Unify principles of two formulae

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In summary, the most mathematical elegant way of converting the numbers on the left (x) to the numbers on the right (y) is to use the formula 1/logBase2(1/x) = y.
  • #1
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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  • #2
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
 
  • #3
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.
 
  • #4
"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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1. What are the two formulae that need to be unified?

The two formulae that need to be unified refer to two separate mathematical equations or principles that have similarities and can be combined or simplified into one formula.

2. Why do we need to unify two formulae?

Unifying two formulae can help simplify and streamline mathematical concepts, making them easier to understand and apply in various situations. It can also help identify relationships between seemingly unrelated equations.

3. How do you unify two formulae?

To unify two formulae, one must first identify the common variables and constants between the two equations. Then, these common elements can be combined or simplified to create a single formula that encompasses both principles.

4. What are the benefits of unifying two formulae?

Unifying two formulae can lead to a deeper understanding of mathematical concepts and relationships. It can also help in solving complex problems more efficiently and provide a more elegant solution.

5. Can two formulae always be unified?

No, not all formulae can be unified. Some equations may have fundamental differences that make unification impossible. However, it is always worth exploring the possibility of unifying two formulae before concluding that it is not possible.

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