"As to computer calculations it’s not only about truncation errors in computation. There is no way to represent sqrt(2) precisely dew to irrational nature of sqrt(2). All numbers n^(1/n) are irrational with the exeption n=1.Who knows, may be one day we’ll be seating in front of the computer with some sort of quantum processor where it’s possible. In expression for probability to go down or up one step for quantum oscillator there is sqrt(2), well this one is just an observation and what about all the rest of “irrationality” after all. (the power of set of rational numbers)/(the power of set of irrational numbers) = 0 if you will. That is how the computer computations weak. Well not really weak, there is a good deal on numerical calculation error estimation and everything…"That’s my quote from another topic. The fun is to come up with some sort of quantum system capable of generating any given rational or irrational number with a given precession. Let’s forget about computation and measurements as they will destroy the quantum system and our beautiful sqrt(2) or whatever.

-1 also satisfies n^(1/n) is rational, as does any number of the form 1/k for nonzero integer k.Would a computer that contained an exact representation of the square root of 2 have infinite information and therefore be thermodynamically impossible to construct?

Yes, n > 1 of course."Would a computer that contained an exact representation of the square root of 2 have infinite information and therefore be thermodynamically impossible to construct?"Yes it would. There is not enought information in the whole Universe to represent sqrt(2) (assuming the Universe is finite).What about any given (finite) precision?

Last edited by EStealth on July 14th, 2004, 10:00 pm, edited 1 time in total.

EStealth,For arbitrary finite precision arithmetic try Numerical Recipes and use your home PC.

Your answer implies that I haven’t ever used it…

If I say that there is not enough information in the Universe to represent any of the irrational number and based on that I say that there is no continuum in the Universe. What it would mean for a QUANT?

- tristanreid
**Posts:**441**Joined:**

QuoteOriginally posted by: EStealthIf I say that there is not enough information in the Universe to represent any of the irrational number and based on that I say that there is no continuum in the Universe. What it would mean for a QUANT?Ummm...that the quant should get a better computer?Maybe the solution is to create a computer that understands all of the properties of sqrt(2). Symbolic computing? As long as there is enough steam in the computer that when you put in an arbitrary number n, it can compute the (n+1)th digit of sqrt(2), and return the nth digit, rounded accordingly.Maybe you can get around those pesky thermodynamics laws by programming concepts instead of brute force. You can certainly improve performance in a calculation by recognizing things like: "these terms cancel", etc.That's why I don't like the conclusions of Penrose's book "The Emporer's New Mind" (or something like that). If I remember correctly, he uses Godel's incompleteness to show that humans will never understand the nature of thought, and will therefore never be able to build computers that can truly think. I say it's just a matter of compartmentalizing, but then I'm a lowly IT drone, not a famous physicist.-t.

Sure, it's easy to represent the square root of two in a computer, I was not precise. It's clearly impossible to represent it as the ratio of two integers, because the integers would be infinitely long and require infinite storage space. I think it's also impossible to represent it with a quantum computer that had probabilistic representations (for example you could represent it in a standard binary integer with a 2 bit that had probability 0.707106781186548 of being set anytime you looked at it). I'm not sure it's impossible, I just asked the question.

Let’s change the course of the topic a bit. I like the idea that there is not enough information in the whole Universe to represent sqrt(2) or any of the irrational number. So the Universe doesn’t have a clue about continuum. It’s our “abstraction”, “approximation” to describe the nature. If there is no continuum in the Universe then everything is quantum perhaps? Quantum space, quantum time… What do you say?

QuoteOriginally posted by: EStealthQuantum space, quantum time… What do you say?I thought that was already the standard model.Anyway, what do you think "space" is? Space is a summary of the probable structural relation of matter. But just because the model contains a description of the contents of space, does not mean there is any such thing as space. After all, nothing can interact with space, but only with other things. I refer to the network across which adjacency is negotiated as "the microstructure of empty space." I refer to the simplest stable, extended, self-shielding network which most people call "space" (and which can be modeled as three-dimensional on medium scales) as "photospace," whereas as tangled space can be referred to as "intranuclear space."My ideas suggest that you should be able to "stretch" wormholes if you are able to shield them from shocks - you know, so that they don't know they are wormholes. But given the absence of accidental wormholes in empirical experience, I decided that the necessary conditions for stretching a wormhole, and the reasons you would want to stretch a wormhole - specifically that conditions would be different from one end to the other - were mutually exclusive.For example, you could trick someone into stepping off a cliff if you set up enough mirrors so that it appeared to him that he was stepping into his own bed. But in the case of a wormhole, the light reflected in your mirrors would have to have propagated across ordinary photospace. So you couldn't use them to trick him into stepping beyond the crystal of ordinary photospace. In other words, if you want to dissipate regional variations by overcoming geographic friction, the meeting of the variations will tell the wormhole it is a wormhole, and that where it leads to is not adjacent to where it leads from. More generally, all you could create would be an explosion, where the energy would assemble itself darwinistically into intervening space.Since I don't know anything about physics and I don't want to learn, I was happy that no accidental wormholes had been observed. But then I recently heard about "quantum entanglement." It seems possible this fits the missing phenomenon which my idea predicted we should have observed. So maybe you can move things through large wormholes when the conditions at both ends are closely controlled, shielded with magnets at enormous energy expense, at identical postions above sea level and in the lunar cycle, and so on. And then you could move large self-shielding matter anomalies - i.e. objects - through them, just like through any photospace.Still, it seems that you would have much better luck moving information-rich or unique objects, rather than massive objects, through the hole. At the very least, you would have to move massive "mirror" object, or a sufficient pattern of them, at the other end. What this would have to trick the massive object into thinking, is that it was near itself along circular paths in euclidean space. I'm tempted to make a .gif animation...

- alexandreC
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Estealth, this is not a mathematical puzzle, this is one of the underlying problems of a great deal of research being carried out in the most advanced Physics institutes (actually, I have a few friends doing this sort of stuff, check out http://www.qubit.org/ )Alex

Last edited by alexandreC on July 16th, 2004, 10:00 pm, edited 1 time in total.

QuoteOriginally posted by: alexandreCMeasure the spin of the electron along the x direction.You will have exactly Sqrt(2)Define "you," "have," and "measure." In words a layman can understand, if possible.I don't see how any "measurement" can produce an irrational number, unless you have that spot marked on the ruler. So far as "you," I can only have Sqrt(2) if things are reduced back to symbols. So far as "have," I am not sure of the exact location of the information copied.

QuoteOriginally posted by: alexandreCFollowing these steps, you have a way of constructiong an exact representation of Sqrt(2).To a guy who doesn't know anything about physics, it seems at the very most you could only use this if you knew you were going to arrive at a rational number. And you could only be sure of your result if you had already performed the necessary calculation. As such, your quantum contraption, while recording sqrt(2) in its subparts, would simply be a symbol for another quantity when taken in its entirety. For all I know, you could use two of these to produce the number 2. But I don't see why you need a quantum computer to do this. Could you produce sqrt(2) as a result? See what I mean?

- alexandreC
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I accept farmer's critics on my attempt of describring a quantum computer in simple terms. New try:The need of a quantum computer for the resolution of this problem lies in the fact that we can construct quantum states that are neither 0 or 1 (which is equivelent to a spin of an electron along x or y).The orientation of the spin of an electron can be compared to the Shrodinger's cat (have you heard abou it?) that is neither death or alive. It is both.Exactly like the cat, the electron's wave funtion is a linear combination of an electon with a spin along x and another one, complitely diferent, with a spin along y, and a third independent electron, along z.Just like the cat is death and alive at the same time, this electron is polarised along x, y, and z at the same time.(Which is completely diferent from saying "the cat is either death or alive, we will know when we open the box".)Now, a clever choice as for the orientation of elecromagnetic fields that select "our" electron from a source (that can be a radioactive element decaying in subparticles, for example), will have the effect of giving the measurement of the spin of the electron along a specified direction (say, x ) a probability of Sqrt(2).Alex

- tristanreid
**Posts:**441**Joined:**

QuoteOriginally posted by: AaronSure, it's easy to represent the square root of two in a computer, I was not precise. It's clearly impossible to represent it as the ratio of two integers, because the integers would be infinitely long and require infinite storage space. I think it's also impossible to represent it with a quantum computer that had probabilistic representations (for example you could represent it in a standard binary integer with a 2 bit that had probability 0.707106781186548 of being set anytime you looked at it). I'm not sure it's impossible, I just asked the question.Let me qualify by first stating that I have very little knowledge of quantum computers, I'm at about 'magazine article' level, and haven't even read one recently. So this post has a danger of wasting your time if you read it. So anyway...That's an interesting question. Perhaps I can skirt the issue of "I have no idea" by asking the following: What if? I'm conjecturing that a computer could be built that somehow uses lengths of rope to perform calculations (putting 2 pieces end to end is adding, for example), and the sqrt(2) operation is just measuring the hypotenuse of a unit triangle. The information is there in the form of the length of rope, but no thermodynamic laws are broken. Let us suppose that we have a way within a quantum computer to perform some calculation, like sqrt, without constructing that operation from other mathematical operations. Then we can suppose that quantum computers have a magical property that allow us to read that result to any 'depth' that we desire. The answer is exactly sqrt(2), and the entanglement of qubits 'holds' that property as a probabilistic representation, but we have to read it to some arbitrary number of digits by measuring it in some way. So in order to do calculations, we keep the qubits in their natural state and make them perform calculations, but it is only in reading them that we lose precision.I guess the real question is whether we could ever construct such a device...do I read too much science fiction?-t.

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