Unify principles of two formulae

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Discussion Overview

The discussion revolves around finding a unified mathematical formula that converts a set of numbers (x) into another set (y), while maintaining certain properties of the transformation. The context includes exploring the elegance of mathematical solutions and the interpretation of specific terms related to the transformation.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant seeks a "master formula" that can convert x to y while preserving the behavior of the numbers, particularly noting different behaviors for values of x closer to 1 versus those closer to 0.
  • Another participant proposes a solution: 1/logBase2(1/x) = y, claiming it effectively addresses the conversion for the entire range of x.
  • There is a call for clarification on what is meant by "unify principles" and the specific meaning of the phrase regarding zeros and their impact on y, suggesting that a clearer definition might lead to a more apparent formula.
  • A participant attempts to refine the explanation of the transformation, stating that "twice as many decimal places in x translates to a geometric halving in y," in an effort to clarify the original intent.
  • One participant acknowledges that they initially had the formula 1/logBase2(1/x) in mind before articulating the transformation properties, indicating a potential evolution in their understanding.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the definition of "unify principles" or the clarity of the transformation properties. There are competing interpretations of the problem and its requirements, and while one solution has been proposed, its acceptance is not universally agreed upon.

Contextual Notes

Limitations include the ambiguity in the terms used to describe the transformation and the lack of a detailed exploration of the proposed formula's validity across the entire range of x values.

Twinbee
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What's the most mathematical elegant way of converting the numbers on the left (x) to the numbers on the right (y). Note that the numbers don't have to be exactly the same, as long as the property of how the numbers act is left intact - so that numbers (x) closer to 1 act as the formula 1/(1-x) = y, while numbers (x) closer to 0 act as the formula that twice as many zeros equals half the amount of y:

In other words, I want to unify these formula to act as one 'master formula'. So there you go, convert x to y:

0.00000000000000000000000000000001 --> 0.0125
0.0000000000000001 --> 0.025
0.00000001 --> 0.05
0.0001 --> 0.1
0.01 --> 0.2
0.1 --> 0.4
0.5 --> 0.75
0.9 --> 4
0.99 --> 40
0.999 --> 400
0.9999 --> 4000
0.99999 --> 40000
 
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Just managed to solve it. For those who don't want to see the answer, it's below.
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1/logBase2(1/x) = y
 
Any "mathematically elegant" way of solving such a problem would start by specifying exactly what "unify principles" means and what "act as the formula that twice as many zeros equals half the amount of y" means. I think once you had done that, the formula would be apparent.
 
"Mathematically elegant" in this context would be perhaps the simplest/smallest way of converting between the two.

The solution I found seems to be the best bet as you can see.

I'll rephrase "act as the formula that twice as many zeros equals half the amount of y".

to:

"so that twice as many decimal places in x translates to a geometric halving in y". If that still doesn't make it clear, then you should see the pattern after a quick scan of the numbers in my first post.

===========================

EDIT:

a: oops, you didn't ask me what "mathematically elegant" was anyway.

b: I'll be honest, and say that in place of "numbers (x) closer to 0 act as the formula that twice as many zeros equals half the amount of y", I originally had 1/logBase2(1/x) = y. It only ocurred to me afterwards that this formula works for the whole range anyway, and is thus the solution.
 
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