Euclidian Irrationality

Originally posted by homeskillet
with that line of thinking, if we cut off your legs and arms you wouldnt be "you".

you are absolutely right. With out my arms or legs I would no longer be "me"
My physical appearance would be different.
My physical abilities would be different.
My mentality would also change, not to mention the emotional trauma.
I would in essence be a different person.

The fallacy there is that you're thinking of atoms as perfect spheres like marbles. Atoms are made of nuclei of protons and neutrons (except hydrogen, of course, which has no neutrons) that are surrounded by electrons. But the electrons do not "orbit" the nuclei, so to speak. The electrons are quanta of energy that exist as waves.

While we tend to think of a hydrogen atom as a single nucleus with a single electron circling it, that is not the case at all. A single atom connot exist alone; you need at least two hydrogen "atoms" to make a hydrogen molecule. For instance, O2 is oxygen and O3 is ozone. In actuality there is no O1 because the electron needs to complete it's wave path. If it traveled in a circle there would be no definable peak or crest to it's wave. So the electron actually performs a figure-8, and needs two nuclei to do that and be a part of the molecule.

In a "square" made up of a metal like, say, aluminum, the nuclei would be stable in their positions, but the electrons are being shared and travel from one nucleus to another.

So, no, a pefect square can't exist at the atomic level since there is no rigid structure available to make a sharp corner. And even a solid block of iron is mostly empty space.

Originally posted by Viszet Oki
How is it with all our wonderful technological advancements, and mathematical prowess here in the 21st century, that we cannot not even rationally define the diagonal of a simple square? Oh sure, a^2 b^2=c^2. Ok, lets look at a simple 1" square.... 1^2 1^2 = c^2

here's where your math breaks down--here's your error:

1 1 = c^2

You removed the "squared" power from the left of your equation, without also removing it from the right hand side in the same step. The line above implies that 1 +1 = 4, which it clearly does not.

***

If you don't like irrational numbers, do what I do, and stay in the real world: use a slide rule!


* kids these days! *

.

We cannot measure anything with 100% accuracy. Nothing. So, square root of 2 is just as close to the actual value as 1 meter is to the idea of 1m.

The actual 1m model, made of platinum, I think, is kept in a safe, at the same temperature. But, it is not really 100% accurate. Neither is the math we use.

[edit on 6-5-2007 by swimmer]

It's not that irrational numbers don't exist, it's that if you expressed them in fraction form and then divided them there is no ending point. The pathagorean theorem doesn't say that the diagonal of a square is infinite, it's just that it has an infinite accuracy. The accuracy of your measurement depends on where you round the srt of 2 to. Math is a way to provide measurements and thus implies a certain amount of accuracy that sometimes isn't definite, as in this case.

Originally posted by Viszet Oki
How is it with all our wonderful technological advancements, and mathematical prowess here in the 21st century, that we cannot not even rationally define the diagonal of a simple square? Oh sure, a^2 b^2=c^2. Ok, lets look at a simple 1" square.... 1^2 1^2 = c^2
1 1 = c^2
2 = c^2
SQRT(2) = SQRT(c^2)
THE SQUARE ROOT OF 2 IS NOT A RATIONAL NUMBER!

First of all, the Pythagorean theorem says that a^2 + b^2 = c^2. This invalidates your entire calculation.

It is quite simple for me to reproduce the proof that sqrt(2) is irrational using elementary mathematics, not to mention a few more advanced proofs (e.g. Eisenstein criterion and Gauss' lemma for the irreducibility of polynomials over the rational numbers applied to p(x) = x^2 - 2). However, I don't think dthis is the problem we're discussing here.

The problem lies with the link between the abstract world of mathematics and Euclidian geometry in particular on the one hand and the tangible world of objects and measurements on the other hand. We use mathematics as a model to describe the world. That is, we can assume that our world satisfies Euclidian geometry and try to deduce some useful things from things. I think that this connection (which is false anyway, according to general relativity, where spacetime is a pseudo-Riemannian four-manifold) is a practical one only, there is no truth attached. It's worth as much as it allows to create useful things.

Usually, we consider measurements as a function from the real world to the real numbers. This is because we think that measurements like length are practically continuous. In real life, they might be not, e.g. by random geometry on Planck length scales, but this is of no interest to us, because lengths that are real numbers are just easy to work with. An abstract theoretical square has a hypotenuse of length sqrt(2), a real-life square can only be modeled by an abstract square and in this model have a hypotenuse of length sqrt(2). Outside our model, the concept of length isn't even defined. Moreover, this connection between model and reality is also clouded further by uncertainties and biases as your ruler will certainly have.

Originally posted by IAF101As for being rational or irrational, if I remember high school properly, any number that can be represented as p/q where q!=0 is a rational number. Now, I cannot show you the proof off hand but I do beleive that it is irrational, by disproving that it is rational.

Well, that theorem you state above isn't that interesting. If x=0, then p=0 and q=pi will do. If x!=0, then p=x*pi and q=pi will do. Maybe you mean something else? It is true that irrational numbers can be approximated arbitrarily well by rational numbers, for example.

The elementary way to prove that sqrt(2) is irrational is indeed by contradiction. There are other methods as well, so if anyone is interested, I can write down a different proof. Also of interest may be the proof which numbers can be constructed by ruler and compass. This amounts to the quadratic completion of rational numbers, e.g. look at chapter 7 of Stewart's Galois Theory.

Originally posted by avingard
It's not that irrational numbers don't exist, it's that if you expressed them in fraction form and then divided them there is no ending point.

No, the irrational numbers are defined as all real numbers (which can be constructed from Q by Dedekind cuts) which are not rational numbers. For example, sqrt(2), pi, e, zeta(3) are all irrational numbers.

I think you mean fractions like 1/3 = 0.33333333... in base 10. However, the infinity of the representation of a rational number like this one depends on the base. If we work in base 3, then 1/3 = 0.1. That we work in base 10 is just a matter of tradition and the infinity of this expansion doesn't mean anything (well, it says something the order modulo 10 of a part of the prime factorization of the denominator, but nothing very deep or significant).

What you are describing is an artifact of how quantities are represented digitally.

For those who understand Quantum computing, a new era is coming, where this will no longer be relevant.

Originally posted by Tuning Spork
For instance, O2 is oxygen and O3 is ozone. In actuality there is no O1 because the electron needs to complete it's wave path. If it traveled in a circle there would be no definable peak or crest to it's wave. So the electron actually performs a figure-8, and needs two nuclei to do that and be a part of the molecule.

Not quite. Monatomic oxygen is still oxygen, and can exist quite handily. O2 and O3 are allotropes of oxygen, but elemental oxygen definitely exists, it just doesn't exist indefinitely, at least not if it can find something to react with.

Originally posted by 777radiant777
My math prof caused a melt down in calc I with this gem

10x=9.99999...
-x=-.99999...
9x=9.0000...

9x/9=9/9

x=1

Ultimately, if you are happy to accept that 1/3*3 is the same as 0.333...*3 then you should be able to accept that 10X equals 10 and not 9.999...

[edit on 8-9-2007 by Woland]

To find out precise length on diagonal - you cant do it by measuring, you can come very close, but not exact - you need algebra.

Lines of the square are not rationale or irrational per se -but ratio of their lengths is irrational. Meaning - that we measure lengths as ratio of segments. There is no need for diagonal to be irrational - you can use length of a diagonal as a unit -hence diagonal would be one with rational length and sides irrational.

Viszet Oki
Now if I take some unbreakable marbles, and pretend these marbles are units of measure, and I then arrange them to form a square that measure four marbles up and down, then a take a couple more marbles to make my diagonal, that diagonal equals four!! According to Pythagoras my diagonal should be the square root of 32 marbles.... which is decidedly not 4!!!

....and: the length of a diagonal in square is equal to sv2 ( D=sv2 ) and s is the length of any side....and we know that v2 = 1,4142135623 ~ 1,41..meaning that length of a diagonal is v2 or Pythagorean constant(1.41) time longer than the side of square.....or:

s=4cm
D=sv2
D=4 x 1.41




[edit on 8-9-2007 by blue bird]

The square root of 2 is

1,4142135623730950488016887242097

reply to post by masterp


It's close, yet still infinitely far away..... There is no decimal representation of the square root of 2.... if there was then it would not be irrational.

OO OO I know! This is like one of them linguistic logic abstract V reality thingemebobs!

*Clears throat*

You are mixing Numbers and Measures. Yeah you use numbers to represent measures but they are really very different, like someone said (I think Dr Love) architects use slide rules not equations to build a 1X1 square.