Can (PI) exist?

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Symbols
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Can (PI) exist?

Postby Symbols » Tue Nov 09, 2010 4:31 am UTC

Concerning the proof of Pi: It uses a bunch of symmetrical polygons inscribed into a circle, right? And as the size of the edges goes up, then the circumference of that polygon will get closer to the value of 2(PI)R? And similarly if you divided by 2R and increase the number of edges, you get closer to just (PI), right?

Then basically, to apply PI in the mathematical concepts that we use for the world around us, we are conceptually imagining the world as a world of polygons, no?

And thus, if we are to imagine this, then that means that (PI) can not exist in this reality of polygons. But if we assume (PI) does exist, then that means (PI) becomes the inherent unit of measure for our reality and that polygons then can not exist.

So polygons and circles can't exist at the same time? :|

Does this suggest that reality might not have an inherent unit of measure? And that the world is an overall illogical construct that we can't ever hope to fully comprehend because we must make inaccurate associations of measure to compare and interpret it?

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Re: Can (PI) exist?

Postby pepsi » Tue Nov 09, 2010 4:47 am UTC

If you'd like you can argue that irrational numbers don't exist in the same way infinity doesn't exist. They are both found through limits so to say one doesn't exist is to say the other doesn't either.

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Re: Can (PI) exist?

Postby nehpest » Tue Nov 09, 2010 5:47 am UTC

Symbols wrote:And thus, if we are to imagine this, then that means that (PI) can not exist in this reality of polygons. But if we assume (PI) does exist, then that means (PI) becomes the inherent unit of measure for our reality and that polygons then can not exist.


This doesn't follow. The Euler constant [imath]e[/imath] is similar to [imath]\pi[/imath] in that it is irrational (and, by the way, transcendental); by your logic, the existence of [imath]e[/imath] would necessarily preclude the existence of [imath]\pi[/imath], since they don't form a rational number in ratio to one another.
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Re: Can (PI) exist?

Postby Eebster the Great » Tue Nov 09, 2010 8:33 am UTC

How do I had continuum?

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Re: Can (PI) exist?

Postby nash1429 » Tue Nov 09, 2010 8:53 am UTC

Mathematical philosophy does not change the reality of the universe we live in. With only 39 digits of pi a circle can be constructed with a diameter equal to that of the known universe accurate to one atom of hydrogen.

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Re: Can (PI) exist?

Postby supremum » Tue Nov 09, 2010 9:32 am UTC

Symbols wrote:Concerning the proof of Pi: It uses a bunch of symmetrical polygons inscribed into a circle, right?


What do you mean "proof of pi?" Do you mean the calculation of the ratio of the circumference of a circle to its diameter? Are you trying to argue that this number does not exist or that it is not the limit of the perimeters of those polygons? The real numbers are defined to be complete, and that sequence can be shown to be a cauchy sequence so it has to have some limit.

Furthermore, there are tons of other ways to calculate pi, so I really don't get what you're trying to say here. It looks like you go from saying "you can calculate pi by using polygons to approximate a circle" to saying "circles don't exist." And then you say some nonsense about "pi becomes the inherent unit of measure for reality"- what are you trying to say here? How the heck did you come to this conclusion?

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Re: Can (PI) exist?

Postby Sagekilla » Tue Nov 09, 2010 5:29 pm UTC

I think what he's forgetting to realize is that you have to look at the infinite limit of cutting up
a circle into straight line segments. If you never take the limit, then yes the circle is just a polygon
with a crap-ton of line segments so that it looks nearly circular. Once you go to infinitely many sides,
then the approximation becomes an equality.

Now, when you start trying to apply this mathematical model of a circle to the world, you'll never get
a true circle. I'm forgetting about the fact that we have things quantized in the real world and so forth,
but the truest circle I could possibly imagine would be a free floating electron (or proton, or neutron)
in space in the ground state.

Now then, reality sure looks continuous to us but if you look at a small enough level you start seeing
discrete things. It's just the discrete units are so small that we see it as something continuous.
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Re: Can (PI) exist?

Postby Eastwinn » Tue Nov 09, 2010 10:37 pm UTC

Circles and polygons don't exist in the real world at all!
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Re: Can (PI) exist?

Postby antonfire » Wed Nov 10, 2010 12:14 am UTC

nehpest wrote:by your logic, the existence of [imath]e[/imath] would necessarily preclude the existence of [imath]\pi[/imath], since they don't form a rational number in ratio to one another.
Probably. Nobody's proved this yet.

Symbols wrote:Does this suggest that reality might not have an inherent unit of measure? And that the world is an overall illogical construct that we can't ever hope to fully comprehend because we must make inaccurate associations of measure to compare and interpret it?
Yes and yes. Neither of these things have much to do with pi being irrational, though.
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Re: Can (PI) exist?

Postby nehpest » Wed Nov 10, 2010 1:49 am UTC

antonfire wrote:
nehpest wrote:by your logic, the existence of [imath]e[/imath] would necessarily preclude the existence of [imath]\pi[/imath], since they don't form a rational number in ratio to one another.
Probably. Nobody's proved this yet.

Symbols wrote:Does this suggest that reality might not have an inherent unit of measure? And that the world is an overall illogical construct that we can't ever hope to fully comprehend because we must make inaccurate associations of measure to compare and interpret it?
Yes and yes. Neither of these things have much to do with pi being irrational, though.


Bolded: That's true, fair enough. It seems pretty unlikely to me, but it is still an open question.

As for the rest: over on the science side of the house, we would tell you that sure, reality can have a fundamental unit of measure, and that what it is depends on what you're measuring. For distance, we're pretty sure that nothing can ever be smaller than the Planck length; if my understanding is correct, that is also the length of which all other lengths are a multiple.
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Re: Can (PI) exist?

Postby letterX » Wed Nov 10, 2010 2:32 am UTC

nehpest wrote:that is also the length of which all other lengths are a multiple.


Can we kill this rumor please? It tends to cause people to have some pretty crazy ideas about exactly what quantization means.

So, let's consider two possible facts:
  • There's a fundamental length, of which all other lengths are multiples
  • Special relativity says that length is relative, depending on your given reference frame

I'll let the audience fight it out with Einstein if they want to stick with fact 1.

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Re: Can (PI) exist?

Postby nehpest » Wed Nov 10, 2010 2:50 am UTC

letterX wrote:
nehpest wrote:that is also the length of which all other lengths are a multiple.


Can we kill this rumor please? It tends to cause people to have some pretty crazy ideas about exactly what quantization means.

So, let's consider two possible facts:
  • There's a fundamental length, of which all other lengths are multiples
  • Special relativity says that length is relative, depending on your given reference frame

I'll let the audience fight it out with Einstein if they want to stick with fact 1.


Wow. D'oh. You're absolutely right. :oops:
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Re: Can (PI) exist?

Postby voidPtr » Wed Nov 10, 2010 3:07 am UTC

I think these ontological discussions are better suited for a debate in a philosophy forum than mathematics. Talking about the "existence" of pi is really no different than talking about the existence of the number 2, a perfect circle, or existence of an abstract idea. Taking it to the extreme, it's no different than debating the existence of...existence.

However from this tool we developed called mathematics, perfect circles and pi exist because we defined them to exist. Debating their existence is kind of like you and your wife naming your baby John and then arguing with each other whether his name is really John.

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Re: Can (PI) exist?

Postby ConMan » Wed Nov 10, 2010 4:39 am UTC

Here's how it works: In Euclidean geometry, the ratio of a circle's circumference to its diameter can be proven to be a constant (i.e. this ratio is independent of the size of the circle). We define this ratio to be pi. As it so happens, approximately seventy bajillion other things can be shown to equal pi, some of which allow us to calculate its decimal representation to arbitrary precision, such as bounding polygons and Ramanujan's formulae. So as a mathematical concept, yes - pi does exist. As a physical "object", then I'd argue that it's a semantic and philosophical debate with no concrete answer, since it depends strongly on how you believe mathematics relates to the real world, and to what extent it's possible to have a "perfect" object in the universe.
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Re: Can (PI) exist?

Postby WarDaft » Wed Nov 10, 2010 4:08 pm UTC

A representation of π can easily exist.

π

There. That's one, there are a lot of others.

As for anything taking on the value of π, as a ratio, or whatever, pretty much no. The universe is finite, and an irrational number can't be the ratio of two finite numbers. Things can behave as if they knew an arbitrarily large amount of the expansion of π, but barring it's use in some renormalization process, you can't have something acting as if it's being influenced by all of π. For one, it would take an infinite amount of time at best to actually verify that it was obeying π to full precision.
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Re: Can (PI) exist?

Postby gmalivuk » Wed Nov 10, 2010 4:51 pm UTC

Well, if you're talking about measurements (rather than counting), then pi exists to exactly the same extent that 2 exists. As in, you cannot determine that something is 2 meters long to any more precision than you can determine that it was pi meters long.
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Re: Can (PI) exist?

Postby WarDaft » Wed Nov 10, 2010 8:44 pm UTC

gmalivuk wrote:Well, if you're talking about measurements (rather than counting), then pi exists to exactly the same extent that 2 exists. As in, you cannot determine that something is 2 meters long to any more precision than you can determine that it was pi meters long.


But we can count to 2 quite easily. Counting to π is either somewhat more difficult, or reduced to a bijection to counting to some natural number by using intermediate values defined in terms of π. If we're only measuring to arbitrary precision, then we're treating 2 like a real without taking advantage of the fact that it's also a member of smaller, easier to discern subsets of the reals.

Of course, π could get a free pass into existence by carefully taking part in some renormalization, something that only works if exactly π is used.
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Re: Can (PI) exist?

Postby Symbols » Wed Nov 10, 2010 9:53 pm UTC

Well...erm...okay. Maybe it does fall under philosophy. I was probably a bit too vague with my intentions for posting. But I guess I was curious if anyone has entertained and attempted to rectify the logical implications of inscribing a polygon infinitely into a circle to obtain a result. See, yes, PI is constant; and it is a limit of mathematical operations, which is fine, and a rather neat idea. It's just the fact that it's irrational that seemingly gargles the logical paradigm of current mathematics (but doesn't at all hurt its practical usefulness in reality).

Related tangent:
See PI is an infinite random sequence. We can not predict what digit will come next in its series without first computing the previous iterations of PI. There is absolutely no number in the current state of mathematics that we can compare PI too. And the fact that PI is incomparable to anything else is also what makes it a random sequence. All we can do is completely abstractly label it under a category of numbers we call irrational, which ends up being a label for math we don't truly understand, and then forget about it. This seems like a lazy idea. As far as mathematics is currently concerned, this nature makes it impossible to comprehend given the way we define and discover the concept with the model of mathematics that we have.

Back to the point:
So what I mean is that given any irrational number, what you really have is not a number of finite digits and length, but an infinite random sequence. It just so happens that PI ends up converging to a constant so that it can be approximated, applied practically, and accepted for what it is in mathematics - a limit to infinity of mathematical operations; but it's not a finite length. It's computable, yes, with infinite time, but that's not what bothers me about it. It's the fact that to use PI in practice we can not use it in its infinite form. By approximating, as we call it, we end up with the circumference of a polygon with edges (a rational number), and not a true circle. Thus, our creation of the concept of PI is amazing and unique, but its application in the state of mathematics means that we can only use the circumference of a polygon (approximated PI) to do calculations with a circle. Thus we can leave equations in an irrational form that just use the symbol PI, but to actually apply PI, we can only work with rational numbers, and are only left to work with edges in our mathematical model.

So when I asked if PI existed, I was merely asking if it exists in the model of mathematics that we have because that's our viewpoint for viewing and understanding our world. As far as this problem is concerned, it begs questions and a solution. So I don't see how this is purely a philosophical problem, rather than an important hole to be fixed in the concept of mathematics. I would be eager to hear if anyone has a theoretical suggestion for fixing the hole.

The only thing I've been able to come up with is that, given time to be assumed as a constant and causality to be true, either everything in the universe is made up of a curve or everything is made up of an edge. Because both couldn't exist in a such a situation.

But let's say that time isn't constant, and then causality is assumed to be uncertain. It could be possible that an edge and a circle could exist at the same time if one of them exists as a constant in an infinite time relative to the other object that doesn't have an infinite time. And it wouldn't matter which one is infinite or not, but just that the object with a flowing time will see the infinite time object as a perfect representation of the limit of its definition. The implications of this, however, suggest that the flowing time will never be able to fully understand every detail of the infinite constant. It has to view this object as an interpretation of whatever detail it is currently focusing on and will only be able to understand it fully if its time goes to infinity as well. The greater implications of this as mathematics views the problem suggests that we are the flowing time and the circle is the infinite time object. Now imagine we go to infinity in time, then our random pattern that we make in the universe becomes constant to another object with a flowing time outside of ours, and the problem goes on like this forever, taken to infinity. Infinity encapsulated in infinity taken infinite times.

And we are left with two conclusions, either reality doesn't make logical sense and we just have to accept it, label everything a philosophical argument, and move on, or circles and edges can't coexist and reality does make sense and we just haven't figured it out yet or may not be able to determine even if this is true, but can constantly improve our understanding to have less logical holes.

I highly suggest assuming and maybe even believing, in the sense of a religion, of the latter because it has at least provided humanity with so much discovery and technological advancement. The former might be a good definition for the word insanity because it is the antithesis of the motivation of a life.

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Re: Can (PI) exist?

Postby WarDaft » Wed Nov 10, 2010 11:02 pm UTC

See PI is an infinite random sequence. We can not predict what digit will come next in its series without first computing the previous iterations of PI. There is absolutely no number in the current state of mathematics that we can compare PI too.
Uh... no, no, and no.

1) Pi is not random. In fact, it's not even known to be normal, which would be pretty important for a random sequence.
2) Yes we can. A spigot algorithm exists for Pi, which means we can start calculating it from any digit we want.
3) Almost all numbers are in the same bucket Pi is in. Or if you meant compare as in < and >, the reals are totally ordered.

That's 3 complete misconceptions in as many sentences.
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Re: Can (PI) exist?

Postby Talith » Wed Nov 10, 2010 11:31 pm UTC

I think your main gripe with pi is that it's an irrational number and the concept of irrational numbers seem in some way contradictory (that's fine by the way, some mathematical objects are conceptually hard to justify to yourself). It might help you understand what pi represents if we first just try and 'build' the irrational numbers and we'll see why we actually need the irrational numbers in order to answer some pretty ordinary questions about the rational numbers.

It sounds like you know what a sequence of numbers is (you gave the example of a sequence of polygons) and you're also fine with rational numbers being a useful, and conceptually consistent set of objects. Convergence is something which we like to see happen to a sequence of numbers, for instance the sequence 1, 0.5, 0.25, 0.125, 0.0625, .... (ie the sequence of reciprocals of the powers of two) pretty obviously gets closer and closer to 0; in fact it gets as close to 0 as we like because if someone says 'I bet you can't get within x of 0 in this sequence' you can show that there will be a point in the sequence where EVERY number after that point will be closer than x to 0. Most sequences don't have this nice property, but when they do, we say it converges to that number (in this case the sequence converges to 0). In this case, our sequence happily converges to a point which is already in the set we're working with (we're working in the rational numbers, which we're happy to work with, and 0 is a rational number).

The main problem with the rational numbers though, is that there are some sequences of rational numbers which don't converge to a rational number, they seem to converge to a number which is in some sense 'in between' the rational numbers. Let's take for example the sequence 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, ... where each successive number adds on an extra digit of the decimal expansion of [imath]\sqrt{2}[/imath]. I hope you're ok with this being a perfectly good sequence of rational numbers, even if the square root of two doesn't exist, this is a sequence of numbers which ALL have a finite decimal expansion (and so they're rational). Now then, this sequence will always be points in the rational numbers and by the looks of it, they're getting closer and closer together the further along in the sequence we go; in fact, we can see that the sequence is obviously converging to a number. The only problem is that this sequence, which we're fine until now with being perfectly well defined, is homing in on a number which we don't like, it's homing in on a number which isn't rational (sqrt(2) is irrational, loads of lovely proofs all over the internet).

What do we do? We've got a sequence which looks fine on the outside, defined in terms of objects we're happy with, but they produce something which we're not happy with. The way we patch up this hole is to invent a collection of objects called the irrational numbers, it solves a lot of problems and turns out to have some wonderful properties. The main property in the context of this post though, is that EVERY sequence of rational numbers will either not converge, or it will converge to a number in this new set together with our old set (ie the real numbers). You might then ask the question "well aren't we just going to have the same problem with this new set, could we get sequences of numbers which converge to a number NOT in the reals? we'd just be left back at square one". Happily no, every convergent sequence in the reals will converge to another real number and this is one of the main reasons we all grow up to love the real numbers (yep, even the irrationals).

It turns out there are a lot of lovely, sometimes beautiful, sequences of rational numbers that converge to pi (I highly recommend researching some of these on wikipedia etc.), it would be a HUGE shame if we didn't let them converge and, by letting them, we produce a whole class of numbers that have some wonderful properties and don't seem to break anything that we already know in the maths we built up using rational numbers. I understand that in the context of the real world, you're not very happy with the concept of infinity or being able to get infinitely close to something, but the beauty of mathematics is that we don't have to apply this stuff to the real world (as it happens, most of this stuff is hugely helpful when we apply it to the real world, which is a nice bonus), we built this stuff up because it helped solve a problem in maths that in some sense was [imath]more[/imath] contradictory than the invention itself.

I generally try and stay away from the philosophy of maths because I just don't think it makes for very useful or even interesting discourse, normally circular arguments, but I think it's important for you to realise that definitions in maths are just that, they're definitions and don't have to apply to the real world because they're very happy with their well-definedness in the beautiful world of mathematics.

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Re: Can (PI) exist?

Postby Eebster the Great » Wed Nov 10, 2010 11:41 pm UTC

There are plenty of times when we use an "exact" value of pi, we just don't use its decimal expansion. The same could be said of any irrational number and therefore pretty much any notable mathematical constant.

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Re: Can (PI) exist?

Postby voidPtr » Thu Nov 11, 2010 4:35 am UTC

To add to Taliths post, you're not the first person to be uncomfortable with irrational numbers. [imath]\sqrt{2}[/imath] is a better example than [imath]\pi[/imath] because it was proved to be an irrational number first, and the proof is easier to understand and doesn't involve calculus.

Early Greek mathematicians were quasi-religious about natural numbers and were convinced that the length of a hypoteneuse of a right-angled triangle with two sides equal to 1, [imath]\sqrt{2}[/imath], should be able to be written as fraction of natural numbers. Let's call them a and b.

Someone eventually came along and took math down a rabbit hole by supposing that [imath]\sqrt{2}=a/b[/imath], with a and b as natural numbers with no common factors ,and asking the question, what properties would the numbers a and b have? What they discovered upset the Greek math world. If [imath]\sqrt{2}=a/b[/imath], then a and b must both be even numbers. However, if a and b are both even, then 2 is a common factor to both a and b. This is a contradiction because then 2 is a common factor to both a and b, but a and b do not have any common factors. Therefore it cannot be true that [imath]\sqrt{2}=a/b[/imath].

Lets put it this way. The natural numbers are the simplest patterns and humans have been using them since prehistoric times. But they're still just patterns. Why do they have to be the only patterns?

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Re: Can (PI) exist?

Postby supremum » Thu Nov 11, 2010 11:08 am UTC

Symbols wrote:It just so happens that PI ends up converging to a constant so that it can be approximated

This is yet another problem with this post. Every nondecreasing sequence of real numbers that is bounded above converges.
So if I have some number of the form
[math]r + \sum_{i=1}^{\infty} \frac{x_i}{10^i}[/math]
where each [imath]x_i[/imath] is less than 10 (think of the xi's as "digits") and r is an integer, we certainly know this number is smaller than the value if each "digit" were 9, which gives
[math]r + \sum_{i=1}^{\infty} \frac{9}{10^i}[/math]
which is r+1. So any number written out in decimal notation (even if there are infinitely many numbers after the decimal point) is the limit of a nondecreasing sequence of real numbers. There are absolutely no convergence issues to worry about here.

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Re: Can (PI) exist?

Postby nehpest » Thu Nov 11, 2010 11:22 am UTC

supremum wrote:
Symbols wrote:It just so happens that PI ends up converging to a constant so that it can be approximated

This is yet another problem with this post. Every nondecreasing sequence of real numbers that is bounded above converges.
So if I have some number of the form
[math]r + \sum_{i=1}^{\infty} \frac{x_i}{10^i}[/math]
where each [imath]x_i[/imath] is less than 10 (think of the xi's as "digits") and r is an integer, we certainly know this number is smaller than the value if each "digit" were 9, which gives
[math]r + \sum_{i=1}^{\infty} \frac{9}{10^i}[/math]
which is r+1. So any number written out in decimal notation (even if there are infinitely many numbers after the decimal point) is the limit of a nondecreasing sequence of real numbers. There are absolutely no convergence issues to worry about here.


I'd just like to point out the appropriateness of someone with your username explaining this concept. Unless OP comes back, I'm not sure how much else can be said about this topic.
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Re: Can (PI) exist?

Postby cjameshuff » Thu Nov 11, 2010 4:07 pm UTC

Irrational numbers are irrational because they can't be represented as a ratio, not because they're somehow outside the realm of logical manipulation.


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