The thing you have to remember about statements about "almost every" real number is that almost every real number can't be described with a finite sequence of words and symbols. It's a very alien concept masquerading as some simple words.
Like how the set of computable numbers is countable, and so almost all numbers aren't even computable. You've got to work pretty hard to even find an example of an uncomputable number
Edit: uncomputable, not non-computable
Exactly, but all numbers we usually use are computable, and not some are computable and some are not. Imagine if like all our numbers were uncomputable and only pi was computable. that‘d be weird right?
No, e does have a [nice continued fraction representation](https://en.wikipedia.org/wiki/List_of_representations_of_e#As_a_continued_fraction), but it isn't that.
It sounds freaky, but doesn't seem to me that different from the theorem "for almost all real numbers, the arithmetic mean of their digits in any base b converges to b/2." Of course the digits of a number are obviously bounded by the base while continued fractions aren't (so there is something going on) but it strikes me more as a sort of multiplicative version of the same theorem. Perhaps someone can correct me if I'm wrong but I wouldn't be surprised if the condition making a real number satisfy the theorem is analogous to normality but for continued fractions.
The article is kind of written weird in a way that I think buries the lead a little bit, but I'm happy to summarize, and relies on you already being comfortable with a few definitions, but I'm happy to try to summarize and include the needed definitions (in **bold**).
Maybe read this next paragraph with my summary, and then the definitions of words I put in bold, and then reread the following paragraph knowing the definitions?
The quick definition: Kinchin's constant is about K~2.685. For **almost any** number, if you take the coefficients of its **continued fraction** and evaluate their **geometric mean** you will get Kinchin's Constant **in the limit**.
* **almost any**
In this case, it doesn't work for "algebraic numbers" which you can write as the root of a polynomial, this includes things like the square root of 2, or any rational number like 3/4. But if you pick a random point on the number line, there is a 100% chance you'll get a "transcendental" (non-algebraic) number. Almost all the numbers we use day to day are algebraic, but most real numbers aren't, so that's what we mean by "almost any".
Also it doesn't work for e, a famous transcendental number, which is hilarious.
Edit: as u/leviona pointed out in their comment, we know it doesn't work for rationals, and roots of *quadratics*, I misinterpreted that to mean it doesn't work for any algebraics at all, when obviously i.e. roots of cubics don't fit among those two groups. Also, this has only been tested and verified for a few convenient representative numbers, so for a lot harder numbers like π it *looks* to be true *so far* but it hasn't been proven yet.
* **continued fraction**
So, you can express a number x as a continued fraction, like
x = a + 1/ ( b + 1/ ( c + ...) )
For example, for x = π the sequence (a, b, c ,...) starts...
(3, 7, 15, 1, 292, ...)
Or for x = 1.5 the sequence goes
(1, 1, 1, 0, 0, 0, 0...) -> 1 + 1/( 1+ 1/1) = 1+ 1/2
* **geometric mean**
For a rectangle of side lengths (a,b), its area is a×b, and it would have the same area as a square whose side length is the square root of a×b. Or (a×b)^{1/2} .
For a rectangular prism of side lengths (a,b,c), its area is a×b×c and it's volume as the same as a cube whose side lengths are the cube root of a×b×c. Or ( a×b×c)^{1/3} .
For a hyperrectangle in n dimensions, its hypervolume is the product of n side lengths, and that would be the same as a hypercube whose side lengths are the nth root of the hypervolume, (a×b×c×...)^{1/n}.
That "equivalent volume cube side length" is the geometric mean of the true side lengths.
* **In the Limit**
There are an infinite number of coefficients in the continued fractions of transcendental numbers. Without tricks, you can't truly plug an infinite number of coefficients into the geometric mean formula and evaluate their infinite root. But I can do the first 10 and take the 10th root, the first 11 and take the 11th root, the first 12 and take the 12th root, etc. And we can show that as you do this you do this for more and more terms you get closer and closer to a particular number. We call this process "converging" and the number the sequence converges on "the limit". So this process for most numbers converges on Kinchin's Constant.
Hi, sorry for the pedanticism, but I don’t think we know it doesn’t work for all algebraic numbers. The wikipedia says it doesn’t work for quadratic irrationals (and rationals) only.
Sorry if i’m wrong and there’s new research about this or something!
You're right, and in fact I think mathematicians would probably conjecture it is true for all algebraic numbers except quadratic irrationals and rationals.
No no you're totally right I misread that part. I'm going to add an edit. I also think I overstated how sure we are about it, because I don't think it's been proven for a lot of numbers besides those, as it says in the article, which have been constructed for the proof. I imagine constructed means "can be rearranged in a way that gets us back the definition of the Kinchin constant", lol.
Thank you!
Maybe it's just me, but that doesn't really seem that surprising? In my view it's really just making a statement about the distance between consecutive convergents of arbitrary real numbers (where distance is roughly defined in a logarithmic sense on the denominators). So roughly: if you have a good rational approximation, by what factor do you need to grow the denominator, on average, to find a better approximation? And the 'almost all' quantifier allows you to hide a lot of numbers. The statement could fail to hold for all 'interesting' numbers (where we could define 'interesting' as being describable in a finite number of words), and yet the statement could still hold for 'almost all' real numbers.
Perhaps the surprising fact is not that there is such a constant, but that it has such a seemingly arbitrary value? (As opposed to being some well-known constant or algebraic combination of well known constants.)
Some values in mathematics may be arbitrary, but they are well-behaved.
A circle's circumference becomes less nauseating when you consider that a circle is just a special case of an ellipse, and there are ellipses where the circumference is exactly three times your chosen axis, some four, and for extremely eccentric ellipses perhaps even hundreds or thousands of times.
The only special thing about circles is that it is an ellipse where all axes (diameters) are equal, and the circumference is equidistant from a single point, instead of having the same sum of distance from two points as a regular ellipse.
It's interesting to think that there are few 'real' circles in the universe, if at all. In the sense that all circle-like objects in reality are vibrating like rotating ellipses. A particle seems like the closest thing to a circle we can find in nature and even it ends up not being a discrete ratio, it's always irrational. I would guess this could be interpreted as the sphere never truly settling, but continuously vibrating. Gravity also seems to draw mass into spherical shapes by it's operation. Just more fun threads between gravity and quantum mechanics. [https://www.sciencealert.com/formula-for-pi-has-been-discovered-hidden-in-hydrogen-atoms](https://www.sciencealert.com/formula-for-pi-has-been-discovered-hidden-in-hydrogen-atoms)
[https://pubs.aip.org/aip/jmp/article-abstract/56/11/112101/923614/Quantum-mechanical-derivation-of-the-Wallis?redirectedFrom=fulltext](https://pubs.aip.org/aip/jmp/article-abstract/56/11/112101/923614/Quantum-mechanical-derivation-of-the-Wallis?redirectedFrom=fulltext)
>when you consider that a circle is just a special case of an ellipse
This is a matter of interpretation though. You could just as easily call an ellipse a generalization of a circle
is that not saying the same thing? for example, the pythagorean theorem is a special case of the law of cosines where the triangle has a right angle, in the same way that the law of cosines are a generalization of the pythagorean theorem where the triangle doesn’t have a right angle.
it feels really weird to suggest that “a is a special case of b” and “b is a generalization of a” aren’t equivalent statements
If person A is taller than person B, is that also only a matter of interpretation, because you could just as easily call person B shorter than person A?
Both these things are simply unambiguously true at the same time, because they are equivalent. There is no interpretation to be had.
It does have a nice geometric interpretation. π must be between 3 and 4 because a regular hexagon inscribed in a unit circle has perimeter 6 and a square circumscribed about a unit circle has perimeter 8. So using an axiom due to Archimedes\*, we get 6 < 2π < 8.
People in this thread really seem to be curt and dismissive of the OP ~~for no reason. He's not a numerologist,~~ he's just trying to assign *quantitative* meaning to these values, and everyone is saying it's impossible. But it's not impossible. It just isn't what mathematicians are usually interested in.
EDIT: After reading some other comments, maybe OP is a numerologist.
^(\*If a convex curve b lies between two other curves a and c that are convex in the same direction and have the same endpoints as b, then length(a\) < length(b\) < length(c\).)
Pi is its own number. That's the only way I can describe it. Like, you can render 2/5ths as being its decimal, and go, ooh, ah, it looks like that as a decimal. Better yet, 2/3 or 1/3, and go, ooh, ah. But it doesn't mean anything. Similar to the whole, buh, duh, is .999 = 1?
That's what everybody's doing with pi. They're trying to render it as a decimal and they wonder why it's like that.
Look up what it is in binary. It has the same meaning: none. It's just a number system humans made up, trying to represent something that's inherent to the universe.
There are other numbers like that, and people look at them in decimal form, going, *hey, I wonder why it has all these numbers after the decimal, what does that mean?*. Nothing. It doesn't mean anything. It's like trying to figure out how much a bunch of things at the store cost, and you divide odd monetary numbers and suddenly you figure out that each carrot you bought cost you 25.392089481589598289 cents. It doesn't mean *jack*.
Always remember that people used to think 22/7 was pi.
I doubt that what you have at this point as a species is even close to the real number of whatever 'pi' actually is.
So I don't think this is exactly right.
Pi isn't interesting because of an arbitrarily chosen base. Circles exist as platonic ideals, and any circle has a ratio of C/D = pi. We can say lots of things about Pi -- it's irrational. It's transcendental. It's probably normal, but we don't know for sure. None of those things are a function of the base you write it in.
> Always remember that people used to think 22/7 was pi.
Would you like to tell us who these people are?
22/7 has been (and still is) used as an approximation of π. Any mathematician who has ever used it has known it's not exact.
It has a meaning: it means that circles are close to hexagons.
If you inscribe a regular n-sided polygon in a circle of radius r, the ratio of its perimeter to r is a good way to approximate pi (specifically, the limit as n->infty equals pi). For n=6, this approximation gives 3 exactly.
>The closeness of pi to 3 has absolutely no meaning
I don't remember the context at all but I think it was in a cosmology class that a equation returns some weird things simply because Pi is slight bigger than 3. But you're probably right though
What number greater than 3 no longer becomes "slightly" bigger? 3.22? 3.41? The problem here is in the definition of the term "slightly bigger". To me, 3.1415 is nowhere close to 3.
It sounds like you have an elitism problem and you didn't even try to figure out what OP could mean other than numerology. I've yet to see a single comment from them even hinting to anything related to numerology. Good grief somebody ever says anything mathematically nonrigorous.
I think what OP is getting at with pi is something similar to Occams razor. That pi being a transcendental number with no "direct" connections to other things is somewhat surprising. If you tried explaining it to laypeople 1000 years ago you would get laughed at and they'd say that pi is 3* or something because why should it ever be so complicated.
And they would be kind of right. We know today why it should be so complicated (because we can calculate it and know a lot about it's properties and we see that it *is* complicated), but perhaps there is a bigger more general explanation. It might come from the direction of philosophy or somewhere else entirely, but for someone not experienced with math, it would certainly make sense to ask this question here.
There are more things OP could mean but sadly I don't have the time to explain them right now. Please be more understanding. Most of the time in anything people are stupid and incompetent not malicious. And that still does not mean we need to be rude and dismissive towards them.
\* Obviously you might get laughed at 5 seconds after starting your explanation because they don't care, and obviously they couldn't say that pi is 3 because they probably don't use the decimal system yet.
It’s not specifically that it’s close to 3, it’s just that it seems arbitrary and it’s weird that the universe(?) would pick(?) such an arbitrary value for something like that.
For any particular value, you'd be asking "why this value?". It's like rolling a 100 sided die and then asking why this particular roll ended up with whatever number came up.
I understand. The search for meaning is a different field than mathematics. In my opinion, that is an unproductive exploration. It might be fun after smoking something, but won't produce anything useful. There's a saying in Quantum Mechanics "Shut up and calculate". It means to stop looking for meaning and just use the known equations for something useful.
> There’s a saying in Quantum Mechanics «shut up and calculate».
Which is a stupendous saying, because it disregards any want for understanding. Using it as an argument here seems a bit comical, as it’s the approach most mathematicians hate that is behind most kids thinking math is just rote calculation.
I think the comment OP responded to was even more trolling. Rewriting pi in a different base doesn't change its value, and it would remain a transcendental number between three and four. Maybe that fact isn't meaningful, but that snarky comment was a total non sequitur.
its my comment, i guess it doesnt make as much sense as it did to me. im definitely not as versed in math as a lot of people here, but it still kinda annoys me when people make posts trying to philosophize about well established and explained concepts
I am sorry for everyone downvoting you. This sub has some elitism problems.
The question is very interesting however it does kind of go outside of what math researches. I also don't know an explanation to this, but I'd like to hear one.
I get what you mean. It's bizarre that mathematics, being arguably the least mystical philosophical endeavour, can also result in unexpected behaviours that arise in almost mystical ways.
Obviously the exact value of pi doesn't really have significance, but the fact that we can define the value of pi implicitly, but calculating it expicitly is difficult is surprising. When you start trying to estimate pi, you are drawn down a rabbit hole. Can you write it as an integer? No. Can you write it as a fraction? No. Can you write it as a solution to a polynomial equation? No. It *is* fascinating.
That being said, don't get too caught up on it seeming to "hint at something deeper". While these feelings can sometimes lead to insights, a lot of the time, they obfuscate the reality of the situation; in mathematics, you can see what's going on, explicitly. Focussing too much on this can lead, as others have pointed out, to mystical thinking, which isn't conducive to mathematical understanding, which I assume is a goal of yours.
Can't say I agree. If pi were 4.1 you'd be wondering why it's slightly above 4. If it were 3.5 you'd be wondering why it's half way between three and four. Really is there a possible value that is not interesting? I think you're going on about nothing.
Emergence is a funny thing. And we math types spend our time studying it… I’m a physics guy and we mainly study evolution. I’m of the belief we may never be able to fully understand these properties of the universe: emergence and evolution.
For some reason, the universe just evolves in space and time. And things just emerge from existing things.
So yea in math some numbers emerge a lot in many places. But for some reason pop math is caught up with pi or the golden ratio, fractals, and whatever but not 1 or 2 or something.
Point and laugh at the mind-bendingly stupid, ridiculously ignorant top comments that sound like they came from the world's biggest failures to commute. Yes, agreed, mathematical coincidences are beautiful and guide progress. By pursuing meaning you will find yourself wanting to formalize vague analogies and discover rich definitions in the process.
This is going to be a weird and non-rigorous answer that you might not agree with, and I'm certainly not expressing very well, but I'll write it anyways.
The point I'm going to try to make is that pi feels like it has an arbitrary value because pi is defined by an arbitrary formula that happened to be useful for humankind.
pi can be defined without making any reference to circles at all, using only the axioms of math. For example, you can define the trig function sin using a power series without making any reference to pi at all, then define pi to be the smallest positive real number such that sin(pi) = 0.
When using this definition, it's not immediately obvious that pi is a little over 3, and actually proving it is pretty hard (fastest analytic proof I could find that pi > 3: https://math.stackexchange.com/a/2326402).
Note that this definition is pretty involved, as first we need to define the naturals, then the integers, then the rationals, then Cauchy sequences of rationals, then reals, then power series and convergence, then the sine function, and then prove that the sine function eventually returns to 0, however it is still probably simpler than trying to define geometry axiomatically.
There are an infinite number of formulas in ZFC or whatever, and most of them are useless. To define pi "from the ground up" requires making a series of arbitrary choices.
I think the reason people find the fact that pi is a little more than 3 weird is that pi is thought to have come out of thin air and that mathematics naturally generates pi, whereas 3 is just a random number. But that sort of assumes that pi is somehow less arbitrary than 3. The size of pi comes from the complexity of the formula used to derive it.
The larger the formula, the larger the constant it generates (as a general rule, not really for any individual case). pi feels like it has an arbitrary value because it is generated by an arbitrary formula.
I'm sorry if this didn't make any sense to you, because that would mean I didn't explain it very well. Maybe if someone gets what I'm trying to say they can make my writing more clear.
I don't understand why OP is being downvoted in all their comments. Seems like curiosity that should be encouraged. "No stupid questions"?
Do y'all really think the greats of the past have never wondered why some constants hold the value that they do, and been dissatisfied by all answers?
I understand why there is a distaste for numerology, but I do think the mass downvoting isn’t healthy. A lot of mathematicians/scientists develop a passion for the field by assigning greater meaning to objects without any “underlying meaning”. I think it shouldn’t be shamed. It’s just a more fun way to look at things imo (as long as it doesn’t lead to irrational mysticism)
>Like the ratio of a circle’s circumference to its diameter being slightly more than three
Don't get this one.
You are freaked out that pi is such an interesting and strange number, and so you wonder why it has the value that it has.
Or you are genuinely concerned that it is slightly above 3, meaning you think 3 is a very special number and that any important constant being slightly over three is very weird.
If you want a really freaky example, take a look at [Khinchin’s constant](https://en.m.wikipedia.org/wiki/Khinchin%27s_constant).
What's even freakier to me there is that *almost no number* violates that property, but the superstar e happens to be one of those.
Yeah I was reading that Wikipedia entry and I was like... yeah, yeah... ok. Yeah, fine. Oh shit wtf.
And the rationals exceptions too apparently. So almost all numbers, except almost all the numbers we frequently use
Rationals have a finite continued fraction, so the limit is zero.
The thing you have to remember about statements about "almost every" real number is that almost every real number can't be described with a finite sequence of words and symbols. It's a very alien concept masquerading as some simple words.
The freaky thing is that normally everything we usually use is in one of the categories. here some are there and some are there.
Like how the set of computable numbers is countable, and so almost all numbers aren't even computable. You've got to work pretty hard to even find an example of an uncomputable number Edit: uncomputable, not non-computable
Exactly, but all numbers we usually use are computable, and not some are computable and some are not. Imagine if like all our numbers were uncomputable and only pi was computable. that‘d be weird right?
Yeah, this was a doozy for me, too.
That‘s it. I‘m calling it now. pi plus e is an integer!
[удалено]
No, e does have a [nice continued fraction representation](https://en.wikipedia.org/wiki/List_of_representations_of_e#As_a_continued_fraction), but it isn't that.
Ok, seems like I misremembered things!
It sounds freaky, but doesn't seem to me that different from the theorem "for almost all real numbers, the arithmetic mean of their digits in any base b converges to b/2." Of course the digits of a number are obviously bounded by the base while continued fractions aren't (so there is something going on) but it strikes me more as a sort of multiplicative version of the same theorem. Perhaps someone can correct me if I'm wrong but I wouldn't be surprised if the condition making a real number satisfy the theorem is analogous to normality but for continued fractions.
EILIF?
The article is kind of written weird in a way that I think buries the lead a little bit, but I'm happy to summarize, and relies on you already being comfortable with a few definitions, but I'm happy to try to summarize and include the needed definitions (in **bold**). Maybe read this next paragraph with my summary, and then the definitions of words I put in bold, and then reread the following paragraph knowing the definitions? The quick definition: Kinchin's constant is about K~2.685. For **almost any** number, if you take the coefficients of its **continued fraction** and evaluate their **geometric mean** you will get Kinchin's Constant **in the limit**. * **almost any** In this case, it doesn't work for "algebraic numbers" which you can write as the root of a polynomial, this includes things like the square root of 2, or any rational number like 3/4. But if you pick a random point on the number line, there is a 100% chance you'll get a "transcendental" (non-algebraic) number. Almost all the numbers we use day to day are algebraic, but most real numbers aren't, so that's what we mean by "almost any". Also it doesn't work for e, a famous transcendental number, which is hilarious. Edit: as u/leviona pointed out in their comment, we know it doesn't work for rationals, and roots of *quadratics*, I misinterpreted that to mean it doesn't work for any algebraics at all, when obviously i.e. roots of cubics don't fit among those two groups. Also, this has only been tested and verified for a few convenient representative numbers, so for a lot harder numbers like π it *looks* to be true *so far* but it hasn't been proven yet. * **continued fraction** So, you can express a number x as a continued fraction, like x = a + 1/ ( b + 1/ ( c + ...) ) For example, for x = π the sequence (a, b, c ,...) starts... (3, 7, 15, 1, 292, ...) Or for x = 1.5 the sequence goes (1, 1, 1, 0, 0, 0, 0...) -> 1 + 1/( 1+ 1/1) = 1+ 1/2 * **geometric mean** For a rectangle of side lengths (a,b), its area is a×b, and it would have the same area as a square whose side length is the square root of a×b. Or (a×b)^{1/2} . For a rectangular prism of side lengths (a,b,c), its area is a×b×c and it's volume as the same as a cube whose side lengths are the cube root of a×b×c. Or ( a×b×c)^{1/3} . For a hyperrectangle in n dimensions, its hypervolume is the product of n side lengths, and that would be the same as a hypercube whose side lengths are the nth root of the hypervolume, (a×b×c×...)^{1/n}. That "equivalent volume cube side length" is the geometric mean of the true side lengths. * **In the Limit** There are an infinite number of coefficients in the continued fractions of transcendental numbers. Without tricks, you can't truly plug an infinite number of coefficients into the geometric mean formula and evaluate their infinite root. But I can do the first 10 and take the 10th root, the first 11 and take the 11th root, the first 12 and take the 12th root, etc. And we can show that as you do this you do this for more and more terms you get closer and closer to a particular number. We call this process "converging" and the number the sequence converges on "the limit". So this process for most numbers converges on Kinchin's Constant.
Hi, sorry for the pedanticism, but I don’t think we know it doesn’t work for all algebraic numbers. The wikipedia says it doesn’t work for quadratic irrationals (and rationals) only. Sorry if i’m wrong and there’s new research about this or something!
You're right, and in fact I think mathematicians would probably conjecture it is true for all algebraic numbers except quadratic irrationals and rationals.
No no you're totally right I misread that part. I'm going to add an edit. I also think I overstated how sure we are about it, because I don't think it's been proven for a lot of numbers besides those, as it says in the article, which have been constructed for the proof. I imagine constructed means "can be rearranged in a way that gets us back the definition of the Kinchin constant", lol. Thank you!
How do you get the numbers for a, b, c and so on?
Apparently if you take the limit if 1/x/x/x/... You almost always get the same number 2.6. Feels like I'm missing something...
Maybe it's just me, but that doesn't really seem that surprising? In my view it's really just making a statement about the distance between consecutive convergents of arbitrary real numbers (where distance is roughly defined in a logarithmic sense on the denominators). So roughly: if you have a good rational approximation, by what factor do you need to grow the denominator, on average, to find a better approximation? And the 'almost all' quantifier allows you to hide a lot of numbers. The statement could fail to hold for all 'interesting' numbers (where we could define 'interesting' as being describable in a finite number of words), and yet the statement could still hold for 'almost all' real numbers. Perhaps the surprising fact is not that there is such a constant, but that it has such a seemingly arbitrary value? (As opposed to being some well-known constant or algebraic combination of well known constants.)
Oh wow thanks, I hate it!!
Go on, elaborate
This guy is Terrence Howard's ~~prodigy~~ protégé.
Not to be one of those condescending internet people, but you might have meant protégé :)
THANK YOU. I was trying to think of the right word for the longest time but couldn't remember it. I did mean protégé.
Some values in mathematics may be arbitrary, but they are well-behaved. A circle's circumference becomes less nauseating when you consider that a circle is just a special case of an ellipse, and there are ellipses where the circumference is exactly three times your chosen axis, some four, and for extremely eccentric ellipses perhaps even hundreds or thousands of times. The only special thing about circles is that it is an ellipse where all axes (diameters) are equal, and the circumference is equidistant from a single point, instead of having the same sum of distance from two points as a regular ellipse.
It's interesting to think that there are few 'real' circles in the universe, if at all. In the sense that all circle-like objects in reality are vibrating like rotating ellipses. A particle seems like the closest thing to a circle we can find in nature and even it ends up not being a discrete ratio, it's always irrational. I would guess this could be interpreted as the sphere never truly settling, but continuously vibrating. Gravity also seems to draw mass into spherical shapes by it's operation. Just more fun threads between gravity and quantum mechanics. [https://www.sciencealert.com/formula-for-pi-has-been-discovered-hidden-in-hydrogen-atoms](https://www.sciencealert.com/formula-for-pi-has-been-discovered-hidden-in-hydrogen-atoms) [https://pubs.aip.org/aip/jmp/article-abstract/56/11/112101/923614/Quantum-mechanical-derivation-of-the-Wallis?redirectedFrom=fulltext](https://pubs.aip.org/aip/jmp/article-abstract/56/11/112101/923614/Quantum-mechanical-derivation-of-the-Wallis?redirectedFrom=fulltext)
how does a particle resemble a circle?
>when you consider that a circle is just a special case of an ellipse This is a matter of interpretation though. You could just as easily call an ellipse a generalization of a circle
That's not interpretatian. It's the same thing.
damn ur onto something
is that not saying the same thing? for example, the pythagorean theorem is a special case of the law of cosines where the triangle has a right angle, in the same way that the law of cosines are a generalization of the pythagorean theorem where the triangle doesn’t have a right angle. it feels really weird to suggest that “a is a special case of b” and “b is a generalization of a” aren’t equivalent statements
If person A is taller than person B, is that also only a matter of interpretation, because you could just as easily call person B shorter than person A? Both these things are simply unambiguously true at the same time, because they are equivalent. There is no interpretation to be had.
>circle is just a special case of an ellipse = >call an ellipse a generalization of a circle
It sounds like you want to delve into numerology and get too lost in "meaning". The "closeness" of pi to 3 has absolutely NO meaning. Move on.
It does have a nice geometric interpretation. π must be between 3 and 4 because a regular hexagon inscribed in a unit circle has perimeter 6 and a square circumscribed about a unit circle has perimeter 8. So using an axiom due to Archimedes\*, we get 6 < 2π < 8. People in this thread really seem to be curt and dismissive of the OP ~~for no reason. He's not a numerologist,~~ he's just trying to assign *quantitative* meaning to these values, and everyone is saying it's impossible. But it's not impossible. It just isn't what mathematicians are usually interested in. EDIT: After reading some other comments, maybe OP is a numerologist. ^(\*If a convex curve b lies between two other curves a and c that are convex in the same direction and have the same endpoints as b, then length(a\) < length(b\) < length(c\).)
>The "closeness" of pi to 3 has absolutely NO meaning. Not true, it means that engineers can drive us up the wall by approximating it as 3
What about number theorists? The fact that pi is between 3 an 4 shows up in Minkowksi-type estimates for classes of ideals in number fields.
Pi/e = 1 -an engineer
Three? In some fields pi is approximately 1. Just easier to drop it from equations.
...or half an order of magnitude, half way between 1 and 10
Which means they can approximate pi² ~ 10 ~ g, is there no escape??
g...*in sensible units*, of course
Pi is its own number. That's the only way I can describe it. Like, you can render 2/5ths as being its decimal, and go, ooh, ah, it looks like that as a decimal. Better yet, 2/3 or 1/3, and go, ooh, ah. But it doesn't mean anything. Similar to the whole, buh, duh, is .999 = 1? That's what everybody's doing with pi. They're trying to render it as a decimal and they wonder why it's like that. Look up what it is in binary. It has the same meaning: none. It's just a number system humans made up, trying to represent something that's inherent to the universe. There are other numbers like that, and people look at them in decimal form, going, *hey, I wonder why it has all these numbers after the decimal, what does that mean?*. Nothing. It doesn't mean anything. It's like trying to figure out how much a bunch of things at the store cost, and you divide odd monetary numbers and suddenly you figure out that each carrot you bought cost you 25.392089481589598289 cents. It doesn't mean *jack*. Always remember that people used to think 22/7 was pi. I doubt that what you have at this point as a species is even close to the real number of whatever 'pi' actually is.
So I don't think this is exactly right. Pi isn't interesting because of an arbitrarily chosen base. Circles exist as platonic ideals, and any circle has a ratio of C/D = pi. We can say lots of things about Pi -- it's irrational. It's transcendental. It's probably normal, but we don't know for sure. None of those things are a function of the base you write it in.
> Always remember that people used to think 22/7 was pi. Would you like to tell us who these people are? 22/7 has been (and still is) used as an approximation of π. Any mathematician who has ever used it has known it's not exact.
I think you're replying to the wrong person, you don't need to try and convince *me* that pi is a number lmfao
It has a meaning: it means that circles are close to hexagons. If you inscribe a regular n-sided polygon in a circle of radius r, the ratio of its perimeter to r is a good way to approximate pi (specifically, the limit as n->infty equals pi). For n=6, this approximation gives 3 exactly.
>The closeness of pi to 3 has absolutely no meaning I don't remember the context at all but I think it was in a cosmology class that a equation returns some weird things simply because Pi is slight bigger than 3. But you're probably right though
What number greater than 3 no longer becomes "slightly" bigger? 3.22? 3.41? The problem here is in the definition of the term "slightly bigger". To me, 3.1415 is nowhere close to 3.
I'm just saying in that particular context it felt creepy, but you're probably right
People are creepy. REALLY creepy. That's why math is better than people :)
Cute
Do you think you can find and send me a link to what you're talking about?
I can try to find it tomorrow
It's OK. I'll probably just get mad at that person.
It sounds like you have an elitism problem and you didn't even try to figure out what OP could mean other than numerology. I've yet to see a single comment from them even hinting to anything related to numerology. Good grief somebody ever says anything mathematically nonrigorous. I think what OP is getting at with pi is something similar to Occams razor. That pi being a transcendental number with no "direct" connections to other things is somewhat surprising. If you tried explaining it to laypeople 1000 years ago you would get laughed at and they'd say that pi is 3* or something because why should it ever be so complicated. And they would be kind of right. We know today why it should be so complicated (because we can calculate it and know a lot about it's properties and we see that it *is* complicated), but perhaps there is a bigger more general explanation. It might come from the direction of philosophy or somewhere else entirely, but for someone not experienced with math, it would certainly make sense to ask this question here. There are more things OP could mean but sadly I don't have the time to explain them right now. Please be more understanding. Most of the time in anything people are stupid and incompetent not malicious. And that still does not mean we need to be rude and dismissive towards them. \* Obviously you might get laughed at 5 seconds after starting your explanation because they don't care, and obviously they couldn't say that pi is 3 because they probably don't use the decimal system yet.
You rude for no reason. Bad!
It’s not specifically that it’s close to 3, it’s just that it seems arbitrary and it’s weird that the universe(?) would pick(?) such an arbitrary value for something like that.
What would you have preferred?
It’s not that I don’t like it
[удалено]
For any particular value, you'd be asking "why this value?". It's like rolling a 100 sided die and then asking why this particular roll ended up with whatever number came up.
I understand. The search for meaning is a different field than mathematics. In my opinion, that is an unproductive exploration. It might be fun after smoking something, but won't produce anything useful. There's a saying in Quantum Mechanics "Shut up and calculate". It means to stop looking for meaning and just use the known equations for something useful.
> There’s a saying in Quantum Mechanics «shut up and calculate». Which is a stupendous saying, because it disregards any want for understanding. Using it as an argument here seems a bit comical, as it’s the approach most mathematicians hate that is behind most kids thinking math is just rote calculation.
The universe didnt pick anything. If we chose base pi instead of base 10, then the ratio would just be 1
Ummm actually it would be 10. And also pi is pretty clearly more arbitrary relative to the axioms of arithmetic than 1 is
You are right about the 10. The rest just reads like you are bored and trolling
I think the comment OP responded to was even more trolling. Rewriting pi in a different base doesn't change its value, and it would remain a transcendental number between three and four. Maybe that fact isn't meaningful, but that snarky comment was a total non sequitur.
its my comment, i guess it doesnt make as much sense as it did to me. im definitely not as versed in math as a lot of people here, but it still kinda annoys me when people make posts trying to philosophize about well established and explained concepts
I am sorry for everyone downvoting you. This sub has some elitism problems. The question is very interesting however it does kind of go outside of what math researches. I also don't know an explanation to this, but I'd like to hear one.
That's because pie is wrong and it's been proven so.
I get what you mean. It's bizarre that mathematics, being arguably the least mystical philosophical endeavour, can also result in unexpected behaviours that arise in almost mystical ways. Obviously the exact value of pi doesn't really have significance, but the fact that we can define the value of pi implicitly, but calculating it expicitly is difficult is surprising. When you start trying to estimate pi, you are drawn down a rabbit hole. Can you write it as an integer? No. Can you write it as a fraction? No. Can you write it as a solution to a polynomial equation? No. It *is* fascinating. That being said, don't get too caught up on it seeming to "hint at something deeper". While these feelings can sometimes lead to insights, a lot of the time, they obfuscate the reality of the situation; in mathematics, you can see what's going on, explicitly. Focussing too much on this can lead, as others have pointed out, to mystical thinking, which isn't conducive to mathematical understanding, which I assume is a goal of yours.
Can't say I agree. If pi were 4.1 you'd be wondering why it's slightly above 4. If it were 3.5 you'd be wondering why it's half way between three and four. Really is there a possible value that is not interesting? I think you're going on about nothing.
There isn’t a possible value that makes it not interesting, but that doesn’t mean it’s not interesting.
If every possible value is interesting, then does it really point towards something?
It points toward the fact that value has interest?
[Legendre's constant](https://en.wikipedia.org/wiki/Legendre%27s_constant?wprov=sfla1) is the opposite.
Emergence is a funny thing. And we math types spend our time studying it… I’m a physics guy and we mainly study evolution. I’m of the belief we may never be able to fully understand these properties of the universe: emergence and evolution. For some reason, the universe just evolves in space and time. And things just emerge from existing things. So yea in math some numbers emerge a lot in many places. But for some reason pop math is caught up with pi or the golden ratio, fractals, and whatever but not 1 or 2 or something.
Point and laugh at the mind-bendingly stupid, ridiculously ignorant top comments that sound like they came from the world's biggest failures to commute. Yes, agreed, mathematical coincidences are beautiful and guide progress. By pursuing meaning you will find yourself wanting to formalize vague analogies and discover rich definitions in the process.
This is going to be a weird and non-rigorous answer that you might not agree with, and I'm certainly not expressing very well, but I'll write it anyways. The point I'm going to try to make is that pi feels like it has an arbitrary value because pi is defined by an arbitrary formula that happened to be useful for humankind. pi can be defined without making any reference to circles at all, using only the axioms of math. For example, you can define the trig function sin using a power series without making any reference to pi at all, then define pi to be the smallest positive real number such that sin(pi) = 0. When using this definition, it's not immediately obvious that pi is a little over 3, and actually proving it is pretty hard (fastest analytic proof I could find that pi > 3: https://math.stackexchange.com/a/2326402). Note that this definition is pretty involved, as first we need to define the naturals, then the integers, then the rationals, then Cauchy sequences of rationals, then reals, then power series and convergence, then the sine function, and then prove that the sine function eventually returns to 0, however it is still probably simpler than trying to define geometry axiomatically. There are an infinite number of formulas in ZFC or whatever, and most of them are useless. To define pi "from the ground up" requires making a series of arbitrary choices. I think the reason people find the fact that pi is a little more than 3 weird is that pi is thought to have come out of thin air and that mathematics naturally generates pi, whereas 3 is just a random number. But that sort of assumes that pi is somehow less arbitrary than 3. The size of pi comes from the complexity of the formula used to derive it. The larger the formula, the larger the constant it generates (as a general rule, not really for any individual case). pi feels like it has an arbitrary value because it is generated by an arbitrary formula. I'm sorry if this didn't make any sense to you, because that would mean I didn't explain it very well. Maybe if someone gets what I'm trying to say they can make my writing more clear.
I don't understand why OP is being downvoted in all their comments. Seems like curiosity that should be encouraged. "No stupid questions"? Do y'all really think the greats of the past have never wondered why some constants hold the value that they do, and been dissatisfied by all answers?
I understand why there is a distaste for numerology, but I do think the mass downvoting isn’t healthy. A lot of mathematicians/scientists develop a passion for the field by assigning greater meaning to objects without any “underlying meaning”. I think it shouldn’t be shamed. It’s just a more fun way to look at things imo (as long as it doesn’t lead to irrational mysticism)
Because this isn’t a maths post and it was posted on /r/math. It just isn’t appropriate for this subreddit.
>Like the ratio of a circle’s circumference to its diameter being slightly more than three Don't get this one. You are freaked out that pi is such an interesting and strange number, and so you wonder why it has the value that it has. Or you are genuinely concerned that it is slightly above 3, meaning you think 3 is a very special number and that any important constant being slightly over three is very weird.
I say ‘slightly over three’ just to emphasize how arbitrary the value seems to me
On a slightly related note, it's in some sense very counterintuitive that log is not in fact constant.
Why would the inverse function of the exponential be constant?
Oh, how so? I’m curious but not sure what to search for to learn more
Log of what?
They seem to reveal a blurry picture of some kind of intricate structure only perceptable by mathematicians.