The Spring, part-1

These are just my thoughts on generalizing additive and multiplicative
operations and extend it to subsequent operations. I'm just trying to
learn the math basics using a different perspective. Please feel free
to correct me. Thanks.

Imagine a spring (S) which is infinitely long and infinitely flexible
and always expands or contracts uniformly throughout its length. The
spring is aligned along the real number line (R).

Bases and control points
------------------------
The spring has two ends called "base end" and "top end". The top end is
always fixed at positive infinity. The base end is movable and is fixed
(set) at some values in R known as bases (B). Every base has a
corresponding control point (CP). Bases occur in levels. Each level has
a set of bases. At zero level we have -oo as base and 0 as CP. The
subsequent levels seem to have the following bases and CPs.

B_n = CP_(n-1)
CP_n = CP_(n-1) + 1^(1/n)

where n is the level number which is a positive integer.
So, we have n bases, n+1 CPs and n+1 springs at level n.

Arithmetic operations using S
-----------------------------
We use the following conventions:

1. x', y', CP' are the values on S marked against x, y, CP on R
respectively.
2. y': x' -> CP, for example, means expand or contract S while keeping
its both ends fixed, until x' coincides with CP on R. The result of the
operation is the value on R pointed to by y'.
3. m': y' => x, for example, means move y' until it points to a value m
on R which is equal to a value m' on S pointed to by x. The result of
the operation is the value on R pointed to by m'.

The results for the following operations are given in a table at the
end of section.

Binary operations, for any x in R
-----------------
Difference: x': y' -> CP
Summation: x': CP' -> y
Mean: m': y' => x

Unary operations, for any x in R
----------------
Inverse: CP': x' -> CP
Double: x': CP' -> x
Half: m': x' => CP

Please use mono-spaced font to view the following table.
+------------+---------+----------+----------------------+
| Base -> | -oo | 0 | 1 |
+------------+---------+----------+----------------------+
| Difference | x-y | x/y | |
+------------+---------+----------+----------------------+
| Summation | x+y | xy | |
+------------+---------+----------+----------------------+
| Mean | (x+y)/2 | sqrt(xy) | |
+------------+---------+----------+----------------------+
| Inverse | 1-x | 1/x | 1 + 1/(x-1)^k |
+------------+---------+----------+----------------------+
| Double | 2x | x^2 | |
+------------+---------+----------+----------------------+
| Half | x/2 | sqrt(x) | |
+------------+---------+----------+----------------------+

.

Relevant Pages

  • Re: The Spring, part-1
    ... operations and extend it to subsequent operations. ... If the spring is intended to model the reals, ... Positive infinity is not a real number; it's an element of the EXTENDED ...
    (sci.math)
  • Re: The Spring, part-1
    ... operations and extend it to subsequent operations. ... If the spring is intended to model the reals, ... Positive infinity is not a real number; it's an element of the EXTENDED ...
    (sci.math)
  • Re: The Spring, part-1
    ... operations and extend it to subsequent operations. ... If the spring is intended to model the reals, ... Positive infinity is not a real number; it's an element of the EXTENDED ...
    (sci.math)
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