Register to reply 
Where are the irrational numbers? 
Share this thread: 
#109
Aug111, 11:31 AM

Mentor
P: 18,293

It's not because we criticize your pointofview, that we can't see it your way... 


#110
Aug111, 11:53 AM

Mentor
P: 18,293

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... 


#111
Aug111, 11:56 AM

Emeritus
Sci Advisor
PF Gold
P: 16,091




#112
Aug111, 12:06 PM

Emeritus
Sci Advisor
PF Gold
P: 16,091

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. 


#113
Aug111, 12:25 PM

P: 460

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 3space. 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 3space. EVERY SINGLE POINT OF 3space 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? 


#114
Aug111, 12:32 PM

P: 460

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. 


#115
Aug111, 12:37 PM

P: 460




#116
Aug111, 12:53 PM

Emeritus
Sci Advisor
PF Gold
P: 16,091

And there is a related theorem: if
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) 


#117
Aug111, 01:01 PM

P: 460

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



#118
Aug111, 01:08 PM

P: 5

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*1035 planckdistance, in meters, or “amount of line” covered by a planckdistance, otherwise known as a plancklength (1.616252*1035)*(1.616252*1035) = 2.61227*1070 = 1planckarea, in square meters, or “amount of surface” covered by a planckarea (1.616252*1035)*(1.616252*1035)*(1.616252*1035) = 4.22208*10105 = 1planckvolume, in cubic meters, or “amount of space” contained in a planckvolume – aka, a planckpoint as a 3d quantized value based on the same 1d quantized value as a plancklength. 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 plancklength 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 plancklength. Regardless of how it’s sliced, a planckvolume is a plancklength from each of its corners (planckvolume would be the smallest unit volume possible, mathematically analogous to a dimensionless point, but still one plancklength 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 pv’s laid one by one next to each other would make a line of pv’s one meter long  a finite number of planckpoints. Using pl’s and pv’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*1043 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 


#119
Aug111, 01:31 PM

P: 460

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... 


#120
Aug311, 08:52 AM

P: 3

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 


#121
Aug311, 09:00 AM

P: 3

Sorry, meant dark energy.



#122
Aug311, 09:03 AM

Mentor
P: 18,293




#123
Aug311, 11:30 AM

P: 662

( 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. 


#124
Aug311, 04:13 PM

HW Helper
P: 805




#125
Aug2111, 01:59 PM

P: 326




Register to reply 
Related Discussions  
Irrational numbers vs. Transcendental numbers  General Math  10  
Pi and irrational numbers  General Math  8  
Irrational Numbers...  General Discussion  15  
Irrational Numbers  Calculus & Beyond Homework  8  
Irrational numbers depends on rational numbers existence  General Math  0 