In the preceding text I promised to answer here in the Appendix the question "How did we find the three answers to the question 'What is 1/3 of 3 on the clock?'"  Before doing so, here is some math that explains some of the comments in that text.  First we illustrate how the associative and equivalency concepts lead us to discover addition:

  3  =  1 + 1 + 1  by definition.
  3  =  (1 + 1) + 1 by association.
  3  =     2   +  1   by equivalency, inasmuch as  2 = 1 + 1.

We know that the number coming before 1 (call it "?") must have the property that ? + 1 = 1.  We have learned to call ? "zero" and say that  0 + 1 = 1.  This logic leads us to observe that the number before 1 has a value of zero.  Look at the following example:

        2   =  1 + (1)  by definition.
        2   =  1 + (1 + 0)  by equivalency and commutation.
        2   =  (1 + 1)  +  0  by association.
        2   =    2   +  0    by equivalency.

What would be the property of the number (call it "x") before 0?

  x + 1   =  0   by the order of numbers.
        1   =  1 + (0)  by equivalency.
        1   =  1 + (x + 1 )   by substitution.
        1   =  (1 + 1) + x    by commutation and association.
        1   =  2 + x  by replacement.
This number has the property of -1.  We would discover by analogy that 11 (on a clock) also has the property of -1, both numbers being two positions before 1.  Look again:

  0  =  x + 1  0  =  -1  +  1
  1  =  2 + x  1  =  2  +  -1

The concept of zero comes about from how we define and order our numbers.

Now back to the question of what, on a clock, is 1/3 of 3?  In the text we showed that there are three answers: 1, 5, and 9, for when checking the results we have:

  1 + 1 + 1  =  3
  5 + 5 + 5  =  15  =  15 - 12  =  3  on the clock.
  9 + 9 + 9  =  27  =  27 - 24  =  3  on the clock.

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