Physics: Decimals can't be infinite because the space between must end.

[CosmoNut]

I've pondered this off and on since high school:

I understand the principle behind the assertion that decimals never end and that you can (theoretically) infinitely halve the space between two objects and they'll never touch. A parabolic curve works on this principle, but really...

If two objects are travelling at each other, at some point the space between them must end so that there is NO space left and they are touching, right? By this notion, I'd think that a value of their distance apart MUST end. So it seems that at some point 0.00000000000000000...1 must become simply 0 when the objects finally touch.

--> <--

Surely someone way smarter than I am can explain this. What am I missing here?

[hardeeharhar]

Things never actually "touch" each other, but that really is besides the point.

Decimals are abstractions in math having nothing to do with reality per se.

Physics takes mathematical abstractions and tries to use its results to predict events -- but attempting to do the opposite -- find physical meaning in math is irrelevent.

[blackbird_1.0]

*head explodes*

[addabox]

Huh. I was just reading a chapter in "Everything and More", a short history of the concept of infinity and how it has been grappled with mathematically since the Greeks (by David Foster Wallace, the notoriously discursive fiction writer and essayist, who apparently is a bit of a math nerd and finally gets to put his love of footnotes to good use).

You're basically describing "Zeno's paradox", AKA "the Dichotomy", of which there are several variants involving distances and times, but which stripped of the word problem narratives (which is a good idea to avoid some of the syntactical confusion arising out of moving between natural language and math) amounts to "successive halving a number approaches but never reaches some limit". Or , if you prefer, "the decimal goes on forever". The most common presentation is "how can I ever really cross the street if I must first go half way, and then half way again, and so on, with no end to the infinite succession of halves?" with "the other side of the street" representing the limit which is approached but never reached.

Another way to talk about the same thing is to consider the infinite density of the number line-- that for any two points a and b we can always define a third point c between a and b such that c =( a +b)/2.

The idea can be applied to any interval you like-- if I increase the resolution of of my measuring stick indefinitely, can I ever be said to have an actual "height", since there does not appear to be a final term in my series of ever more minute slices of space?

A good chunk of mathematical history is the story of coming to terms with theses "infinitesimals"-- the arbitrarily small amounts that separate the sum of a convergent series from its limit (which is to say a mathematically rigorous treatment of infinity), and I have neither the math nor the vB code typographical chops to represent the actual equations that deal with all this, so I heartily recommend Wallace's book (although for my money the math gets pretty dense at times).

However, it should be noted that a lot of the "paradoxical" nature of Zeno lies in the way natural language is ambiguous where mathematical language is not, and some shifty sliding between the two.

That is, Zeno is inviting us to regard an interval in the real world as being "made of" the infinite convergent series 1/2 + 1/4 + 1/8 + 1/16....., approaching but never reaching 1, whereas in fact in the real world the interval "1" (AKA "crossing the street") can be understood to simply exist (in the manner that things in the real world are allowed to do, as opposed to the abstract rigor of mathematical space), and that same convergent series approximately describes the a priori quantity "1". It's the inversion of the abstract descriptor with the thing itself that makes for the apparent weirdness. (For any mathematicians in the house I hope I'm not making to much of a hash of this, it does take a book to really nail down the movement from Zeno to transfinite math, and a lot of it isn't really "intuitively" graspable outside of the math itself).

[giant]

well, .9999... == 1

Edit: don't mistake this to mean that it will reach this point. See other comments.

[blackbird_1.0]

I'm good with regular math, but anything that takes me beyond algebra makes my head hurt. :)

[addabox]

Originally posted by giant
well, .9999... == 1

Right, but the important thing to consider in CosmoNut's question is that in the case of "paradoxically" applying that to the real world you have to bear in mind that ".9999999...." is not a feature of the world, it is a feature of a mathematical description of the world.

So that "the reason" that .9999......"is equivalent" to 1 is a series of internally consistent mathematical statements regarding the abstract interval 0 to 1 on the number line, but if we start thinking of the distance to cross the street as being somehow "made" of (as opposed to "satisfactorily described by) .99999...... we immediately get into trouble.

CosmoNut is actually describing the inverse of Zeno's paradox, something like "since I already know that space can be successfully transversed, and mathematics purports to describe space, how is it that a finite world of closed intervals simultaneously supports infinite descriptors in what appears to be the same interval?"

Again, the problem arising if we are not very careful to distinguish between real world phenomena like "things colliding" and abstract descriptions of things colliding that move endlessly toward a finer and finer description of the point of collision.

[Hiro]

Originally posted by CosmoNut
I've pondered this off and on since high school:

I understand the principle behind the assertion that decimals never end and that you can (theoretically) infinitely halve the space between two objects and they'll never touch. A parabolic curve works on this principle, but really...

If two objects are travelling at each other, at some point the space between them must end so that there is NO space left and they are touching, right? By this notion, I'd think that a value of their distance apart MUST end. So it seems that at some point 0.00000000000000000...1 must become simply 0 when the objects finally touch.

--> <--

Surely someone way smarter than I am can explain this. What am I missing here?

Here's the rub for the two objects traveling towards each other bit. They may be moving towards each other, but they are always aiming for a point short of the other object. The closer they get the slower they move towards the new version of the aimpoint which is updated to always be halfway there. The objects following the halfway there game never intend to touch, they always dynamically adjust their travel to conform to the halfway there game rules.

This way the seeming paradox is gone when we think of objects actually traveling towards each other that are not playing the halfway there game. If the objects were meant to actually touch each other they would not ever worry about the halfway there aimpoint. Then any smartass that tries to tell you the motion to get them to touch is mathematically impossible is full of crap because the smartass would have to introduce a bogus assumption that would eventually force a division by zero someplace in the real world math to force the paradoxical halfway there idea onto a constant motion object not playing the halfway there game. That bogus assumption and division by zero being the happy contradiction that that allows you to safely ignore the halfway there paradox when you are moving someplace in the real world.

[Hiro]

Originally posted by hardeeharhar
Things never actually "touch" each other, but that really is besides the point.

Decimals are abstractions in math having nothing to do with reality per se.

Physics takes mathematical abstractions and tries to use its results to predict events -- but attempting to do the opposite -- find physical meaning in math is irrelevent.

Au contreaire! Physic IS math! Chemistry IS physics. Everything else follows off that. The problem is when someone make an assumption that is valid in one system but not another for some reason or another. Then all hell breaks loose, but the rules are all still purely mathematical and consistent. We can get to areas which we don't understand well enough yet and consistency starts to fall apart there, but that all comes back to assumption validity.

[hardeeharhar]

No. All fields are aesthetics on fundamental thoughts that human's are capable of having, but this does not mean that one field is the same as the others. They build upon each other, but they are not the same. Chemistry is NOT Physics, rather it uses Physics.

That is the study of Chemistry builds upon the study of Physics.

Remember the fields have nothing to do with reality all that much either, as in, the discplines are human inventions, and the scopes we limit them to are artificial, but that is another discussion for another time...

[benzene]

Good clarification hardeharhar.

Concerning "smallest possible distances":
If you subscribe to string theory, there is a smallest possible distance, the size of the individual string. (link (http://www.columbia.edu/cu/record/23/18/14.html))
This implies that nature is in fact, digital, not analog.

String theory has been maligned a bit these days though. I'm not qualified to make any statement beyond "that's intriguing", in any case.

[CosmoNut]

So...has anyone actually given a real answer yet, or is it just a "paradox" and we leave it at that?

Oh, and Mac_Doll, you were right. ;)

[addabox]

Originally posted by CosmoNut
So...has anyone actually given a real answer yet, or is it just a "paradox" and we leave it at that?

Oh, and Mac_Doll, you were right. ;)

The answer is: objects in motion do not move through the decimal system. They move through real space. Cutting space into ever finer slices is a mathematical process intended to closely approximate the motion of objects in space, but it is not the motion itself, or the space itself.

So on the number line you can have one infinite series that approaches an upper limit of, say, 2, and another that approaches a lower limit of 2, and they never meet, despite getting ever closer forever.

But that is an abstraction designed to help us describe the world. Two baseballs hurtling at one another are described by such numbers, but not constrained. They are not obliged to perform Zeno's endless halving of the remaining space, because the space between them is not being generated by a formula-- it already exists, the point that the baseballs make contact is a real point in real space, and the mathematical description is after the fact.

[Marvin]

Originally posted by addabox
how is it that a finite world of closed intervals simultaneously supports infinite descriptors in what appears to be the same interval?

I think that's it in a nutshell. It's true that if you do continuously halve the distance you travel, you will never reach where you are going but you just don't do that in the real world. If a street is ten steps wide and you take ten steps, you will reach the other side, assuming you aren't run over while you were too busy pondering the nature of the universe instead of watching for oncoming traffic.

Fractals follow a similar principle of bounding infinite detail in a finite geometric space.

[sunilraman]

[QUOTE]Originally posted by CosmoNut
So...has anyone actually given a real answer yet, or is it just a "paradox" and we leave it at that?


Addabox and Benzene have nailed it down pretty well. Your question is about what is the smallest possible space between two objects.

For example, one idea of what an "object" is:
http://www.columbia.edu/cu/record/23/18/26b.gif

There is a way to mathematically try and model that, but like we learnt in college(?) scientific models are just that - a way of trying to describe a situation, in this case, trying to describe mathematically a physics case.

So coming back to the question of the smallest possible space between two objects, we go down to what is the physical nature of those objects? Are you talking about atoms? Subatomic particles popping in and out of our dimension quantum-mechanics style? Then there is the String theory earlier mentioned which means "everything is connected there is actually no empty space"...

1. The above paragraph is my armchair-layman's interpretation of stuff.

2. I'm more inclined to disrespect the maths side of things because that's just a way to represent the "real world" whereas physics actually gets down to the nitty gritty of what IS the "real world". Maths for the sake of maths is too abstract for my liking.

[sunilraman]

[QUOTE]Originally posted by CosmoNut
.....So it seems that at some point 0.00000000000000000...1 must become simply 0 when the objects finally touch....Surely someone way smarter than I am can explain this. What am I missing here?


What is missing is what the mathematical description of the empty space is. If 0.000000000000000000000000000000000...001 = 0 is true on a physical level then some assumption has been made of that "empty space" - you have "put something in there (that something could be nothing*)" for 0.000000000...00001 = 0 if you are talking about objects occupying physical space.

*So we go back to essentially the physics question of "what is empty space", or "what would you put there" for 0.00000000000.0001 = 0 to be true.

Then there is the question of what is your definition of two objects "touching"

Come to think of it, even if we look at the atomic level of things and not go deeper into subatomic particle and string theory, CosmoNut, have you done "electron orbitals" at college? Unless this theory has been disproven recently, see, in high school we are taught that electrons "orbit around the nucleus made up of protons and neutrons".

This is actually not true. http://en.wikipedia.org/wiki/Electron_configuration

http://upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Electron_orbitals.png/350px-Electron_orbitals.png

You see, in an atom, electrons occupy what is known as a "probability region" or "electron orbital space". That is, at any one point in time, we cannot predict where the electron actually is. It's not like predicting where the moon is as it goes around the earth. The "probability region" is only defined when you use some way to "observe" the electrons. For example in the image above the 3d spaces defined are where the electrons are most likely to be for those atoms (not going into shells and subshells too complex)

So two objects touching on the atomic level is an interesting question because if you take it as the electron orbit being the outer edge of an object, there is no such thing as an atom being a nice round ball so you can define it as touching another atom of being a nice round ball. In other words atoms are not nice round balls...!

1. Again, probably gonna get slammed by more in-the-know people.

2. But man, my first-year chemistry classes at college were
"fun" when I learnt about electron orbitals. It was like, WTF? Atoms are not these nice round balls...?!!!

[hardeeharhar]

Um...

We don't know what any multi-electron atoms' electron probability maps actually look like. There are higher order approximations that appear to imply that our use of the hydrogen orbitals are at least qualitatively correct for ATOMS, but when push comes to shove, QM calculations on MOLECULES are mostly high level approximations of systems of atoms.

Hydrogen filled with any number of electrons still looks like a ball -- as the electrons tend to avoid each other.

Regardless, when you push upon an object the electrons of your skin and the object start filling shared molecular orbitals pair wise until the energy of the next highest orbital is greater than the work being done to push the two objects together. Simply, the electron clouds of your finger and key repel each other.

[audiopollution]

A few personal observances that objects do actually touch:

1) That stupid Subaru totally ruined the back of my car.
2) A punch can, in fact, cause a broken nose.
3) You should always pay attention to the bat, while playing baseball with kids.

[benzene]

This thread is actually quite funny to read.

Hardeharhar hit the nail on the head. Since QM states electron clouds in terms of potentials, the idea of "distance apart" for macroscopic items is by definition, also quite fuzzy. The electron clouds will start to interact, and it is through this repulsion that we can exert force. There is a distinct, mathematically described potential that an electron belonging to an atom in my mouse will spend some of its time in my hand (even if they're technically separated by thousands of angstroms).

If you wanted to really get technical, you could measure distances by measuring between nuclei, but with brownian motion and the potential for quantum tunneling even that's rather sketchy. The point at which your objects become sufficiently small to truely "measure between them", you end up with a whole other level of weirdness.

Moral of the story: you can't be glib when talking about anything on a subatomic scale.

[addabox]

Originally posted by benzene
This thread is actually quite funny to read.

Hardeharhar hit the nail on the head. Since QM states electron clouds in terms of potentials, the idea of "distance apart" for macroscopic items is by definition, also quite fuzzy. The electron clouds will start to interact, and it is through this repulsion that we can exert force. There is a distinct, mathematically described potential that an electron belonging to an atom in my mouse will spend some of its time in my hand (even if they're technically separated by thousands of angstroms).

If you wanted to really get technical, you could measure distances by measuring between nuclei, but with brownian motion and the potential for quantum tunneling even that's rather sketchy. The point at which your objects become sufficiently small to truely "measure between them", you end up with a whole other level of weirdness.

Moral of the story: you can't be glib when talking about anything on a subatomic scale.

Let's not make the mistake of implying that the instrumental imprecision of defining the smallest possible unit of space time to be used in defining the boundary between objects has anything to do with transcendental numbers, except insofar transcendental numbers might be used to describe some aspect of such a unit.

The infinitude of irrationals and such is of a very different order of epistemology than the "fuzziness" of the world at very small scales, which in fact the point of "paradox" in the original post.

[hardeeharhar]

Again Mathematical abstractions have no physical meaning in and of themselves.

[Hiro]

Again. The universe is just one vastly large set of mathematically describable interactions. How the subsets of mathematical interactions are are parsed into disciplines doesn't change that.

Abstractions are just a set of artificial assumptions about the actual interactions which may or may not be correct. As long as your assumptions are correct for the scale you are examining, your described outcomes will be correct as well. All physical sciences are engaged in discovering ever more correct abstractions (assumptions). The fact we don't know them all doesn't mean something at a less granular level doesn't actually happen - like objects touching.

Everything else is obfuscating bullshit hiding reality behind a layer of insider lingo. Go ahead an believe otherwise if you want but that won't change the fact that things actually touch.

[hardeeharhar]

Not all math is applicable to "reality."

Period.

You can continue to deny this, but if you ever have had a conversation with a real mathematician doing research in modern topics, you will quickly realize that math has advanced far beyond simple descriptions of reality.

Math doesn't depend upon confirmation in the real world, and that is a testament to its history and proofs.

Continuing to deny this prevents you from understanding the depth of math.

Edit: You don't discover assumptions. Science is actually founded on the principle of minimizing the number of assumptions needed to understand a system. In fact, the exact opposite of what you argue...

[hardeeharhar]

I should also say that it is a semantic point with regard to objects touching.

Is your cut off at the 70% orbital overlap level or the 50%?

[midwinter]

This is kind of where this is going, but I've always kind of assumed that "objects" don't really "touch" at all, and that to assume that they did was to presume the "thingness" of the object itself.

[sunilraman]

[QUOTE]Originally posted by hardeeharhar
I should also say that it is a semantic point with regard to objects touching...Is your cut off at the 70% orbital overlap level or the 50%?


Precisely why I brought in the point of not having to go into quarks and string theory and all that, and brought up orbital probabilities*. What is touching? What is empty space?

I say (I assume, define personally) that things touch because there is evidence of electron repulsion. I touch my mouse because I feel a force between the mouse surface and my hand surface brought about by repulsion of the two molecular electrons. I define/ assume touching as the feeling of repulsive forces.

%Of orbital overlap of quantum electron probalities of molecules is beyond our ability to measure at this stage, right?

*I forgot electron orbitals that we know about come about from starting with the Hydrogen atom. At this stage we still can't work out electron probability 3d-space for molecules? Wow... lots of work still to be done.

[sunilraman]

[QUOTE]Originally posted by midwinter
This is kind of where this is going, but I've always kind of assumed that "objects" don't really "touch" at all, and that to assume that they did was to presume the "thingness" of the object itself.


Currently in our science the "thingness" of the object is assumed/ proven by the process of observation - ie, there is someway to prove that it is actually there. Eg. the pattern observable when atoms are smashed together, you can then "prove" that subatomic particles such as bosons and mesons and those kind of particles exist.

It "started" at a Newtonian level, we know the moon is a "thing" that exists because we can observe it, and also we can observe the effect of forces exterted on/ exerted by the "thing". In this case gravitational forces and light reflection is proof of the "thingness" of the moon.

About objects touching, that goes back to semantics as hardeeharhar says of how you define touching -- electron orbital probabilities, string theory, and again, how to define empty space....

[sunilraman]

[QUOTE]Originally posted by hardeeharhar
Not all math is applicable to "reality."

Period.

You can continue to deny this, but if you ever have had a conversation with a real mathematician doing research in modern topics, you will quickly realize that math has advanced far beyond simple descriptions of reality.

Math doesn't depend upon confirmation in the real world, and that is a testament to its history and proofs.

Continuing to deny this prevents you from understanding the depth of math.


This is great that you have mentioned this. Math for Math's sake does exist as a field of study in and of itself.

Using Maths for physics problems is Applied Math, but there is Math just for Math -- something I feel is so abstract, but I suppose I'm glad people do it because eventually with all those theorems and stuff it can be Applied to useful problems in the "real world".

Yes -- a "real mathematician" doing advanced Masters and phDs and post-doctoral work in Maths researches all sorts of stuff, especially the super weird crap of trying prove theorems that have not been proven yet. Theorems and proofs and all that jazz. Mindf*ck stuff if you're not in that field.

Just look at this from our Apple cheerleaders Wolfram Research:
http://mathworld.wolfram.com/FermatsLittleTheorem.html

Eg. Fermat's Theorem in this case. On the page there is no application to the "real world" although Applied Mathematicians have probably used it in various areas. There are links to other stuff there that's "pure mathematics"
("SEE ALSO: Binomial Theorem, Carmichael Number, Chinese Hypothesis, Composite Number, Compositeness Test, Euler's Theorem, Fermat's Little Theorem Converse, Fermat Pseudoprime, Modulo Multiplication Group, Pratt Certificate, Primality Test, Prime Number, Pseudoprime, Relatively Prime, Totient Function, Wieferich Prime, Wilson's Theorem, Witness".)

But, they can send a man to the moon and solve hundreds-year-old theorems but why can't they make my feet smell nice :D

[sunilraman]

[QUOTE]Originally posted by addabox
Let's not make the mistake of implying that the instrumental imprecision of defining the smallest possible unit of space time to be used in defining the boundary between objects has anything to do with transcendental numbers, except insofar transcendental numbers might be used to describe some aspect of such a unit. The infinitude of irrationals and such is of a very different order of epistemology than the "fuzziness" of the world at very small scales, which in fact the point of "paradox" in the original post.


I agree to the extent of what I understand. CosmoNut is trying to ask what he is "missing". He has not defined a "proffessional" Paradox like the kinds the "scientific community" is dealing with.

CosmoNut, IMHO, this is what you are "missing"

1. You propose a situation whereby two objects are moving towards each other, each time moving half the distance between them. Yes, at some point in time, the space between them will become sooooo small that essentially Newtonian physics is no longer relevant, we get into the Quantum physics realm and all the "weirdness" associated with it - which we have tried to describe above. This small distance is not as small as you think, we can think in terms of one Angstrom. Ten billion angstroms equal 1 meter. So AFAIK there is no specific definition where Quantum physics take over but certainly once you hit 1 Angstrom and less Quantum mechanics and the weirdness of atomic- and subatomic-particle interactions start to apply. At this stage the "Newtonian-style view" of an electron orbiting the nucleus like the earth around the sun is total rubbish.

2. You made a huge jump in tying together the maths side of things and the physics side of things. You started with saying, let's assume 0.000000000000....000001 approximates to essentially zero. Then you JUMP to the conclusion of these numbers relating to a scale of space between two objects. Then you jump again to the conclusion that "I approximate 0.0000000.....0000001 to zero therefore the real-world space between two objects at that level must also be zero". So like it has been said before, currently in our understanding there is a Physics "real world" situation and we use Applied Maths to tackle the issue. Going the other way round does not make sense in this case because you are taking an abstracted Mathematical idea and then "dumping" it onto the real world. The title of this thread itself is problematic because the way you defined it -- taking an abstracted mathematical situation and then "duct-taping" it onto a physical situation.

3. "Decimals can't be infinite" ... Who says so? If we just look at the Mathematical implications of that, again this is the third issue of what you are "missing". Why is 1+1 = 2? Only because by convention. Mathematicians do all sorts of weird stuff in the "pure maths" area. For example, depending on what they are trying to do and the conventions of the field they are working with, they can say Decimals are infinite or decimals are not, they can say all sorts of things, depending on the problem they are working on. Just take Pi - we think of it as 22/7 and it just goes on forever. But actually 22/7 is GREATER than Pi.
http://en.wikipedia.org/wiki/A_simple_proof_that_22/7_exceeds_pi
I would assume then from this that those programs that use computers to derive all the values of pie, do not just take 22 and divide it by 7. They use other formulas to work out the millions of digits or whatever. Stupid irrational numbers, so irrational :p


Disclaimer: Again, as is my understanding at this point in time and I can't believe my brain is still handling at least 10% of this mindf*ck stuff.

[sunilraman]

http://mathworld.wolfram.com/ZenosParadoxes.html
"The dichotomy paradox leads to the following mathematical joke. A mathematician, a physicist and an engineer were asked to answer the following question. A group of boys are lined up on one wall of a dance hall, and an equal number of girls are lined up on the opposite wall. Both groups are then instructed to advance toward each other by one quarter the distance separating them every ten seconds (i.e., if they are distance d apart at time 0, they are d/2 at t==10, d/4 at t==20, d/8 at t==30, and so on.) When do they meet at the center of the dance hall? The mathematician said they would never actually meet because the series is infinite. The physicist said they would meet when time equals infinity. The engineer said that within one minute they would be close enough for all practical purposes."

I've taken the engineer-style approach in my statement no.1 above. At the end of the day it comes down to our individual/ fields-of-study approaches, assumptions, perceptions and observations of distance and time.

[sunilraman]

Extra Credit: Two objects are 1 metre apart. They move towards each other, by one quarter the distance separating them every one second (i.e., if they are distance d apart at time 0, they are d/2 at t==1, d/4 at t==2, d/8 at t==3, and so on.) Given d = 1 metre At what time t will the distance between the two objects be 1 Angstrom (d/10billion)?

Which of you 1337 pe0pl3s can answer this :smokey: (my head would burn up in an instant the second I tried to solve this). I suspect this is an easy question though for Maths peoples. I also suspect the answer is *not* 10 billion seconds.

[Hiro]

Originally posted by hardeeharhar
Not all math is applicable to "reality."

Period.

You can continue to deny this, but if you ever have had a conversation with a real mathematician doing research in modern topics, you will quickly realize that math has advanced far beyond simple descriptions of reality.

Math doesn't depend upon confirmation in the real world, and that is a testament to its history and proofs.

Continuing to deny this prevents you from understanding the depth of math.

Duh!!!

Will you quit trying to conjure up bogus points to argue against? Then state them as if I am daft because I must believe them? They are completely your own!!!

I say all reality is describable by math, even though we don't know how to do all those descriptions yet. Which has no direct conflict with your Not all math is applicable to "reality.". That's like saying, right after someone points out all squares are rectangles, -- Not all rectangles are squares!!! OMG this is so wrong!!! -- You might as well claim the price of kumquats in Jakarta disproves my point! :p

I couldn't give a rats ass about pure mathematics for the sake of pure mathematics (or the price of kumquats in Jakarta), this thread happens to be about applied math and a seeming paradox that really isn't a paradox at all.

Edit: You don't discover assumptions. Science is actually founded on the principle of minimizing the number of assumptions needed to understand a system. In fact, the exact opposite of what you argue...

Semantics. Assumptions in an equation, known "factors" in an equation, both are exactly the same thing because in physical reality we don't ever actually "KNOW" everything about everything. The terminology all just depends on which technical philosopher you most recently happen to agree with.

The entire scientific method hinges on testing the hypothesis to enough technocrats satisfaction that we then collectively "bless" the latest "discovery" of something that has been there all along. We treat the blessed idea as a "fact" for lack of anything better and employ it as an "assumption" in just about every use except when we describe it directly, then we label it as a "theory" in the finest print possible and get on with making the next "discovery" that we can use as an assumption in the NEXT "discovery"! See that last sentence? Whatever we "discover" today is just one of our assumptions in what we are trying to "discover" tomorrow.

Anyone who argues with that hasn't been reading or publishing enough lately!

[Hiro]

Originally posted by sunilraman
Extra Credit: Two objects are 1 metre apart. They move towards each other, by one quarter the distance separating them every one second (i.e., if they are distance d apart at time 0, they are d/2 at t==1, d/4 at t==2, d/8 at t==3, and so on.) Given d = 1 metre At what time t will the distance between the two objects be 1 Angstrom (d/10billion)?

Which of you 1337 pe0pl3s can answer this :smokey: (my head would burn up in an instant the second I tried to solve this). I suspect this is an easy question though for Maths peoples. I also suspect the answer is *not* 10 billion seconds.

Between the 33rd and 34th seconds the transition will occur. When in between is entirely dependent on the acceleration characteristics applied to the objects between the sample points. Figuring the remaining distances at the whole seconds is just binary vs base 10 math: 2^(-n) <= 10^(-10) solve for n. 10 seconds and a spreadsheet do that quite effectively.

[CosmoNut]

Okay, so what the answer seems to be is that the numerical cannot be tied to the physical in this case. Not only that, but the actual definition of two objects touching is up for debate.

I did learn something new, however: I'd never heard about the 22/7 thing being approximately equal to pi. Interesting.

[midwinter]

NERDS!!!


;)

[benzene]

Originally posted by hardeharhar
I should also say that it is a semantic point with regard to objects touching.

Is your cut off at the 70% orbital overlap level or the 50%

My point exactly, put more succinctly.

Originally posted by Hiro
I say all reality is describable by math, even though we don't know how to do all those descriptions yet. Which has no direct conflict with your Not all math is applicable to "reality."

This is why I said lots of people have a problem with string theory. The issue is that string theory is primarily a mathematical abstraction and is so complex that just about any observation, whether valid or not, could potentially be explained. (Sort of a deus ex machina, but we're not getting into that here).

Most of the phenomena that string theory propose we're completely unable to measure at this point, which has led many to criticize string theory as untenable. It's mathematical abstraction taken beyond the point of description, into a realm of creation, almost. Every now and then, the mathematical description has to be tempered with reality.

Slashdot had an article on this some time ago (http://www.nwfdailynews.com/articleArchive/jun2006/notevenwrong.php).

[sunilraman]

[QUOTE]Originally posted by Hiro
Between the 33rd and 34th seconds the transition will occur. When in between is entirely dependent on the acceleration characteristics applied to the objects between the sample points. Figuring the remaining distances at the whole seconds is just binary vs base 10 math: 2^(-n) <= 10^(-10) solve for n. 10 seconds and a spreadsheet do that quite effectively.


Lets assume acceleration to be constant. Since it is d/2 at 1sec, d/4 at 2sec, d/8 at 3sec, decceleration can be calculated. Extra Credit Part 2: What is the deceleration in metres per second squared?

Cool. Extra credit for Part 1 to you. Yay! But partial extra credit only. Show your working on how you solve for n ..!

I went for a swim earlier and it dawned on me, yeah, just a matter of 2^x = 10billion. Then just incrementally guess x and calculate the result.

Using a spreadsheet and "guessing" x and seeing how close it is to 10billion is kinda cheating though. Did you just do this?

How would you solve for x without using a spreadsheet/ calculator? :smokey: Show your working on paper (since symbols don't show up properly on these pages), scan it, and post it. :D :D :D

[sunilraman]

[QUOTE]Originally posted by midwinter
NERDS!!! ;)


*sigh* after all the sex, drugs and trance music this is the 1% of my brain and knowledge I use to have that is left over to be barely able to process this. Chemistry, electron orbitals, polynomial maths, Newtonian physics, integration and differentiation. I've totally forgotten how to calculate acceleration/deceleration... Phew :err: :\

[addabox]

I'm just dumbfounded that a question about Zeno's Paradox went all sub-atomic and shit.

[sunilraman]

[QUOTE]Originally posted by addabox
I'm just dumbfounded that a question about Zeno's Paradox went all sub-atomic and shit.


"all sub-atomic and shit" :lol: Well, we just came to a pragmatic approach. Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

Will they ever touch? Well, around 33seconds, the distance between the two objects is about 1 Angstrom, one-ten-billionth of a metre. Quantum mechanics start to come into play then. From 33seconds onwards that's where it gets "all sub-atomic and shit" because the definition of "touching" then comes into play.

[sunilraman]

Just asking this again because the question was buried in a post above:

Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

What is the rate of deceleration of each object in metres/(second squared) ?? I Just can't bloody remember how to calculate it.

[midwinter]

Originally posted by sunilraman
[QUOTE]
*sigh* after all the sex, drugs and trance music this is the 1% of my brain and knowledge I use to have that is left over to be barely able to process this. Chemistry, electron orbitals, polynomial maths, Newtonian physics, integration and differentiation. I've totally forgotten how to calculate acceleration/deceleration... Phew :err: :\

Well, all I want is to be able to go to my internet discussion forum for rumors about upcoming Apple Computer products without bumping into a bunch of nerds! ;)

[sunilraman]

ROFL :lol: :lol: :lol:

[hardeeharhar]

In response to how would one calculate how many steps you would have to take to get 1 A separation:

log (10^10)/log(2) = 10*1/(log(2)) = 33.2192809 (looking up on a log table ;))

[sunilraman]

Whoa... Cool. Logarithms. Takes me back to high skool maths. Completely forgotten all that stuff. Don't do drugs, kids!! :embarrass :lol: :smokey: Now, hardeeharhar, can you answer the deceleration rate question?

[hardeeharhar]

Originally posted by sunilraman
Whoa... Cool. Logarithms. Takes me back to high skool maths. Completely forgotten all that stuff. Don't do drugs, kids!! :embarrass :lol: :smokey: Now, hardeeharhar, can you answer the deceleration rate question?

If I had a spread sheet in front of me (Kaleidagraph really, since it can do derivatives)...

[Hiro]

Originally posted by hardeeharhar
In response to how would one calculate how many steps you would have to take to get 1 A separation:

log (10^10)/log(2) = 10*1/(log(2)) = 33.2192809 (looking up on a log table ;))


Ahhh. I see someone making assumptions about the motion!! (I thought scientists didn't do that in your world) Do we know if the motion is smooth, or is it really herky-jerky like the earlier mentioned kids who would take steps and pause??? You can only determine the part to the right of the decimal if you know the equations of motion hence my earlier answer that it is sometime between the 33rd and 34th second. Is it OK to be sloppy when the equations are "nice"? But all lingoistic when writing in prose?

[Hiro]

Originally posted by sunilraman
Using a spreadsheet and "guessing" x and seeing how close it is to 10billion is kinda cheating though. Did you just do this?

Actually I already knew 2^33 is about 8 billion and 2^34 is about 16 billion. I used the spreadsheet to verify I hadn't screwed up my decimal places.

Three columns, one for n {looked at 28-42}, one for 2^(-n) and one for 1^(-10). Then did a quick visual comparison to double check the intervals where the values crossed.

[hardeeharhar]

Originally posted by Hiro
Ahhh. I see someone making assumptions about the motion!! (I thought scientists didn't do that in your world) Do we know if the motion is smooth, or is it really herky-jerky like the earlier mentioned kids who would take steps and pause??? You can only determine the part to the right of the decimal if you know the equations of motion hence my earlier answer that it is sometime between the 33rd and 34th second. Is it OK to be sloppy when the equations are "nice"? But all lingoistic when writing in prose?

It doesn't matter if the steps are jerky or nice and neat and decimilic. The formula still works.

[Splinemodel]

Originally posted by CosmoNut
I've pondered this off and on since high school:
...

To some extent this is the rationale behind quantum mechanics, which is a subject where I'm not even a casual expert, and I won't make much of a claim.

But I will say that, at the subatomic level, energy tends to operate in quantized amounts. This has been observed in many classic experiments of the 20th century.

[Splinemodel]

Originally posted by hardeeharhar
Not all math is applicable to "reality."

Period.

You can continue to deny this, but if you ever have had a conversation with a real mathematician doing research in modern topics, you will quickly realize that math has advanced far beyond simple descriptions of reality. . .

The best part about math is that "imaginary numbers" are very much part of reality.

[SpcMs]

Originally posted by sunilraman
[QUOTE]Originally posted by addabox
I'm just dumbfounded that a question about Zeno's Paradox went all sub-atomic and shit.


"all sub-atomic and shit" :lol: Well, we just came to a pragmatic approach. Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

Will they ever touch? Well, around 33seconds, the distance between the two objects is about 1 Angstrom, one-ten-billionth of a metre. Quantum mechanics start to come into play then. From 33seconds onwards that's where it gets "all sub-atomic and shit" because the definition of "touching" then comes into play.
Yeah, but a pragmatic approach is virtually pointless when tackling a mathematics problem. Sure the math problem is placed in the real world to make it more approachable, but it has no consequence whatsoever as to the actual explenation. Which is, of course, that an infinite addition of infinite small numbers does not get you an infinite large number. This may go against your language-based intuition, but should not go against you mathematical intuition.

The easiest way to 'prove' this, is trying to prove the opposite: that the number will grow to infinite. Well, let's try:

1 + .5 + .25 + .125 + ...
Now, everybody can see this number will never grow bigger than 10. Or 5. Or 2. But it will grow infinitely close to 2 (meaning you can prove it will grow bigger than any number smaller than 2, no matter how close to 2 it is), for all practical ánd mathematical matters (and here is where the two meet) making it equal to 2.

Edit: i never had to explain math in english so excuse me for any weird wording.

[addabox]

Originally posted by Splinemodel
To some extent this is the rationale behind quantum mechanics, which is a subject where I'm not even a casual expert, and I won't make much of a claim.

But I will say that, at the subatomic level, energy tends to operate in quantized amounts. This has been observed in many classic experiments of the 20th century.

Gaaaaahhhh! Zeno's Paradox has nothing to do with quantum mechanics!

Here it is again, restated: Since an arrow flying towards a target must traverse half the distance, then half the distance remaining, and half again, and we can continue to halve the remaining distances forever, the arrow therefore can never reach its target.

This is not resolved by talking about quarks, or electron clouds, or the uncertainty principle, or anything in that neck of the woods. The arrow is not imagined to enter, in the last pico seconds before hitting the target, some kind of boundary ambiguity that resolves the paradox, because that is not remotely what the paradox is about-- unless somebody is prepared to argue that, in fact, the arrow never does reach the target and continues to move through an infinite series of bifurcations, forever.

Which it doesn't, and is in no way a condition suggested by quantum mechanics.

The paradox is simply, and again, about the difference between an infinite sequence tending toward limit x, where x is the distance to the target, and the real world movement of the arrow. The paradox hinges on the ambiguity of natural language being switched for mathematical description, not on some inherent ambiguity in the idea of "things touching".

[midwinter]

Originally posted by addabox
Gaaaaahhhh! Zeno's Paradox has nothing to do with quantum mechanics!

Here it is again, restated: Since an arrow flying towards a target must traverse half the distance, then half the distance remaining, and half again, and we can continue to half the remaining distances forever, the arrow therefore can never reach its target.

This is not resolved by talking about quarks, or electron clouds, or the uncertainty principle, or anything in that neck of the woods. The arrow is not imagined to enter, in the last pico seconds before hitting the target, some kind of boundary ambiguity that resolves the paradox, because that is not remotely what the paradox is about-- unless somebody is prepared to argue that, in fact, the arrow never does reach the target and continues to move through an infinite series of bifurcations, forever.

Which it doesn't, and is in no way a condition suggested by quantum mechanics.

The paradox is simply, and again, about the difference between an infinite sequence tending toward limit x, where x is the distance to the target, and the real world movement of the arrow. The paradox hinges on the ambiguity of natural language being switched for mathematical description, not on some inherent ambiguity in the idea of "things touching".

What utter horeshit. Clearly, you have only been getting one side of the Zeno's Paradox debate, and that, no doubt, is coming from mainstream math, which has a vested interest in maintaining its hegemony. And they will brook no dissent, as your snarky look down your nose just indicated. Whether you like it or not, teh relationship between Zeno's paradox and quantum mechanics is a subject of debate, and sticking your fingers in your ears isn't going to make it go away. If I have some time, I'll dig up some of the papers I have on paleo-Zeno's Paradoxic change over time. You'll find, once you read up on the subject, that the "hockey stick" graph of the frequency of halfway points as the distance between objects decreases is actually a distortion and based on a limited number of experiments that have never been repeated.

[addabox]

Originally posted by midwinter
What utter horeshit. Clearly, you have only been getting one side of the Zeno's Paradox debate, and that, no doubt, is coming from mainstream math, which has a vested interest in maintaining its hegemony. And they will brook no dissent, as your snarky look down your nose just indicated. Whether you like it or not, teh relationship between Zeno's paradox and quantum mechanics is a subject of debate, and sticking your fingers in your ears isn't going to make it go away. If I have some time, I'll dig up some of the papers I have on paleo-Zeno's Paradoxic change over time. You'll find, once you read up on the subject, that the "hockey stick" graph of the frequency of halfway points as the distance between objects decreases is actually a distortion and based on a limited number of experiments that have never been repeated.

:lol: You owe me a keyboard, sans snorted out coffee.

[max_naylor]

I'm glad I've given up maths in college. Although I am doing physics. Poo.

[SpcMs]

Originally posted by midwinter
What utter horeshit. Clearly, you have only been getting one side of the Zeno's Paradox debate, and that, no doubt, is coming from mainstream math, which has a vested interest in maintaining its hegemony. And they will brook no dissent, as your snarky look down your nose just indicated. Whether you like it or not, teh relationship between Zeno's paradox and quantum mechanics is a subject of debate, and sticking your fingers in your ears isn't going to make it go away.
I'm sorry, but that's just bull. Zeno's Paradox was all about the 'mainstream math' problem as discribed by Addabox and myself, and has nothing to do with mechanics. Sure, today it is used to illustrate some theoretical concepts about things 'touching', but is completely unrelated. It's an interesting and worthwile topic, but does nothing to 'solve' or 'answer' Zeno's Paradox.

[addabox]

Originally posted by SpcMs
I'm sorry, but that's just bull. Zeno's Paradox was all about the 'mainstream math' problem as discribed by Addabox and myself, and has nothing to do with mechanics. Sure, today it is used to illustrate some theoretical concepts about things 'touching', but is completely unrelated. It's an interesting and worthwile topic, but does nothing to 'solve' or 'answer' Zeno's Paradox.

Sorry, SpcMs, Midwinter is just having some fun with the tone of another thread about global warming. He's just really good at imitating the style of an obtuse loon. (Hmmmm....... or is he?)

[midwinter]

Originally posted by addabox
:lol: You owe me a keyboard, sans snorted out coffee.

I would send you one, but it would never get there.

[midwinter]

Originally posted by SpcMs
I'm sorry, but that's just bull. Zeno's Paradox was all about the 'mainstream math' problem as discribed by Addabox and myself, and has nothing to do with mechanics. Sure, today it is used to illustrate some theoretical concepts about things 'touching', but is completely unrelated. It's an interesting and worthwile topic, but does nothing to 'solve' or 'answer' Zeno's Paradox.

See? This is how it starts (http://www.renewamerica.us/analyses/050317hutchison.htm).


a) "You are not qualified to question math,"

b) "All mathematicians say X; therefore, anyone who questions X is not of math,"

c) "You must be motivated by political loyalties, economic vested interests, or religious beliefs if you question X,"

d) "You must be ignorant or lacking intellectual honesty if you question X,"

e) You desire a personal insult that demonizes you and casts you in the role of a monster, an enemy, or a fool.


I see you've added a new one:

f) you don't seem to know what the hell you're talking about.

[addabox]

And so it has begun. The post-objective truth era begins to spread through the forums.

Soon, we can question whether the "Mac" actually exists at all, or if it is just a liberal hobby-horse propped up by a cabal of MS hating terror lovers.

[midwinter]

Originally posted by addabox
And so it has begun. The post-objective truth era begins to spread through the forums.

Soon, we can question whether the "Mac" actually exists at all, or if it is just a liberal hobby-horse propped up by a cabal of MS hating terror lovers.

Oh it's far worse than that, my friend. Postmodern mathematicians are destroying traditional liberal education and indoctrinating our children with their so-called "mathematical relativism." All you have to do is look at the debate over Fermat's last theorem to see it in its fullest flowering, where it's become a kind of Rube Goldberg-esque game to see how complicated the answer can be. Anybody with half a brain knows the solution is "equals 4."

We must be ever-vigilant about this if we are to preserve our way of life, because these people truly do hate America and all she stands for.

[addabox]

Originally posted by midwinter
Oh it's far worse than that, my friend. Postmodern mathematicians are destroying traditional liberal education and indoctrinating our children with their so-called "mathematical relativism." All you have to do is look at the debate over Fermat's last theorem to see it in its fullest flowering, where it's become a kind of Rube Goldberg-esque game to see how complicated the answer can be. Anybody with half a brain knows the solution is "equals 4."

We must be ever-vigilant about this if we are to preserve our way of life, because these people truly do hate America and all she stands for.

Two keyboards.

[Hiro]

Originally posted by hardeeharhar
It doesn't matter if the steps are jerky or nice and neat and decimilic. The formula still works.

Not if there are any kinks in the derivative curves, or if accel is non-linear between sample points.

I find the debate on Zeno's utterances infinitely, even limitlessly more satisfying at the moment though.

[Splinemodel]

Originally posted by addabox
Gaaaaahhhh! Zeno's Paradox has nothing to do with quantum mechanics!
Here it is again, restated: Since an arrow flying towards a target must traverse half the distance, then half the distance remaining, and half again, and we can continue to halve the remaining distances forever, the arrow therefore can never reach its target.

When did I ever mention Zeno's Paradox? I was responding to the author of the thread, and happened to notice that none of the physics-based answers really touched upon the basic idea that energy is released in distinct packets.

Beyond that, if there's a universal speed limit, i.e. the speed of light, and it happens to be a quantifiable amount, then what prevents the quantization of minimum amount of energy? If I'm not playing along with your little game, I apologize, but I'm not interested in Zeno.

[skatman]

You're discovering the difference between mathematics and physics.
Mathematics is ONLY concerned with being consistent with itself.
Physics is concerned with explanations of physical phenomena that is consistent with our observations.


Originally posted by CosmoNut
I've pondered this off and on since high school:

I understand the principle behind the assertion that decimals never end and that you can (theoretically) infinitely halve the space between two objects and they'll never touch. A parabolic curve works on this principle, but really...

If two objects are travelling at each other, at some point the space between them must end so that there is NO space left and they are touching, right? By this notion, I'd think that a value of their distance apart MUST end. So it seems that at some point 0.00000000000000000...1 must become simply 0 when the objects finally touch.

--> <--

Surely someone way smarter than I am can explain this. What am I missing here?

[skatman]

Originally posted by sunilraman
Just asking this again because the question was buried in a post above:

Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

What is the rate of deceleration of each object in metres/(second squared) ?? I Just can't bloody remember how to calculate it.

The distance between objects at time = t (sec) is 1/ (2^t) meters.
Acceleration is second derivative of distance function.
That's twice the deceleration of each object.
I'll let you ponder the simple derivative. :-)

[hardeeharhar]

Originally posted by Hiro
Not if there are any kinks in the derivative curves, or if accel is non-linear between sample points.

I find the debate on Zeno's utterances infinitely, even limitlessly more satisfying at the moment though.

The acceleration between points doesn't matter (remember the pathway independence concept in all physical sciences)...

But really whatever.

[addabox]

Originally posted by Splinemodel
When did I ever mention Zeno's Paradox? I was responding to the author of the thread, and happened to notice that none of the physics-based answers really touched upon the basic idea that energy is released in distinct packets.

Beyond that, if there's a universal speed limit, i.e. the speed of light, and it happens to be a quantifiable amount, then what prevents the quantization of minimum amount of energy? If I'm not playing along with your little game, I apologize, but I'm not interested in Zeno.

Sigh.

Please read CosmoNut's original post again. It's in two parts:

1) CosmoNut notes that he understands that the distance between two objects can be halved repeatedly, forever, yielding a number that constantly approaches, but never reaches, zero (or, to put it another way, the sum of these numbers approach, but never equal, the original distance).

2) He then wonders how that it is that objects may be observed to traverse this seemingly "infinite" space (the space getting smaller between two objects approaching each other part isn't necessary, the principle is the same for any finite interval).

This is Zeno's paradox. Sorry it doesn't float your boat, but that's what it is. It isn't a physics problem, and pointing that out isn't "my little game". Everybody is free to talk about anything they want, but CosmoNut's original question is very definitely "So, what's up with Zeno's paradox?"

[sunilraman]

[QUOTE]Originally posted by skatman
The distance between objects at time = t (sec) is 1/ (2^t) meters. Acceleration is second derivative of distance function.
That's twice the deceleration of each object.
I'll let you ponder the simple derivative. :-)


Sorry skatman, can't process this stuff. Why can't anyone tell me what the deceleration of the object in metres per second squared?

*All I remember is that if you have an acceleration curve, you differentiate at a certain point you get velocity.... or something like that.

[sunilraman]

[QUOTE]Originally posted by addabox
Two keyboards.

[QUOTE]Originally posted by midwinter
I would send you one, but it would never get there.


ROFLMAO :lol: :lol: :lol: :lol:

[sunilraman]

edit: Zeno's paradox is somehow related but Let's go back to CosmoNut's question and the title of this thread.

"Physics: Decimals can't be infinite because the space between must end."

[QUOTE]Originally posted by CosmoNut
I understand the principle behind the assertion that decimals never end and that you can (theoretically) infinitely halve the space between two objects and they'll never touch. A parabolic curve works on this principle, but really...

[QUOTE]Originally posted by CosmoNut
If two objects are travelling at each other, at some point the space between them must end so that there is NO space left and they are touching, right? By this notion, I'd think that a value of their distance apart MUST end. So it seems that at some point 0.00000000000000000...1 must become simply 0 when the objects finally touch.


You all are arguing in a totally different space of thought as to what CosmoNut is saying. In the first part he is talking about a series of infinitestimally small numbers, then jams on the idea to a real world physical idea.

Thus any discussion here should involve both physics and applied maths

In CosmoNut's second part, he confuses the issue further by saying that he makes an assumption 0.0000.....0001 must equal zero, therefore two objects extremely super-close to each other must therefore also have zero space between them.

Yes Zeno's paradox is related in some way, but it is a bit different in that it talks about an arrow flying to the air, at each stage (slice of time) it is not moving, therefore on the whole it never moves at all. The paradox is proven false in general by looking at it and saying, you can't just decide that time is discrete and slice it up the way you might do with an infinitely small series of numbers. Ie. 0.00000000....0000000001 sec is not equal to zero.


So we go back to try and explain in a way that deals with the *PHYSICS* aspect of CosmoNut's question, eg two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc. Will they ever touch? Well, around 33seconds, the distance between the two objects is about 1 Angstrom, one-ten-billionth of a metre. Quantum mechanics start to come into play then. From 33seconds onwards that's where it gets "all sub-atomic and shit" because the definition of "touching" then comes into play.

Similar in a way to disproving Zeno's paradox of the arrow is saying that just because you can split numbers into superbly small amounts doesn't mean that space or time in "the real world" works that way. Physics people are still exploring what the smallest amount of space is (they don't just say 0.000000....00000000001 metres must equal zero) because subatomic space just doesn't work that way. Similarly, they don't just cut up time into superbly small amounts and then say 0.0000000.....0001 seconds must be zero. If they did this, then it would create a situation where time did not exist at all. First and foremost this violates Newtonian physics. It also affects quantum mechanics because for example, the position of an electron in an atom is not static. It pops here and there (mostly within the electron orbital) OVER a certain amount, however small, of time.


So essentially, we go back to the point that in Maths you can do a lot in pure maths that doesn't have to relate to the real world. However, when you use certain aspects of Maths and try and apply it to the real world, eg. CosmoNut's question and Zeno's paradox, it comes straight up against the Physics view of the world. Whether you're coming from the Maths side of things or the Physics side of things the moral of the story is that when combining the two you can't just simply BASH TOGETHER one idea from Maths and another idea from Physics and hope it flies (pun unintended wrt Zeno's arrow)

[sunilraman]

[QUOTE]Originally posted by max_naylor
I'm glad I've given up maths in college. Although I am doing physics. Poo.


Can you answer my deceleration question please? Nobody else here feels like doing it: Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

What is the rate of deceleration of each object in metres/(second squared)?

[midwinter]

Originally posted by sunilraman
therefore two objects extremely super-close to each other must therefore also have zero space between them.

Is "extremely super-close" some high-falutin' technical math/physics term?

[sunilraman]

[QUOTE]Originally posted by Hiro
Not if there are any kinks in the derivative curves, or if accel is non-linear between sample points.


In my question, I was asking how long it takes for the distance between the two objects to equal 1 angstrom. Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

For the sake of the problem, it was assumed that the pattern of movement as described above is constant. No kinks :)

[sunilraman]

[QUOTE]Originally posted by midwinter
Is "extremely super-close" some high-falutin' technical math/physics term?

:lol: :err: Okay now you owe addabox two keyboards, and you owe ME one keyboard and some nasal- sinus- cleanser-spray.

[Hiro]

Originally posted by hardeeharhar
The acceleration between points doesn't matter (remember the pathway independence concept in all physical sciences)...

But really whatever.

Now you just exposed yourself as espousing bullcrap. How much else have you made up? Lingo can't save you from that gaffe. :lol:

The fatal counterexample: an acceleration step function. No movement until 33.5 sec (later than your log result), then constant velocity (instantaneous infinite accel) and immediate decel back to zero velocity. Your log function computes an incorrect time because it implicitly assumes a smooth non-kinked acceleration function. It can't take into account that movement may not even commence until after the smooth functions result, leading to an obvious contradiction.

The fact infinite accel is theoretically impossible is of little consequence since there are still an infinite number of other accelerations that can also be applied after 33.5 sec that will get our object to the correct point at 34 sec, but it only takes a single counterexample to prove your statement false.

[Hiro]

Originally posted by sunilraman
[QUOTE]Originally posted by max_naylor
I'm glad I've given up maths in college. Although I am doing physics. Poo.


Can you answer my deceleration question please? Nobody else here feels like doing it: Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

What is the rate of deceleration of each object in metres/(second squared)?

The acceleration changes over time.

The proof of this goes something like this (and it's easier to shof for the whole system rather than one side at a time):

If we assume constant acceleration the equations of motion are as such:

The total collective distance traveled is 1-2^(-t); 1/2 at t=1, 3/4 at t=2, etc

also distance travelled = Vo + 1/2At^2

making 1-2^(-t) = Vo + 1/2At^2

Vo happens to be zero as no motion at time zero so that term drops out leaving 1-2^(-t) = 1/2At^2

simplifying gets us

2 [ 1- 2^(-t) ] = A
t^2


Here we see A changes as t changes. A contradiction of the equations assumption that acceleration is constant.


I don't remember off the top off my head how to derive the integrated acceleration equation which would allow us to solve for an instantaneous acceleration and velocity. I'll leave that exercise to other brave souls. Maybe hhh wants to redeem himself. :p

[FormerLurker]

Originally posted by midwinter
Is "extremely super-close" some high-falutin' technical math/physics term? This thread is seriously in need of the proper engineering terminology, which is:

"Close Enough".

[Mac_Doll]

Bwahahaha!! You started this thread, CosmoNut, and like you and I predicted, all the lovely math nerds came out of the woodwork. :D I love it.

I say to Hell with Math... I'm an art major. ;)

[max_naylor]

Originally posted by sunilraman
Originally posted by max_naylor
I'm glad I've given up maths in college. Although I am doing physics. Poo.


Can you answer my deceleration question please? Nobody else here feels like doing it: Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

What is the rate of deceleration of each object in metres/(second squared)?

I'm sorry I have no idea what your talking about. I haven't actually started college yet, I've just left secondary school! So what you're asking is beyond the scope of a GCSE textbook by the sound of it. I hope someone else has managed to answer your question. (I meant sixth form college, I'm only 16 on the 21st!!!, college means uni over there doesn't it? Ask me in two years time and I may be able to tell you ;))

[sunilraman]

[QUOTE]Originally posted by Mac_Doll
Bwahahaha!! You started this thread, CosmoNut, and like you and I predicted, all the lovely math nerds came out of the woodwork. :D I love it.....I say to Hell with Math... I'm an art major. ;)


Heh. All the bloody art majors at my college/university only had classes 3 days out of 5, most of those only like 9am-1pm or 1pm-6pm or something like that...!

I had 5 full days in a week. A few 3 hour practicals each week.

I majored in Biology[Molecular] - 3/4s of my degree. The other 1/4 was IT. SmallTalk/ Ada95/ SQL/RDBMS theory. At least the SQL/RDBMS had some practicals in Access. The coding stuff, oh had one C programming class. A pity they didn't teach anything practical like ASP or Cold Fusion. Heh. PHP was not even on the horizon back then.

*sigh* My IT stuff at Uni was high-scoring, but I did only a few cherry-picked subjects here and there. Not enough to really make a career out of it except for web design for a few years!!! Now though with server admin and MCSE CCNA and whatever the hell the whole IT industry looks weird to me.

Well, that's what happens when you major in Molecular Biology (enzymes, DNA, RNA, protein folding, blah blah blah) and end up HATING it and trying to cobble together a career in Web Design....

Not to mention all the self-learning Photoshop and Illustrator and GOOD graphic design AND self-learning about art movements (Dada, Surrealism, etc) :\ Did some bad (bad as in bad not bad-ass) Lightwave 3D stuff for a bit....

Yeah, F**K Math. Last class I did was back in '95. At college/uni the only Math we needed in Biology was more the Biochemistry stuff like calculating concentrations of mixtures and chemicals and stuff.

[sunilraman]

This discussion would not be complete without the discussion of Planck Units. http://en.wikipedia.org/wiki/Planck_units

"...At lengths and times of less than approximately one Planck unit, quantum theory as presently understood no longer applies."

Smallest length in the universe - 1 Planck unit approx. 1.61624 × 10^-35 metres

Smallest time slice in the universe - 1 Planck unit approx. 5.39121 × 10^-44 seconds

Just wanted to add this to the discussion as one of the Physics' view of super-duper-tiny (yes this is a highly technical term) amounts of time and length.

[hardeeharhar]

Originally posted by Hiro
Now you just exposed yourself as espousing bullcrap. How much else have you made up? Lingo can't save you from that gaffe. :lol:

The fatal counterexample: an acceleration step function. No movement until 33.5 sec (later than your log result), then constant velocity (instantaneous infinite accel) and immediate decel back to zero velocity. Your log function computes an incorrect time because it implicitly assumes a smooth non-kinked acceleration function. It can't take into account that movement may not even commence until after the smooth functions result, leading to an obvious contradiction.

The fact infinite accel is theoretically impossible is of little consequence since there are still an infinite number of other accelerations that can also be applied after 33.5 sec that will get our object to the correct point at 34 sec, but it only takes a single counterexample to prove your statement false.

What is your problem?

You are trying far too hard to prove me incorrect. It is actually pathetic.

My statement is completely true: acceleration doesn't matter if the object ends up at the same spot at the same time.

Oh, and the distance dependence of one of the moving objects (the one moving in the positive direction) is:

d= 1/2*(1-2^-t)

The derivative gives us the instantaneous velocity:

dd/dt=V(t) = (1/2)*(2^-t) ln(2)

The second derivative gives us the acceleration:

dV(t)/dt = -(ln(2)/2)^2*(2^-t)

So yes, as hiro said the acceleration is not constant, but of course we knew that without doing the math - constant acceleration never lets an object in motion come to rest at t= infinity.

Sorry it took me so long to respond, I had a long day in lab yesterday.

Edit:

Similar results for object moving in negative direction:
d=1-1/(1-2^-t)
V(t)= -(1/2)*(2^-t) ln(2)
A(t)= (ln(2)/2)^2*(2^-t)

[skatman]

Originally posted by sunilraman
[QUOTE]Originally posted by skatman
The distance between objects at time = t (sec) is 1/ (2^t) meters. Acceleration is second derivative of distance function.
That's twice the deceleration of each object.
I'll let you ponder the simple derivative. :-)


Sorry skatman, can't process this stuff. Why can't anyone tell me what the deceleration of the object in metres per second squared?

*All I remember is that if you have an acceleration curve, you differentiate at a certain point you get velocity.... or something like that.

Time derivative of the acceleration function is called a "jerk".
To get the velocity, you need to integrate the acceleration function with respect to time.

[Hiro]

Originally posted by hardeeharhar
What is your problem?

You are trying far too hard to prove me incorrect. It is actually pathetic.

My statement is completely true: acceleration doesn't matter if the object ends up at the same spot at the same time.

What you say in the last half of the last quoted sentence is true. But that isn't what your value of t=33.2192809 represents from this post:

In response to how would one calculate how many steps you would have to take to get 1 A separation:

log (10^10)/log(2) = 10*1/(log(2)) = 33.2192809 (looking up on a log table ;))

That is explicitly for some intermediate point between time step sample points that happens to be at 1 Angstrom distance.

It's easy to be pissy and indignant when you move the goal posts, but when everyone looks you are someplace "over there" (wherever that is) waving your arms furiously, while the goal is really still right where it is all along.

[hardeeharhar]

Who is using lingo now?

Goal post, waving arms?

Come on, any intelligent reader can come to the realization that quite literally 33.something is in between 33 and 34. If you parse the steps in units then you have to round up. If you parse the steps in decimals, then you can provide how ever many decimals you feel like providing. Regardless, my answer is absolutely correct to the nth place, and I have left it up to the Intelligent reader to figure out where they parse.

(honestly, I think you are just pissed because you forgot how logarythms work)...

[Mac_Doll]

sunilraman: Oooh, you're a Biology major. I was one too, before I changed it to Art. I was going into Zoology. I figured that my artistic talent is more rare than knowledge of the animal kingdom. I still adore science though.

[Hiro]

Originally posted by hardeeharhar
Who is using lingo now?

Goal post, waving arms?

Come on, any intelligent reader can come to the realization that quite literally 33.something is in between 33 and 34. If you parse the steps in units then you have to round up. If you parse the steps in decimals, then you can provide how ever many decimals you feel like providing. Regardless, my answer is absolutely correct to the nth place, and I have left it up to the Intelligent reader to figure out where they parse.

(honestly, I think you are just pissed because you forgot how logarythms work)...

I don't know if I should admire your single-minded persistence in the face of adversity, or pity you for abject self-inflicted cluelessness. It's kinda sad because you obviously have some smarts, but have your ego filters so tight you can't see, or at least acknowledge, your error.

[hardeeharhar]

Originally posted by Hiro
I don't know if I should admire your single-minded persistence in the face of adversity, or pity you for abject self-inflicted cluelessness. It's kinda sad because you obviously have some smarts, but have your ego filters so tight you can't see, or at least acknowledge, your error.

Aren't we the condescending one?

These statements from someone who has attempted to suggest my answer was a representation of something more than just an answer.

Edit: Removed gratuitous go away.... You should know better than to continue...

[Hiro]

Originally posted by hardeeharhar
Aren't we the condescending one?

These statements from someone who has attempted to suggest my answer was a representation of something more than just an answer.

Edit: Removed gratuitous go away.... You should know better than to continue...

Hit a nerve did I? Accuracy sucks, doesn't it.

[hardeeharhar]

Originally posted by Hiro
Hit a nerve did I? Accuracy sucks, doesn't it.

No, not really. As someone who has been called condescending throughout my adult life, I enjoy spreading the love.

[sunilraman]

[QUOTE]Originally posted by Mac_Doll
sunilraman: Oooh, you're a Biology major. I was one too, before I changed it to Art. I was going into Zoology. I figured that my artistic talent is more rare than knowledge of the animal kingdom. I still adore science though.


Heh. It was kind of weird. On one hand I had read and watched A LOT of sci-fi growing up (though not as much as most on these boards ;)) so I was like DNA-this and genetic-that and that was all cool with the Molecular Biology and Biochemistry stuff.

On the other hand I was doing Botany, Ecology and (very basic) Zoology. To like, save the world, man..... like, dude..... check out these flowers, man.... whoaa.... One of the Ecology 101 assignments was to go off to an island off Brisbane, Australia (Stradbroke Island) and count the number of trees or some shite like that. And I was, how do we save the world doing this nonsense? Like, dude, where's the environmental activism?

So yeah in my fourth and final year of college I pursued the genetics part heavily and a lot of mice died in the name of science and a lot of frog tadpoles were genetically mutilated via DNA type injections. When I graduated I was like, f**K this and did web design for a few years.

2003 I actually worked at Greenpeace in Australia doing web admin/ design and was finally, like Fighting The Power, man.....

Then I went off and wanted to be an artist and 2004 was all over the place. My artistic talent was too heavily tied in with my emotions and spiritual outlook. Not sustainable :( Making a living from my artistic talent was incredibly stressful, dude.

2003 at Greenpeace was a kinda cool job though. Not as hippy as you think, and not as corporate-environmentalism as you might fear. But my parents just did not understand WTF I was doing and that it was like, a real job, man.... :rolleyes: :no:

[sunilraman]

Weird thing. I'm like an idiot savant sometimes, knowing some stuff about this here and this that but can't do other things. I'm happy to have somehow "predicted" the 50-cent-Apple thing and brought the infinite-numbers discussion here into the subatomic physics realm. But a lot of empty spaces in my knowledge and understanding. Like you peoples that did the derivative thingy to get an expression for the deceleration of the objects, that was 1337, man....

Anyway, love you all, spread the AppleInsider love.... Hiro and HHH, let's not fight man, let's spread the intellectual love, man.... yeahhhh Have a spliff between you two and all will be chilled man.... yeahhhh :smokey:

[Hiro]

Originally posted by hardeeharhar
No, not really. As someone who has been called condescending throughout my adult life, I enjoy spreading the love.

Just two peas in a pod... ;)

[Delectric]

Originally posted by blackbird_1.0
*head explodes*

yeah im with ya on that one. lol.