studied every night, got nothing less than 95 on all the homework, but never got higher than 80 on the tests.  That gave me the B needed to graduate.  So, I am not good with mathematics, but am relatively good with arithmetic.  Got an A in statistics without much problem.  And all that, because, given their huge and increasing place in world society, I want to have a reasonably good understanding in how Cryptos work

I will dig into the concepts of Group, Field, etc.  I do solicit suggestions of where to look.

Thank you again for your time and patience.