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April 29

Origin of a formula constant

I have discovered by an accurate empirical method that a constant needed in a predictive mathematical formula is 0.986093(7). In the SI metric system, the constants generally turn out to have simple origins, e.g., 2, pi, 4/3 times pi and the like, or fundamental constants such as the speed of light c, which I believe is not relevant here. Can anybody spot the components of 0.9860937 ? I have not been able to deduce it. ```` Dionne Court (talk) 14:35, 29 April 2024 (UTC)[reply]

2^{-0.{\overline {02}}}

, or, if you like,

1/{\sqrt[{99}]{4}}

. —Amble (talk) 16:37, 29 April 2024 (UTC)[reply]

OK, I'll bite. How did you get that? Greglocock (talk) 00:36, 30 April 2024 (UTC)[reply]

Nothing too clever — I just guessed there should be good matches with

n^{-1/p}

for some integer n and p. I tested n by brute force, and for each n, used

p=-\log(n)/\log(x)

, rounded to the nearest whole number. There are many possible answers, with n=4 as the first one. Although I took the (7) at the end to be an uncertainty level, and looking again, that may not be what OP intended. —Amble (talk) 03:51, 30 April 2024 (UTC)[reply]

I included brackets around the 7 to indicate that due to random experimental error, the value is centred around 7 but could be anything. Thus you could infer that the 3 before it is accurate (if I made no error in procedure).

The n^-p form doesn't help much because I cannot imagine any reason why it should be so. Its a bit like how we were taught in school to use 22/7 as pi - there is no reason for it, it is just a coincidence that 22/7 evaluates to equal pi to 3 places. It gives no insight into WHY pi equals 3.1416..... ```` Dionne Court (talk) 11:30, 30 April 2024 (UTC)[reply]

Right. The n^-p form is a silly example. The point is that you can produce a matching value in an infinity of ways. Without more information about the underlying process, there's no possible way to know which (if any) may be relevant. The notation I'm familiar with uses parentheses to indicate the standard uncertainty in the final digit (or digits), as shown in the NIST page here: [1]. --Amble (talk) 17:18, 30 April 2024 (UTC)[reply]

You may be interested in this inverse symbolic calculator, though it doesn't seem to recognize your constant. Staecker (talk) 23:31, 29 April 2024 (UTC)[reply]

I love that calculator. I wish I had known about it before. Thanks. ```` Dionne Court (talk) 00:56, 30 April 2024 (UTC)[reply]

I used to work with a lot of predictive formulas and nearly all of them had constants. They were calculated by using large populations and working out a best fit regression formula. The resulting constants were not based on pi or the speed of light or anything recognizable. They were based on average biological processes of humans, such as how many creatinine is cleared by the bladder on average. 12.116.29.106 (talk) 12:40, 30 April 2024 (UTC)[reply]

The problem I have been working on is not a biological process, as you may realise from the six-digit precision of the constant. It is within the realm of physics. In physics one seeks to discover why the constants are what they are and thus understand the process.

In complex biological process, the constants could be any weird thing, as dozens or hundreds of sub-processes are involved. In physics, the constants are usually very simple combinations of integers, pi, squared or rooted, trig identities, whatever. ```` Dionne Court (talk) 02:29, 1 May 2024 (UTC)[reply]

That formula above uses less symbols when printed than the given constant, I think that's pretty impressive. Too many of these approximations are worse 22/7 = 3.1428 with four symbols is only just about shorter if one says the 28 is close to 16. NadVolum (talk) 17:15, 30 April 2024 (UTC)[reply]

355/113 gets pi to six figures past the decimal point, which is eight symbols if you count the 3 and the period. The fraction is only seven characters if you write it without spaces. --Trovatore (talk) 23:23, 30 April 2024 (UTC)[reply]

Of course π itself is just one symbol.... --Trovatore (talk) 23:29, 30 April 2024 (UTC) [reply]

I think they told us in high school to use 22/7 so that we could more easily spot where algebraic cancellation could be used by dumping unnecessary precision. If it was about saving digits, they could have just told us to memorise 3.142. ```` Dionne Court (talk) 02:36, 1 May 2024 (UTC)[reply]

Yeah, I remember problems were often rigged to have a factor of 7 that you could cancel upon approximating pi as 22/7. With 3.14 that would be much harder. Double sharp (talk) 05:48, 1 May 2024 (UTC)[reply]