Constructing Lengths with Irrational Numbers

[gravenewworld]

This might be a stupid question, but how can you construct something that has an irrational length? For example if you make a right triangle with the 2 sides=1 the hypotenuse is sqrt(2). How can sqrt(2) be a length if that number goes on for ever and never repeats?

 

[matt grime]

Equally, how can you construct something that has a rational side?

Real numbers are not physical lengths. The reals are used to model such quantities, but they ar idealized.

If you wish, think on it this way:

IF I could construct a length of *exactly* 1 unit, and then construct another, and IF I were able to stick these together at exactly 90 degrees and IF I were able to then construct a straight edge joining the resultant ends and IF I were to ignore all bending of spacetime and everything else, then the constructed length *might* be thought of as of length sqrt(2) units.

Your dislike of infinitely long decimal expansions with no repetitions is just that, a dislike, it has no physical bearing since numbers are not physical objects.

Suppose I were to construct a perfect circle of radius 1, then I've just constructed an area of exactly pi units^2. And pi is transcendental, never mind just irrational.

 

[NeutronStar]

Real Quantities vs. Mathematical Abstractions

This might be a stupid question, but how can you construct something that has an irrational length? For example if you make a right triangle with the 2 sides=1 the hypotenuse is sqrt(2). How can sqrt(2) be a length if that number goes on for ever and never repeats?

Don't worry too much about it. Irrational numbers are just an inconsistent fabrication of abstract mathematics. Irrational lengths can't exist in the real world. There is no way that you are going to construct a perfect circle or perfect triangle in the real universe. Planck's constant and the Heisenberg uncertainty principle will prevent it. There simply aren't any "physical" irrational lengths in the universe. They don't exist. Even the universe rounds off irrational quantities.

The universe behaves more like a physicist than a mathematician.

I've found that the best way to think of irrational numbers are in terms of the self-referenced situations that give rise to them. Thinking of them in this way we can clearly see that these quantities are merely reflections of a self-referenced situation (like putting two mirrors back-to-back). The images go on forever (or at least they appear to). We know that in the real universe even those images in "perfect" mirrors would have to end at the resolution of a photon. So again, the universe even rounds off this situation.

The universe is great at rounding things off.

By the way, abstract mathematics is inconsistent in that they treat irrational numbers as being both, infinite decimal expansions, and precise calculus limits.

In other words, Cantor's famous diagonal proof that the set of real numbers has a larger cardinality than the set of natural numbers depends on the infinite expansion of decimal numbers. Yet, by formal definition these real numbers are said to be equal to their calculus limits. That's an inconsistency in logic. After all, if we take the real numbers to be equal to the calculus limits of their decimal expansions then each real number is a finite quantity.

In other words, we could simply represent each real number by a symbol such as pi, or e, or the square root of 2, or whatever symbol we wish to use.

Well, if we do this look what happens!

Let S_n be a symbol for a limit of a real decimal expansion. Then the real numbers can be listed as S₁, S₂, S₃, ... and so on.

In other words, I've just put the real numbers into a direct bijection with the natural numbers proving that they have the same cardinality! And I did this by using the formal definition of mathematics that the real numbers are equal to the limit of their decimal expansions!

Mathematicians want to have their cake and eat it too. They want to claim that the decimal expansion of real numbers are equal to their limits, yet at the same time they want to claim that the cardinality of the set of real numbers is somehow larger than the cardinality of the natural numbers.

Either 0.999... does not equal 1 and Cantor's conclusions about cardinality is correct.

Or

0.999... does equal 1 and Cantor was wrong! The set of natural numbers and the set of real numbers have precisely the same cardinality.

Both of the above can't be true simultaneously! Mathematical formalism is inconsistent. The mathematical community needs to decide which way they are going to do things and stick to their guns. In the meantime they're making me dizzy!

In short, don't take abstract mathematics too literally!

 

[HallsofIvy]

" They want to claim that the decimal expansion of real numbers are equal to their limits, yet at the same time they want to claim that the cardinality of the set of real numbers is somehow larger than the cardinality of the natural numbers."

No, mathematicians don't "claim" that- that is the DEFINITION of "decimal expansion".

And it does not conflict in any way with the fact that the cardinality of the set of real numbers is larger than the cardinality of the natural numbers.

"In short, don't take abstract mathematics too literally! "

Certainly not if you don't understand it.

 

[shmoe]

Oh my. You know if you're uncomfortable with decimal expansions, you can take the diagonal argument and replace all the decimal expansions that appear like 0.a_{i1}a_{i2}a_{i3}\ldots with
\lim_{k\rightarrow\infty}\sum_{j=1}^{k}\frac{a_{ij}}{10^j}

and everything will work out just fine. It's understood (by most) that this is really what we mean by the decimal expansion, which is used for notational convenience.

Let Sn be a symbol for a limit of a real decimal expansion. Then the real numbers can be listed as S1, S2, S3, ... and so on.

I suppose you're able to offer some sort of proof that every real number is on your list there?

 

[matt grime]

And provide a definition of self referenced, as well as contra-proofs of all the other methods of showing that the reals are uncountable. Of course, that is assuming you have ever bothered learning from all the posts explaining to you what the real numbers are, cos it it doesn't appear you have.

Noticed that you're the only person to ever claim that mathematics ought to be physics, Neutron?

Your alleged proof of the enumeration of the reals is again laughably inaccurate, not to say circular - what n in the sense of S_n corresponds to the limit of :

\sum_r 10^{-(r!)}

without assuming an enumeration.


(A) "little learning" does indeed appear to be a dangerous thing.

 

[quantumdude]

Since our mathematicians are correcting the errors in math and logic, I'll correct this bit.

Planck's constant and the Heisenberg uncertainty principle will prevent it. There simply aren't any "physical" irrational lengths in the universe. They don't exist. Even the universe rounds off irrational quantities.

You appeal to the uncertainty principle, and then you turn right around and make a claim which that very principle says is impossible. You claim to know for certain that irrational lengths do not exist. How can that be? Can you measure the length of an object exactly, through the quantum fuzziness, to rule out any possibility of an irrational length? And can you perform such a measurement on every single body in the universe?

Until you can do that, you cannot claim that Heisenberg is on your side here.

The universe behaves more like a physicist than a mathematician.

And apparently you behave like neither.

 

[fourier jr]

any algebraic number can be constructed; that includes irrationals like sqrt(3), etc. (but not e or pi or anyone of an uncountable number of other transcendental #s) i don't know what exactly the problem is though. is it that the technology (etc) doesn't exist to be able to construct irrational lengths in the real world, rather than just in theory?

(getting back to the original topic, although that physics talk is interesting)

 

[matt grime]

The concept of length is almost entirely relative, it is arguably meaningless to say that anything in the real world is some exact length: construct something for me "exactly" 1.5 metres long, for instance.

 

[NeutronStar]

I'll be on my way,...

Well, I was going to respond to some concerns but I've just received a warning via email from a super mentor stating that my thoughts are not welcome here.

Hmmmm?

I must say that I'm shocked. I had no clue that I was that threatening. I must have touched on some pretty sensitive topics.

I will gladly leave this forum as I prefer to share ideas with more open-minded people anyway. However, before I leave I would like to respond to one thing that Matt said,…

Noticed that you're the only person to ever claim that mathematics ought to be physics, Neutron?

I never made any such claim. However, it would seem to me that scientists would indeed be concerned that any mathematics that they use does indeed correctly model the quantitative nature of the universe.

It is also mathematicians who seem to claim that mathematics is so closely related to physics. I see many physics students on here that are asking for justification (such as this very thread) of how something like an irrational number can "exist". In other words, what does it mean to have an irrational quantity. I believe that the answer that I gave is an answer that most distinguished physicists would agree with.

It seems ironic to me that this forum is called the "Physics Forum" yet it is obviously very much weighted in the direction of abstract mathematics. So much so that when people try to talk about topics from a physical point of view all the mathematicians jump into the thread and try to deny these ideas in favor of mathematical abstractions.

As I see it, this whole forum is just a bunch of abstract mathematicians trying to make physics into what THEY want it to be!

Seriously, you people should change the name of this place to the "Abstract Mathematics Forum", and then put up a sign:

WARNING: No Physicists Allowed![/color]

 

[matt grime]

At the risk of posting something to which you will be unable to reply:

Who here is saying the results in physics are wrong?

You've cited the Heisenberg uncertainty principle. Do you know where that derives from? It is a mathematical result (to do with integrals and norms and things in Fourier analysis), and might be considered contradictory to people's intuition about the real world as they experience it. And is at any rate a direct consequence of a system you reject as unphyiscal.

The number systems we use are a construct of the mathematics we use (define e or phi without it). They may be used to model the real world, to some degree of precision. But beyond seemingly wishing to misapply these results, I really don't see your issue.

Don't accept the construction of the reals as possible, whatever, but that doesn't stop us proving results about them as abstract objects - that is all they are after all. I mean give me a physical manifestation of e, pi, sqrt(2), or for that matter almost anything.

And that doesn't stop physicists using them as a tool in their models.

So, are Dirac, Heisenberg, Bohr all wrong because they used the reals?

Very few people are bothered with the fundamentals of mathematics - exactly what a set is, what a real number is, and so on, because we all accept the same intuitive definitions of these things (after all, we all know what a function is even if we can't write down a solid definition), however sometimes people, and generally the crackpots at that, suddenly have marvellous new "insights" that aren't at all. Fortunately we can fall back on solid foundations if we need to.

None of us has said what sqrt(2) "is", in any physical sense - it isnt't a physical object. That has no bearing on reality which existed before we figured out any of the maths you object to.

In one thread you said that if we throw away the empty set (as we ought to, in your opinion) we must throw away the irrationals. How? The irrationals were known to the greeks, and they had no zero, never mind the concept of the empty set, which you credit Cantor with inventing anyway.

 

[quantumdude]

Well, I was going to respond to some concerns but I've just received a warning via email from a super mentor stating that my thoughts are not welcome here.

I didn't say that. I said that your personal theories are not welcome here. This is a place for scholarly discussion, not stream-of-consciousness speculation and handwaving.

I must say that I'm shocked. I had no clue that I was that threatening. I must have touched on some pretty sensitive topics.

You aren't threatening, and the topic is not sensitive. It's just that this is a moderated website, and we do not choose to host half-baked rubbish here. It's a waste of bandwidth. We especially don't take too kindly to such rubbish when it's posted in response to a question. So, I have 2 choices. I can either go around deleting your posts that are based on your mathematical, logical, and scientific errors, or I can issue a warning in the hopes that you will correct the deficiency yourself. I chose the latter.

I will gladly leave this forum as I prefer to share ideas with more open-minded people anyway.

But to accept the ideas that you've presented here, a person would have to be so open-minded that his brain falls out. If you're serious about physics, then why not stick around and try to learn from the members here?

 

[jcsd]

Maths is the language of physics, if you want to learn physics you got to speak the lingo.

 

[Gokul43201]

A response to the "physics" in this post :

Don't worry too much about it. Irrational numbers are just an inconsistent fabrication of abstract mathematics. Irrational lengths can't exist in the real world.

And rational lengths can ?

There is no way that you are going to construct a perfect circle or perfect triangle in the real universe. Planck's constant and the Heisenberg uncertainty principle will prevent it.

Planck's constant does not prevent anything; it is just a physical constant, incapable of preventing or promoting. Uncertainty only prevents you from knowing whether or not you've constructed a perfect something. Nevertheless, what suggests to you that even Planck's constant is a rational ?

There simply aren't any "physical" irrational lengths in the universe. They don't exist.

And how does the Uncertainty Principle allow "physical" rational lengths ?

Even the universe rounds off irrational quantities.

Really ? This makes absolutely no sense...but by this same logic, you could perhaps explain why c, h, and G are not simple rationals. Perhaps then you'll think about what a "quantity" really is.

The universe behaves more like a physicist than a mathematician.

I'm curious what exactly your expertise in physics is. The way you throw about the Uncertainty Principle (which incidentally, is the favorite tool of the typical physics crank) to support your view of the "inconsistencies" in Peano or ZFC sure seems a strange thing for physicist to do.

WARNING: No Physicists Allowed![/color]

 

[selfAdjoint]

As Baez has pointed out, you can express all your lengths as rational multiples of some transcendental number (pi, or 1/e, or whichever you like). That way all lengths will be transcendental, and you won't be able to tell the difference!

 

[Galileo]

But to accept the ideas that you've presented here, a person would have to be so open-minded that his brain falls out.