# Where are the irrational numbers?

by smolloy
Tags: irrational, numbers
Mentor
P: 18,062
 Quote by agentredlum 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...
 Mentor P: 18,062 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...
Emeritus
PF Gold
P: 16,092
 Quote by agentredlum 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?
Emeritus
PF Gold
P: 16,092
 Quote by agentredlum 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.
P: 460
 Quote by micromass 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?
P: 460
 Quote by Hurkyl 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.
P: 460
 Quote by Hurkyl 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.
Emeritus
PF Gold
P: 16,092
 Quote by agentredlum 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)
 P: 460 Check out bullet #4 of your post. What do you think about my powers of observation now?
P: 460
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...
 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
 P: 3 Sorry, meant dark energy.
Mentor
P: 18,062
 Quote by 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
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
P: 662
 Quote by gb7nash 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.
HW Helper
P: 805
 Quote by Bacle 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.
P: 326
 Quote by micromass 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.

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