Counting back and predecessor function

Counting back by one we can denote by ‘n.

So 2 = ‘3.

1 = ”3

Counting back leads us to subtraction.

x-0 = x

x-y’ = ‘(x-y)

The latter equation says if x is our anchor, then counting back one more is the same as counting back one from the previous difference.

4-3 – 4 – 2’ = ‘(4-2) = ‘2 = 1

If we used B for counting back by one

B(x) = ‘x

We can then define the B(x,y) function by

B(x,0) = x

B(x,y’) = ‘B(x,y)

where y must be less than or equal to x.   Alternatively,

we could say that B has to be non-zero to continue.  If we start y at 0, the stopping condition can be stated as stop when y equals x.  So we don’t have to have less than or equal worked out yet.

This is the function given above for subtraction using the minus sign.

The B symbol makes clear that the structure of the B function is recursive.

If we continued to go backward when B reaches zero, then we would need to create negative numbers.  This requires defining negative numbers as an ordered pair of a sign and size, and then redefining B on such ordered pairs.  B is then a new function on these new types of inputs.


About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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