Strange constant. Apparently useless.

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Rhombic
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Strange constant. Apparently useless.

Postby Rhombic » Fri Jan 03, 2014 6:54 pm UTC

I was thinking about the result of the infinite sums of n^-2, which converges, so I immediately supposed that the infinite sums of n^-n would converge too.
And it does. It converges to an apparently dull number, an obscure number: 1.291285997... (continues, the number has been rounded)
After searching that number in the browser, I saw that there's one thing called Sophomore's dream, but still no way to write this number as a product, sum, division or whatever of either π, e, i or square root of whatever. This probably has some importance in mathematics since it must be either a constant or a combination of already-known constants, and both of those possibilities are particularly attractive (if it was (which it isn't) (e+pi)^(e-2), there must be some reason for that, related to the natural numbers and to maths itself).

So here goes the QUESTION:
Can it be represented as a combination of variables?

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Re: Strange constant. Apparently useless.

Postby fishfry » Fri Jan 03, 2014 7:57 pm UTC

The real number system has an uncountable infinity of totally useless constants. The one you picked isn't especially interesting, since it can be uniquely characterized in a finite string of mathematical symbols. That makes it computable. Loosely speaking, the computable numbers are the ones that could possibly have individual names, like pi or the square root of 2 or your interesting number. I think the sum of 1/n^n is interesting. But note that I just described that number in a few characters, "the sum of 1/n^n". So even though it takes infinitely many decimal digits to specify it exactly; nevertheless we still do have a finite unique characterization of it. We have it under control in a universe based on elementary logic, the axioms of set theory, and finite-length proofs.

http://en.wikipedia.org/wiki/Computable_number

What interests me are the non-computable numbers. These are all the real numbers that can never have any finite description. No formula, algorithm, or even a name! Now those are the useless real numbers. But without them, the real line would be full of holes. "Almost all" numbers are non-computable, in the usual technical sense that the set of exceptions has measure zero.

The "purpose" of the non-computable numbers is simply to plug up all the holes in the real numbers so that we can call it a continuum. But that's still an important job. I don't think there are any useless real numbers. Each real number plugs a specific hole in the real line. Take one real number away, and the real line no longer models the continuous Euclidean line.

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Re: Strange constant. Apparently useless.

Postby notzeb » Fri Jan 03, 2014 8:23 pm UTC

According to the Inverse Symbolic Calculator, there isn't any simple relationship with other constants.
Zµ«V­jÕ«ZµjÖ­Zµ«VµjÕ­ZµkV­ZÕ«VµjÖ­Zµ«V­jÕ«ZµjÖ­ZÕ«VµjÕ­ZµkV­ZÕ«VµjÖ­Zµ«V­jÕ«ZµjÖ­ZÕ«VµjÕ­ZµkV­ZÕ«ZµjÖ­Zµ«V­jÕ«ZµjÖ­ZÕ«VµjÕ­Z

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Re: Strange constant. Apparently useless.

Postby Sizik » Fri Jan 03, 2014 8:46 pm UTC

fishfry wrote:These are all the real numbers that can never have any finite description. No formula, algorithm, or even a name!


Incorrect, see Chaitin's constant.
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Re: Strange constant. Apparently useless.

Postby MartianInvader » Fri Jan 03, 2014 10:52 pm UTC

Heh, someone confused the computable numbers with the definable numbers.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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Re: Strange constant. Apparently useless.

Postby fishfry » Sat Jan 04, 2014 4:28 am UTC

MartianInvader wrote:Heh, someone confused the computable numbers with the definable numbers.


I thought I used the phrase "speaking loosely" at some point but maybe not. I didn't want to add an unnecessary layer of technical detail to what I wrote by mentioning the distinction between the computable and the definable numbers. My basic point is valid in either case, that there are uncountably many real numbers that can never be named or characterized in finitely many symbols.

Since two people jumped on that, I plead guilty as charged and offer this humble disclaimer.

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Re: Strange constant. Apparently useless.

Postby Rhombic » Sun Jan 05, 2014 10:58 am UTC

Hmmmm... nonetheless, I checked that Inverse Symbolic Calculator does not recognise (pi + e)*log(2) either, so there are some combinations which still are out of reach. Bearing in mind that you can mathematically express the constant, anyway, I believe that there must be some sort of explanation or something.
Can anyone get me around 30 digits or so, please?

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Re: Strange constant. Apparently useless.

Postby jestingrabbit » Sun Jan 05, 2014 11:43 am UTC

Rhombic wrote:Hmmmm... nonetheless, I checked that Inverse Symbolic Calculator does not recognise (pi + e)*log(2) either, so there are some combinations which still are out of reach. Bearing in mind that you can mathematically express the constant, anyway, I believe that there must be some sort of explanation or something.
Can anyone get me around 30 digits or so, please?


http://oeis.org/A073009

Always try there for these sorts of queries.
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Re: Strange constant. Apparently useless.

Postby Rhombic » Sun Jan 05, 2014 9:39 pm UTC

jestingrabbit wrote:
Rhombic wrote:Hmmmm... nonetheless, I checked that Inverse Symbolic Calculator does not recognise (pi + e)*log(2) either, so there are some combinations which still are out of reach. Bearing in mind that you can mathematically express the constant, anyway, I believe that there must be some sort of explanation or something.
Can anyone get me around 30 digits or so, please?


http://oeis.org/A073009

Always try there for these sorts of queries.


Oh thanks! I had forgotten about the possibility of finding it as a list at OEIS. Funny, the only relationship to mathematics is the sum I mentioned and that it is the definite integral between 0 and 1 of one function, being 1/K (let's call this number K in this topic) the definite integral of another function, according to Wikipedia and the few sources I found to check if there was anything else to it.

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Re: Strange constant. Apparently useless.

Postby quarkcosh1 » Fri Jan 10, 2014 8:09 pm UTC

There is nothing dull about that number. It is approximately equal to (1/sinh^-1(1))^2. The square root of that number is equal to the w boson/z boson mass ratio (it could be the inverse of this but I can't remember).

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Re: Strange constant. Apparently useless.

Postby gmalivuk » Fri Jan 10, 2014 10:17 pm UTC

But an uncountable infinity of numbers are *also* approximately equal to (1/sinh^-1(1))^2 and whose square roots are approximately equal to that mass ratio.

Being one of an uncountable infinity is hardly sufficient to make something interesting.
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Re: Strange constant. Apparently useless.

Postby quarkcosh1 » Sat Jan 11, 2014 12:00 am UTC

gmalivuk wrote:But an uncountable infinity of numbers are *also* approximately equal to (1/sinh^-1(1))^2 and whose square roots are approximately equal to that mass ratio.

Being one of an uncountable infinity is hardly sufficient to make something interesting.


Let's see some examples of these other numbers and functions that approximate this value. Many of them have a much longer description than this one. Being in a uncountable infinity means nothing if the function is too long to be of any interest.

Edit: Let me further clarify what I mean by this. An infinity is something with no limit so you can remove the infinity by setting a limit. Limiting yourself to short functions is an example of a limit.

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Re: Strange constant. Apparently useless.

Postby Meteoric » Sat Jan 11, 2014 1:55 am UTC

quarkcosh1 wrote:Let's see some examples of these other numbers

"The sum of 1 and .3"
"The average ratio between deadlift and squat strength for an individual training both"
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Re: Strange constant. Apparently useless.

Postby doogly » Sat Jan 11, 2014 3:45 am UTC

Meteoric wrote:"The average ratio between deadlift and squat strength for an individual training both"

Pffft, maybe the weak gauge sector bears a relationship to your lifts...
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Re: Strange constant. Apparently useless.

Postby gmalivuk » Sat Jan 11, 2014 4:08 am UTC

quarkcosh1 wrote:Let's see some examples of these other numbers and functions that approximate this value. Many of them have a much longer description than this one. Being in a uncountable infinity means nothing if the function is too long to be of any interest.
The constant in question is 1.291285997..., and (1/sinh^-1(1))^2 is 1.287300498...

So, 1.29 is closer, *and* has a much shorter description.
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Re: Strange constant. Apparently useless.

Postby quarkcosh1 » Sat Jan 11, 2014 2:54 pm UTC

gmalivuk wrote:
quarkcosh1 wrote:Let's see some examples of these other numbers and functions that approximate this value. Many of them have a much longer description than this one. Being in a uncountable infinity means nothing if the function is too long to be of any interest.
The constant in question is 1.291285997..., and (1/sinh^-1(1))^2 is 1.287300498...

So, 1.29 is closer, *and* has a much shorter description.


The description is only shorter in base 10 and it is also an obvious description. My description has an advantage that it only assumes the numbers -1 through 2 exist.

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Re: Strange constant. Apparently useless.

Postby gmalivuk » Sat Jan 11, 2014 3:04 pm UTC

Your description is only shorter in integer bases or bases greater than 2, because there is no terminating representation of 2 in transcendental bases between 1 and 2. Also your description is only short when you let the definition of sinh do all the heavy lifting for you. When we name constants and functions, we always get a shorter description of things that depend on them, but that isn't anything special. If I decide to name 1.29 something special, then I can always express it exactly in however many characters that name requires.
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Re: Strange constant. Apparently useless.

Postby quarkcosh1 » Sat Jan 11, 2014 3:22 pm UTC

gmalivuk wrote:Your description is only shorter in integer bases or bases greater than 2, because there is no terminating representation of 2 in transcendental bases between 1 and 2. Also your description is only short when you let the definition of sinh do all the heavy lifting for you. When we name constants and functions, we always get a shorter description of things that depend on them, but that isn't anything special. If I decide to name 1.29 something special, then I can always express it exactly in however many characters that name requires.


Ok I admit sinh does all the heavy lifting for me but I feel that this is justified due to the many connections that sin and sinh have in math and physics.

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Re: Strange constant. Apparently useless.

Postby doogly » Sat Jan 11, 2014 5:55 pm UTC

No one's saying you're a bad person for including it in your library of functions you can just call. It's just that when you are counting the number of steps to carry out an operation, they're not free.
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Re: Strange constant. Apparently useless.

Postby Rhombic » Sat Jan 25, 2014 3:00 pm UTC

Well, 22/7 is very similar to 3.141592653... but that doesn't mean that 22/7 is not dull. It actually is quite dull and uninteresting.
However, I find it peculiar that while many other sums converge to either combinations of pi, e, log(2), log(3), etc., this one does not. For example, 1-1/2+1/3-1/4+1/5-1/6+1/7-1/8+1/9-1/10... converges to log(2). Not approximately log(2), but log(2) itself.

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Re: Strange constant. Apparently useless.

Postby skullturf » Mon Jan 27, 2014 4:07 pm UTC

This is a little subjective, but personally, I think it could be argued that it's *more* surprising when the value of an infinite sum turns out to be, say, sqrt(2*e) or log(pi) or something similarly related to "known" constants.

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Re: Strange constant. Apparently useless.

Postby Bloopy » Tue Jan 28, 2014 2:14 am UTC

Here's one such 'surprise':

http://mathworld.wolfram.com/PerfectPower.html
As shown by Goldbach, the sum of reciprocals of perfect powers (excluding 1) with duplications converges,
Image


n-n is what I guess you'd call the diagonal of the above, ie. when m=k. I doubt this gets us any closer to a closed form for the constant, as otherwise I'd expect someone to have found it by now.

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Re: Strange constant. Apparently useless.

Postby sjorford » Tue Jan 28, 2014 5:02 pm UTC

Bloopy wrote:
As shown by Goldbach, the sum of reciprocals of perfect powers (excluding 1) with duplications converges,
Image

Useless but pretty fact: the partial sum for 2 <= k <= 33 and 2 <= m <= 33 ≈ 0.9696969696

EDIT: Actually, it looks like this is just because the sum where 2 <= k <= ∞ and 2 <= m <= n = (n - 1) / n. Never mind.

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Re: Strange constant. Apparently useless.

Postby quarkcosh1 » Thu Feb 13, 2014 10:02 pm UTC

doogly wrote:No one's saying you're a bad person for including it in your library of functions you can just call. It's just that when you are counting the number of steps to carry out an operation, they're not free.


The number of steps it takes to calculate a trigonomic function depends on context. On a regular computer you need to convert it to the taylor series in order to calculate it but in a geometric context all you have to do is take the ratio of 2 sides of a triangle.

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Re: Strange constant. Apparently useless.

Postby doogly » Thu Feb 13, 2014 10:12 pm UTC

You mean, in a geometric context where the sides of the triangle are known. Which is to say, where someone has already done the trigonometry for you, and now you are just giving a name to the ratio. You aren't actually calling the function in a meaningful sense there.
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Re: Strange constant. Apparently useless.

Postby quarkcosh1 » Fri Feb 14, 2014 1:39 am UTC

doogly wrote:You mean, in a geometric context where the sides of the triangle are known. Which is to say, where someone has already done the trigonometry for you, and now you are just giving a name to the ratio. You aren't actually calling the function in a meaningful sense there.


All you have to do is draw a triangle with 2 sides or angles equal so that the 1 ratio applies. Since so many things in physics are symmetrical this isn't asking much. The way you are calling the function is by performing a measurement on the triangle. So in other words you don't need traditional computers for this because everything can be done with geometry and measurement.

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Re: Strange constant. Apparently useless.

Postby Carlington » Fri Feb 14, 2014 1:52 am UTC

I feel like you're missing the point that was being made in the first place, quarkcosh. If I'm reading properly, all that's being said is that sinh() is only shorter because you're writing it down as sinh(). The actual act of calling sinh() requires you to do the necessary calculations involved, so it effectively expands out to two exponentiations, a subtraction, and a division.

Basically, instead of counting the steps as:
  1. Do sinh(x)

You should be counting the steps as:
  1. Find ex
  2. Take the reciprocal of ex (i.e. Find e-x)
  3. Subtract e-x from ex
  4. Divide this result by 2

Four steps, rather than one. That's what's meant by "letting sinh() do the heavy lifting." - you're using the presence of sinh() to disguise the fact that there's more within that black box.
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Re: Strange constant. Apparently useless.

Postby gmalivuk » Fri Feb 14, 2014 1:55 am UTC

Nevermind the fact that finding ex isn't exactly a simple operation in itself. There are a number of beautiful equations for it, but all of them require rather a lot of computation if what you want in the end is a specific number to some degree of precision.
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Re: Strange constant. Apparently useless.

Postby doogly » Fri Feb 14, 2014 1:59 am UTC

quarkcosh1 wrote:
doogly wrote:You mean, in a geometric context where the sides of the triangle are known. Which is to say, where someone has already done the trigonometry for you, and now you are just giving a name to the ratio. You aren't actually calling the function in a meaningful sense there.


All you have to do is draw a triangle with 2 sides or angles equal so that the 1 ratio applies. Since so many things in physics are symmetrical this isn't asking much. The way you are calling the function is by performing a measurement on the triangle. So in other words you don't need traditional computers for this because everything can be done with geometry and measurement.

How do you propose to draw that triangle precisely? It certainly can't be a geometric construction, with some ruler and straightedge, for an arbitrary angle.
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Re: Strange constant. Apparently useless.

Postby Carlington » Fri Feb 14, 2014 2:28 am UTC

gmalivuk wrote:Nevermind the fact that finding ex isn't exactly a simple operation in itself. There are a number of beautiful equations for it, but all of them require rather a lot of computation if what you want in the end is a specific number to some degree of precision.


I suppose at this point it becomes relevant to consider who is performing the operations. A person who is sufficiently well-trained in mathematics can work out an expression for Sn and then take the limit as n→∞, but a computer won't be as skilled at finding the pattern and simplifying it to an expression for Sn, nor will it be any good at evaluating infinite sums. So, for a person with moderate education, finding ex is as many steps as it takes to do an infinite sum, but for a computer it's potentially infinite steps, or at least as many steps as it takes to get within some margin of error.

EDIT: It occurs to me that a person can't work out the infinite sum for ex very much better than a computer, after all. That was the "something" that was bugging me during writing my post, but I didn't work out what it was until after I hit submit, of course.
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Re: Strange constant. Apparently useless.

Postby Rhombic » Mon Feb 17, 2014 10:24 pm UTC

Now this is actually great: I found in http://xkcd.com/1047/ a great approximation. Just check it out, it's at the bottom, right hand side. Looks like xkcd has already thought about this... again, it's only an approximation.

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Re: Strange constant. Apparently useless.

Postby cyanyoshi » Tue Feb 18, 2014 3:52 am UTC

I don't know exactly how exp and sinh are practically calculated (or very much about computer science for that matter), but the Taylor series representation of sinh(x) is basically exp(x) without the even terms, so it seems like it would converge faster. On the other hand, the fact that exp(x+n)=exp(x)*e^n might be particularly useful in case those partial sums get too big.

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Re: Strange constant. Apparently useless.

Postby ConMan » Tue Feb 18, 2014 4:57 am UTC

cyanyoshi wrote:I don't know exactly how exp and sinh are practically calculated (or very much about computer science for that matter), but the Taylor series representation of sinh(x) is basically exp(x) without the even terms, so it seems like it would converge faster. On the other hand, the fact that exp(x+n)=exp(x)*e^n might be particularly useful in case those partial sums get too big.

Very broadly, that's what usually happens - perform some transformations to get to a certain form, then apply a Taylor series once you know it'll converge quickly. Sometimes with the use of "magic numbers" to make things even faster, as in the Quake 3 implementation of 1/sqrt(x), although that uses Newton's approximation rather than Taylor series.
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Re: Strange constant. Apparently useless.

Postby Rhombic » Sun Mar 23, 2014 10:24 pm UTC

Just in case, the constant is near eπ-(3+6π)


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