Where are the irrational numbers?

[micromass]

I disregard everything and pick the number in the 'middle'.

OK, so what is the point in the middle?? You have an infinite product which extends to both sides, there is no middle...

Yeeess, I know it's dangerous but IMHO so are the subtle points of arithmetic.

Arithmetic is not dangerous. It's very well defined. You just need to follow the rules.

 

[agentredlum]

OK, so what is the point in the middle?? You have an infinite product which extends to both sides, there is no middle...
Arithmetic is not dangerous. It's very well defined. You just need to follow the rules.

I thought that it was obvious i was using middle sarcastially bcause i put quotes around it. Then there was the biggrin...

Just because its not easy to FIND 2/3 in that infinite product doesn't mean it isn't THERE. You can write it down first THEN surround it however you wish, just make sure it works.

 

[micromass]

I thought that it was obvious i was using middle sarcastially bcause i put quotes around it. Then there was the biggrin...

Just because its not easy to FIND 2/3 in that infinite product doesn't mean it isn't THERE. You can write it down first THEN surround it however you wish, just make sure it works.

You can write everything you want to. But it's useless if you can't evaluate it properly...

 

[Hurkyl]

I'm not sure if you know what 'taking a step back' and' looking at the big picture' means.

It means, among other things, to stop focusing on minor details and evaluate what you're doing and why.

So let's try it. You are very fixated on the idea of trying to apply the word "fraction" to a lot of situations where it doesn't fit.

Can you explain to me your motivation for this fixation? Can you describe what you are having to do in order to follow this fixation? Is this approach really a good way to achieve whatever goals you have?



I hate to say things that way, but it seriously looks like you are being self-destructive -- you're not only resisting attempts to examine the idea in your head, but you are actively rejecting knowledge that could be useful for the purpose.

This latest post looks like you are trying to wrap your head around the fact that the rationals are dense in the reals, or possibly that the reals are the Cauchy completion of the rational numbers.

Are you going to continue crippling yourself by refusing to move beyond thoughts like "\sqrt{2} is a fraction in some sense" -- or are you going to start examining just what that "some sense" really is, and try to explain it in terms of existing mathematical ideas or even to devise new mathematical ideas invented just for this purpose, should no existing ones apply?




I'm reminded of the time I first digested the notion of "tangent vector to a manifold". I remember actively disliking the definition I had seen. But rather than just mentally rejecting the notion I had seen, I set out trying to define what I thought a tangent vector ought to be. By the time I had worked out all the difficulties and had something I was happy with, I had pretty much written down exactly what I had originally rejected.

 

[agentredlum]

It means, among other things, to stop focusing on minor details and evaluate what you're doing and why.

So let's try it. You are very fixated on the idea of trying to apply the word "fraction" to a lot of situations where it doesn't fit.

Can you explain to me your motivation for this fixation? Can you describe what you are having to do in order to follow this fixation? Is this approach really a good way to achieve whatever goals you have?
I hate to say things that way, but it seriously looks like you are being self-destructive -- you're not only resisting attempts to examine the idea in your head, but you are actively rejecting knowledge that could be useful for the purpose.

This latest post looks like you are trying to wrap your head around the fact that the rationals are dense in the reals, or possibly that the reals are the Cauchy completion of the rational numbers.

Are you going to continue crippling yourself by refusing to move beyond thoughts like "\sqrt{2} is a fraction in some sense" -- or are you going to start examining just what that "some sense" really is, and try to explain it in terms of existing mathematical ideas or even to devise new mathematical ideas invented just for this purpose, should no existing ones apply?

I'm reminded of the time I first digested the notion of "tangent vector to a manifold". I remember actively disliking the definition I had seen. But rather than just mentally rejecting the notion I had seen, I set out trying to define what I thought a tangent vector ought to be. By the time I had worked out all the difficulties and had something I was happy with, I had pretty much written down exactly what I had originally rejected.

Oh man...

I REJECT NOTHING, I QUESTION EVERYTHING. Questioning everything is not the same as rejecting anything. Wouldn't you think I was a fool if I accepted everything without question? Making my point does not mean i have to reject yours. You think it does but I am not responsible for that. I can see it your way and agree it's useful. You can't see it my way even after i ask for a little latitude. That's not fair.

 

[agentredlum]

You can write everything you want to. But it's useless if you can't evaluate it properly...

I don't agree with that. Just because a person does not see a use for it doesn't mean it isn't useful.

I am tempted to Quote Faraday here when they asked him "Of what use is electricity?" and he replied, "Of what use is a newborn baby?"

Keep an open mind, that's all I'm asking.

 

[agentredlum]

I'm reminded of the time I first digested the notion of "tangent vector to a manifold". I remember actively disliking the definition I had seen. But rather than just mentally rejecting the notion I had seen, I set out trying to define what I thought a tangent vector ought to be. By the time I had worked out all the difficulties and had something I was happy with, I had pretty much written down exactly what I had originally rejected.

Why did you stop? You should have kept going. If everything you knew up to that point made you question a definition then if i were you i would continue seeking confirmation from multiple sources before i concluded that i was wrong.

As a matter of fact, i would bring into question the very system itself that allowed me to make an erroneous assumption in the first place. I woudn't reject it, but an eyebrow would certainly be raised, and i would think long and hard about that.

 

[micromass]

I don't agree with that. Just because a person does not see a use for it doesn't mean it isn't useful.

I am tempted to Quote Faraday here when they asked him "Of what use is electricity?" and he replied, "Of what use is a newborn baby?"

Keep an open mind, that's all I'm asking.

Having an open mind in mathematics is really not a good idea...

When I'm doing research, I always like to be my own biggest critic. I criticize every step I take, and I put everything into question. Handwaving and non-rigorous arguments are ok, but they need to be formalized soon.
Once you've satisfied your own critics, only then can you present your work to somebody else. The point being that this other person criticizes your work again and shows possible flaws in your work.

In short: being skeptic in mathematics is a very good thing!

 

[micromass]

Oh man...

I REJECT NOTHING, I QUESTION EVERYTHING. Questioning everything is not the same as rejecting anything. Wouldn't you think I was a fool if I accepted everything without question? Making my point does not mean i have to reject yours. You think it does but I am not responsible for that. I can see it your way and agree it's useful. You can't see it my way even after i ask for a little latitude. That's not fair.

We do understand your point very well. But we see the possible flaws and mistakes too, and we point that out to you.
It's not because we criticize your point-of-view, that we can't see it your way...

 

[micromass]

Do check out http://en.wikipedia.org/wiki/Supernatural_numbers
This can be generalized to superrational numbers (not sure of the term), in which arbitrary infinite fractions can be studied.
However, I'm very unsure how (or if) the reals can be embedded in the superrationals...

 

[Hurkyl]

I REJECT NOTHING, I QUESTION EVERYTHING.

So, what were the results of questioning the notion of fraction you've been trying to push onto the thread?

 

[Hurkyl]

Why did you stop? You should have kept going. If everything you knew up to that point made you question a definition then if i were you i would continue seeking confirmation from multiple sources before i concluded that i was wrong.

As a matter of fact, i would bring into question the very system itself that allowed me to make an erroneous assumption in the first place. I woudn't reject it, but an eyebrow would certainly be raised, and i would think long and hard about that.

You're serious? You would reject your own reasoning in favor of clinging desperately to a first impression, even when the textbook agrees with your reasoning? And then when you finally give up your first impression, you would try and blame everyone else for filling your head with misleading thoughts?


I stopped because I was interested in learning differential geometry, and had concluded a useful exercise that shed insight onto the concept of tangent vector and demonstrated my own thoughts on the matter were already in alignment with the textbook despite my first impression. There is no expectation of utility in being contrary for the sake of being contrary.

 

[agentredlum]

We do understand your point very well. But we see the possible flaws and mistakes too, and we point that out to you.
It's not because we criticize your point-of-view, that we can't see it your way...

WOOOOOOOW! Finally a little respect, thank you, it means a LOT to me.

Oh yeah, of course there are many flaws, no question. However I do not reject any idea because of a few flaws.

I am fascinated by alternative ways of thimking. Example, in Stewart's calculus text in the section on alternating series, there are a few words about some famous mathematician proving that an infinite alternating series can be arranged to give ANY sum. I don't have the book anymore so i can't quote it verbatim but i remember the fascination i felt and the main idea.

Years ago i went to lunch with my math professor, who was an awsome teacher, and we talked and he gave an example i find fascinating even to this day.

He said a point has no length, height or width. Take a point and translate it to the right until you get a line segment, use your pencil if you like. That line segment is an infinite collection of points that have no length, wdth or height. Take the line segment and translate it up until you get a plane. Now that plane can be considered an infinite collection of line segments. Translate the plane out of the paper until you get a rectangular box. That box can be considered an infinite collection of planes. Now translate that box until it fills up all space.

Now that is mind boggling! You have just used something that has no length, width, height to (IN SOME SENSE) construct all 3-space. Is it rigorous, absolutely not. Is this thought experiment interesting, imho absolutely yes!

Later in a Linear Algebra course i learned the most amazing thing, the first day of class, from the same professor.

0x + 0y + 0z = 0

This is the equation of all 3-space. EVERY SINGLE POINT OF 3-space satisfies this equation.

Now, that is mind boggling and something Anton's Linalg book did not mention. Apparrantly, out of NOTHING you get EVERYTHING. Is it rigorous? no. Is it fascinating? Definitely yes!

So you see why i am not too eager to reject ideas?

 

[agentredlum]

So, what were the results of questioning the notion of fraction you've been trying to push onto the thread?

That there is more going on here than simple definitions could account for.

I'm not trying to 'push' anything on anybody. I would be hypocrite if i didn't accept scrutiny of my opinions. I'm just 'floating' it out there sort of like a colorful balloon with the word WARNING! on it.

 

[agentredlum]

You're serious? You would reject your own reasoning in favor of clinging desperately to a first impression, even when the textbook agrees with your reasoning? And then when you finally give up your first impression, you would try and blame everyone else for filling your head with misleading thoughts?I stopped because I was interested in learning differential geometry, and had concluded a useful exercise that shed insight onto the concept of tangent vector and demonstrated my own thoughts on the matter were already in alignment with the textbook despite my first impression. There is no expectation of utility in being contrary for the sake of being contrary.

If you convinced yourself that your first impression was superficial then i don't blame you for stopping, i would have done the same.

 

[Hurkyl]

Example, in Stewart's calculus text in the section on alternating series, there are a few words about some famous mathematician proving that an infinite alternating series can be arranged to give ANY sum. I don't have the book anymore so i can't quote it verbatim but i remember the fascination i felt and the main idea.

If, for a given infinite sequence of real numbers:

• The positive terms converge to zero
• The positive terms add to +\infty
• The negative terms converge to zero
• The positive terms add to -\infty

Then for any extended real number a, there exists a permutation of the sequence whose infinite sum converges to a.

And there is a related theorem: if

• The positive terms add to a finite number
• The negative terms add to a finite number

(this case is called "absolute convergence")

Then all permutations of the sequence sum to the same number.



This is rather important, since people like to rearrange sums arbitrarily, and these two facts not only tell you either a sum behaves 'perfectly' under rearrangement or it is capable of misbehaving in the worst way possible, but they also give you a very, very good way to tell which is which.

('perfect' is, of course, subject to the situation. Sometimes you want a sum that behaves badly under rearrangement)




One particular example of rearranging having actual practical importance (rather than just being a neat example) is double summations -- it is really, really, really convenient to think of it as just having a set of numbers to add up without having to pay attention to how they're arranged and in what order they are being summed. You can only get away with it in the case of absolute convergence.

(e.g. the sum might be given as adding up the rows first, then adding the results -- but it might be easier to instead add up the columns first, or sometimes adding up along diagonals is the way to go)

 

[agentredlum]

Check out bullet #4 of your post. What do you think about my powers of observation now?

 

[cb174503]

If the volume is equal to the numerical value of the Planck length then i am interested to know the dimensions of this geometrical solid.

An example: Suppose volume is a cube with side Planck length. Then the numerical value you get is about 10^-105. This is a lot smaller than the quantized value. Problems like this arise with any kind of geometrical object, including marshmellows, you can't even make a square with side Planck length without violating quantization.

Lets make the problem simpler so i can make my point without confusion. Pretend PL is .5 and you have a quantized system that does not allow smaller values. Then someone asks,'what happens if you square PL?'
You get .25 which is a smaller value than PL. 'No problem' you say, 'I'll just put .25 into my system as an exception.' What if you square the exception? then you get a smaller number still. You can repeat this process ad infinitem. You always get a number smaller than the previous and greater than zero. So you are rapidly creating an INFINITE number of exceptions approaching zero in your quantized system that was designed to have a FINITE number of points between zero and one. This is one of the reasons IMHO i believe quantization is dead.

--------------------------
Re: Agentredlum's example about size and dimensionality above:
I hope everyone will forgive me more of my words... I did not expect a geometric challenge to where those doggone irrational numbers are. At any rate:

1.616252*10-35 Planck-distance, in meters,
or “amount of line” covered by a Planck-distance, otherwise known as a Planck-length

(1.616252*10-35)*(1.616252*10-35) = 2.61227*10-70 = 1planck-area, in square meters,
or “amount of surface” covered by a Planck-area

(1.616252*10-35)*(1.616252*10-35)*(1.616252*10-35) = 4.22208*10-105 = 1planck-volume, in cubic meters, or “amount of space” contained in a Planck-volume – aka, a Planck-point as a 3-d quantized value based on the same 1-d quantized value as a Planck-length.

There isn't a reason to proceed with the calculation to higher dimensions because there is no evidence that there are any. However, if there were, the same logic would apply.

Math based on a number line with a finite number of points where those points are defined based on a Planck-length yields dimensional sizes which are comparably finite. Progressively larger, but not infinitely so. Only 70 orders of magnitude larger from a “line” to a volume… The difference with Agentredlum’s example is merely a difference in point of view, but an important difference.

So...the three standard issue multiplications above are different dimensional measurements using the same value, a Planck-length. Regardless of how it’s sliced, a Planck-volume is a Planck-length from each of its corners (planck-volume would be the smallest unit volume possible, mathematically analogous to a dimensionless point, but still one Planck-length per edge, visualized as a cube). There is no conflict in measurement from the “line” to the volume; it is but the same value measured in different dimensions. The dimensions being measured, however, are vastly different. Depending on the number of dimensions you wish to discuss, the value appears to get smaller according to the exponent, but in reality the dimensions are becoming larger. The same progression occurs whether the end volumes are cubes, spheres or marshmallows or universes – or, for that matter, however many dimensions you wish to consider. Also, with no infinite outcomes, which is unlike using math where infinite values exist in a line segment.

Intriguingly, 1.616252*10 to the +35th p-v’s laid one by one next to each other would make a line of p-v’s one meter long - a finite number of Planck-points. Using p-l’s and p-v’s suggests a means of distinguishing size between dimensions – where using traditional math to try and measure size differences between dimensions doesn’t work well, if at all. It is possible the CERN machine (LHC) may find real world evidence related to the question.

Speaking of dimensions, I’ve read that some consider time to be a 4th dimension. Usually it’s referred to as an extension of the third dimension, similar to the third as an extension of the second. I don’t think that’s the case… rather, time is an extra linear dimension similar to the single dimension but not as an extension of the three we experience. It has already been suggested that time is discrete in structure at a scale of about 1*10-43 seconds. Others have considered this question in relation the Zeno paradox and have come to the solution that time isn’t something we’re “in” that can be visualized statically like “in the present instant”. They assert time is something we and our world are passing through dynamically with no stop actions in the discrete instants. To me that is unsatisfying as it seems to mix discrete and continuous structure - but hard to refute.
cb

 

[agentredlum]

If the volume is equal to the numerical value of the Planck length then i am interested to know the dimensions of this geometrical solid.

An example: Suppose volume is a cube with side Planck length. Then the numerical value you get is about 10^-105. This is a lot smaller than the quantized value. Problems like this arise with any kind of geometrical object, including marshmellows, you can't even make a square with side Planck length without violating quantization.

Lets make the problem simpler so i can make my point without confusion. Pretend PL is .5 and you have a quantized system that does not allow smaller values. Then someone asks,'what happens if you square PL?'
You get .25 which is a smaller value than PL. 'No problem' you say, 'I'll just put .25 into my system as an exception.' What if you square the exception? then you get a smaller number still. You can repeat this process ad infinitem. You always get a number smaller than the previous and greater than zero. So you are rapidly creating an INFINITE number of exceptions approaching zero in your quantized system that was designed to have a FINITE number of points between zero and one. This is one of the reasons IMHO i believe quantization is dead.

--------------------------
Re: Agentredlum's example about size and dimensionality above:
I hope everyone will forgive me more of my words... I did not expect a geometric challenge to where those doggone irrational numbers are. At any rate:

1.616252*10-35 Planck-distance, in meters,
or “amount of line” covered by a Planck-distance, otherwise known as a Planck-length

(1.616252*10-35)*(1.616252*10-35) = 2.61227*10-70 = 1planck-area, in square meters,
or “amount of surface” covered by a Planck-area

(1.616252*10-35)*(1.616252*10-35)*(1.616252*10-35) = 4.22208*10-105 = 1planck-volume, in cubic meters, or “amount of space” contained in a Planck-volume – aka, a Planck-point as a 3-d quantized value based on the same 1-d quantized value as a Planck-length.

There isn't a reason to proceed with the calculation to higher dimensions because there is no evidence that there are any. However, if there were, the same logic would apply.

Math based on a number line with a finite number of points where those points are defined based on a Planck-length yields dimensional sizes which are comparably finite. Progressively larger, but not infinitely so. Only 70 orders of magnitude larger from a “line” to a volume… The difference with Agentredlum’s example is merely a difference in point of view, but an important difference.

So...the three standard issue multiplications above are different dimensional measurements using the same value, a Planck-length. Regardless of how it’s sliced, a Planck-volume is a Planck-length from each of its corners (planck-volume would be the smallest unit volume possible, mathematically analogous to a dimensionless point, but still one Planck-length per edge, visualized as a cube). There is no conflict in measurement from the “line” to the volume; it is but the same value measured in different dimensions. The dimensions being measured, however, are vastly different. Depending on the number of dimensions you wish to discuss, the value appears to get smaller according to the exponent, but in reality the dimensions are becoming larger. The same progression occurs whether the end volumes are cubes, spheres or marshmallows or universes – or, for that matter, however many dimensions you wish to consider. Also, with no infinite outcomes, which is unlike using math where infinite values exist in a line segment.

Intriguingly, 1.616252*10 to the +35th p-v’s laid one by one next to each other would make a line of p-v’s one meter long - a finite number of Planck-points. Using p-l’s and p-v’s suggests a means of distinguishing size between dimensions – where using traditional math to try and measure size differences between dimensions doesn’t work well, if at all. It is possible the CERN machine (LHC) may find real world evidence related to the question.

Speaking of dimensions, I’ve read that some consider time to be a 4th dimension. Usually it’s referred to as an extension of the third dimension, similar to the third as an extension of the second. I don’t think that’s the case… rather, time is an extra linear dimension similar to the single dimension but not as an extension of the three we experience. It has already been suggested that time is discrete in structure at a scale of about 1*10-43 seconds. Others have considered this question in relation the Zeno paradox and have come to the solution that time isn’t something we’re “in” that can be visualized statically like “in the present instant”. They assert time is something we and our world are passing through dynamically with no stop actions in the discrete instants. To me that is unsatisfying as it seems to mix discrete and continuous structure - but hard to refute.
cb

Very good point that even though the numbers are decreasing, the DIMENSIONS are getting larger in the sense you can do more with them and on an intuitive level.

I also agree with you that it is disquieting to mix discrete and continuous, and brings into question the motivation behind such an endeavor. Are they using facts to fit the theory?, or are they using the theory to change the facts? Or are they doing both whenever it suits them? Or maybe it's a misunderstanding and they're doing neither?

Like I said before, I hope you succeed in your attempt to quantize, many are still working on this so you could too.

If you call your Planck Length 'one' then squaring, cubing, etc. don't present the problem I mentioned. Your meter would have about 10^35 PL. like you mention above. After all the standard length of 1m is comepletely arbitrary. Why not define PL as 'one meter'?

Then the distance of my face to the monitor is 10^35 meters...

 

[cant_count]

Thank you for suggesting Cantor - very interesting. Another quick question: there are integers, rational numbers, irrationals (which can be expressed using polynomials), transcendentals (which can't, but still can be defined, e.g. pi and e). What about numbers that cannot be defined in any way whatsoever? Does such a category have a name? And are there more of them than anything else? Are they ever used in mathematics?
(If they are still transcendentals, they'd need to be distinguished from the definable ones, I'd be thinking.) Perhaps they can't be discussed because they're undetectable or unprovable by definition? But there should be more of them than anything else, right? (Like dark matter ..!)
- Just wondering

 

[cant_count]

Sorry, meant dark energy.

 

[micromass]

Thank you for suggesting Cantor - very interesting. Another quick question: there are integers, rational numbers, irrationals (which can be expressed using polynomials), transcendentals (which can't, but still can be defined, e.g. pi and e). What about numbers that cannot be defined in any way whatsoever? Does such a category have a name? And are there more of them than anything else? Are they ever used in mathematics?
(If they are still transcendentals, they'd need to be distinguished from the definable ones, I'd be thinking.) Perhaps they can't be discussed because they're undetectable or unprovable by definition? But there should be more of them than anything else, right? (Like dark matter ..!)
- Just wondering

Your intuition is correct! There are indeed things like undefinable and uncomputable numbers! Check out http://en.wikipedia.org/wiki/Definable_real_number and http://en.wikipedia.org/wiki/Computable_number

 

[Bacle]

I've heard of this, though my knowledge on it is very limited. This is more of a physics question than mathematical question though. In the sense of the real numbers, time and length are not discrete.



In terms of preciseness, this would probably be better. However, you would need some kind of mechanism/scheme to determine the actually length of a Planck. This would require insane detail, since if you're off by just a tiny bit, the error between the actual length and the observed length will amplify when measuring visible objects.

I don't know if this is what you're looking for, but there are efforts to "algebraize" parts of calculus and analysis; specifically with ideas like those of algebraic derivations on rings
( maps associated with a ring that satisfy the Leibniz property), which are(intended?) discrete substitutes of the derivative, which do not assume the continuum. These algebraic substitutes are also used to solve differential equations in differential Galois theory. There are also areas like discrete differential geometry which try to do the same.

 

[gb7nash]

I don't know if this is what you're looking for, but there are efforts to "algebraize" parts of calculus and analysis; specifically with ideas like those of algebraic derivations on rings
( maps associated with a ring that satisfy the Leibniz property), which are(intended?) discrete substitutes of the derivative, which do not assume the continuum. These algebraic substitutes are also used to solve differential equations in differential Galois theory. There are also areas like discrete differential geometry which try to do the same.

No, I was talking about Planck measurements in real life. I'll be the first to admit I know very little about it, but I think this is more of a physics problem than a calculus problem.

 

[Bourbaki1123]

Your intuition is correct! There are indeed things like undefinable and uncomputable numbers! Check out http://en.wikipedia.org/wiki/Definable_real_number and http://en.wikipedia.org/wiki/Computable_number

I just want to point out that in fact the set of computable numbers has measure zero (which I think is mentioned in the article, but it's an important point so I want to emphasize it), so almost all numbers in R have no algorithm for arbitrary precision decimal approximation. This is something people seem to often fail to take into consideration, it is actually pretty counter intuitive before you have some grounding in the theory of computation.