Euclidian Irrationality

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!
But I can take my ruler and draw the same square and measure the diagonal,
and the number I get is of course a rational number.

why is this? Why does euclidean(linear) geometry not really work in the "real" world? So if the Pythagorean theorem doesn't quite work, what does?

Come to think of it, where in nature do we even find a perfect square?

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!!!
What the *&$s going on?
What does this imply?

I don't know, what do you think?

Does it have something to do with Zeno's Paradox, perhaps? Wild guess, not that mathy.

I think this is only an issue when one believes that there's some cosmic significance to the square root of 2 being non-rational, such as Pythagoras did. In the real world and in mathematics in general, it's just a non-rational number. Really not much more to it. If we had a different name for a number that cannot be expressed as the ratio of two intergers, other than irrational, than I suspect the issue wouldn't get the airplay it does.

A common misunderstanding that leads to the fact that Euclidean geometry does not describe real world is that some people believe this geometry is absolutely accurate. It' not, results that are given by math correspond to the properties of real world objects only with a finite accuracy. If you take it into account that sqrt(2) can be approximated by a rational number with any desired accuracy there is no paradox.

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

OK, so who has the theorem to get a rational (real) hypotenuse of a perfect square? Because certainly I can take out my ruler, draw one, then measure said hypotenuse and get a rational measurement. I don't know about any cosmic significance, but if the entirety of out technological achievements is based on something... irrational. Perhaps somewhere in this simplest form of inaccuracy, physics could stumble upon something new. I mean they are called irrational numbers because they do not really exist, right? I can draw a square and that diagonal definitely exists. So why can I not predict a number that exists with some formula? I don't want approximations. I want solid undeniable accuracy.

The definition of a rational number in mathematics is very delimited, and has nothing to do with the number being "real" or involving a mental state we would ordinarily call "rational." A rational number is one that can be expressed as the "ratio" of two intergers.

en.wikipedia.org...

So 2 is a rational number because it can be express as 2/1, 4/2, 8/4, etc.

The problem Pythagoras and the boys had with the square of 2, or square root if you will, is that the hypoteneuse cannot be expressed as the ratio of two intergers. It was a problem for them specifically because Pythagoras thought that this "irrational" number, a number that cannot be expressed as a ratio of two intergers, upset his mathematical cosmology.

en.wikipedia.org...

Here's the wiki on irrational numbers:

en.wikipedia.org...

I think that Pythagoras's legacy in placing a significance to rational numbers that has no real practical weight is what makes even modern folk ponder the "irrationality" of irrational numbers.

I still want someone to give me the formula to find the true hypotenuse of a triangle 1x1. I want the answer to be a real number. There is no such thing as the square root of two. No number times itself will equal two. Therefore something is wrong with the equation. What is want is the equation that will give me the real answer. Unless.... there is no such thing as a perfect square...

Irrational numbers are perfectly real. They just cannot be expressed as a fraction, like 1/2 or 3/4.

I beg to differ... give me a handful of marbles in the amount of the square root of two.... I'm really starting to doubt the validity of fractions even. I.E. half a pie, well it is no longer a pie. It is a whole new thing. The fraction is just our way of describing it. Take the same example all the way down to an atom.
If I have an atom of a particular element, then I try to give you half of it, it is no longer the same element it was to begin with. I propose our entire mathematical system is in reality not quite true... But just an imaginary system for our minds to better understand the universe, but who can we truly understand the true nature of the universe if our tools for doing such are askew? Or maybe I'm just crazy.

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

Originally posted by Viszet Oki
I propose our entire mathematical system is in reality not quite true... But just an imaginary system for our minds to better understand the universe, but who can we truly understand the true nature of the universe if our tools for doing such are askew? Or maybe I'm just crazy.

Robert Anton Wilson talked about that all the time. All the forms of science that we think are explaining the universe are in fact just describing how the human mind works. He use to say "There is no mathematics, there is only neuro-mathematics. There is no physics, there is only neuro-physics" etc.

Vas

I beg to differ... give me a handful of marbles in the amount of the square root of two.... I'm really starting to doubt the validity of fractions even. I.E. half a pie, well it is no longer a pie. It is a whole new thing. The fraction is just our way of describing it. Take the same example all the way down to an atom.

Don’t forget that mathematics is just an abstraction. For your marbles example, only natural numbers(ex: 1 2 and 3) are meaningful. But for your pie example, fractions would represent that the piece was taken by cutting a whole pie into two. While one may not be able to see how there can be negative something (negative numbers), they are still meaningful if for example calculating how much money someone owes for instance. While i, sqrt(-1), the imaginary number may not mean anything in an intuitive sense, it can still be treated as a number and used for practical applications, although mostly indirectly. It works fine in equations such as e^(iPI)=-1. Mathematics doesn’t have to really mean anything, it just needs to be logically consistent.

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!

Yes. Irrational numbers do exist, they just aren't defined well with our base-10 number system.

Originally posted by Viszet Oki
But I can take my ruler and draw the same square and measure the diagonal,
and the number I get is of course a rational number.

No, it won't be rational, if it's exact. But it's impossible to be infinitely accurate, so you will lose a little accuracy and therefore end up with an approximate number.

Originally posted by 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!!!

Uh... The diagonal of a square will always be larger than its sides. If the sides are each four marbles long, then the diagonal must be longer than four.

Drat! You stole all of my thunder, Johnmike. But I do have one more thing to add.

Originally posted by Johnmike

Originally posted by 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!!!

Uh... The diagonal of a square will always be larger than its sides. If the sides are each four marbles long, then the diagonal must be longer than four.

Viszet,
Look at those four marbles on the diagonal. That hypotenuse appears to be four marble diameters in length, but it is not. The edges of the diagonal marbles are not touching each other along the line of the hypotenuse. That empty space is the differance between 4 and 5.656854 (the sq rt of 32).

Originally posted by Viszet Oki
So why can I not predict a number that exists with some formula? I don't want approximations. I want solid undeniable accuracy.

You can,

where delta is the accuracy of your measurement.

It is the diagonal that exists in real world, not the number, numbers are used to measure its properties with certain accuracy: 1%, 7%, 10%. There is no such thing as undeniable accuracy.

Viszet,
Look at those four marbles on the diagonal. That hypotenuse appears to be four marble diameters in length, but it is not. The edges of the diagonal marbles are not touching each other along the line of the hypotenuse. That empty space is the differance between 4 and 5.656854 (the sq rt of 32).

I did realize that after the fact, the four marble did not touch... the only way to make them touch to bend the legs, which means I no longer can have a right triangle. I use marbles to represent atoms in my own head. Does this mean at the atomic level, using the same element of course, that a true perfect square cannot exist...
How about this little bit of off the wall thinking. 1 + 1 = 3.
You have the first object. Then you have the other object. Thirdly you have the set of the two object together which is an entirely different entity. All for a total of three different entities. Now, take this principle and expand it. 1 + 1 +1 = 7... and so on. This would mean the number 2 does not really exist in this funcky world. We all sorts of "mystical" numbers popping up. The kabbalistic tree of life I two pillars on the outside with the middle pillar a combination of the two male and female principles..... Don't mean to esoteric, but just because modern science doesn't give these things much merit does not mean they don't exist, look at what modern science used to claim impossible that we now know to be true....

...No, in a perfect square, the diagonal will always be longer than each of the sides.

Well I read a couple of the intial posts and then stopped but to address the main topic in this thread.

What makes you think that your square is accurate ? What makes you think your ruler is in anyway correct ? Or any ruler for that matter. When you increase the resolution you will see that the number approaches --> 2^0.5 . Hell, if you dont beleive me get onto AUTOCAD and then draw a square with 1 unit sides and ask it to dimension the diagonal. You will see that the value is 1.414... .

As 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 .
Any a google search, does reveal this site with a proof: www.mathacademy.com...

What makes you think that your square is accurate ? What makes you think your ruler is in anyway correct ? Or any ruler for that matter. When you increase the resolution you will see that the number approaches --> 2^0.5 . Hell, if you dont beleive me get onto AUTOCAD and then draw a square with 1 unit sides and ask it to dimension the diagonal. You will see that the value is 1.414...

i do not assume my square, or ruler to be accurate, it is a "theoretic" suggestion. I'm sure a perfect square drawn with a laser by the most advanced super-computer there is would fall short of perfectly accurate. Back to the example of atomic level, a square built of atoms, indestructible spheres all of the same size, in the diagonal there is space between the spheres. In my example though, the spheres are my unit of measure. There can be no half sphere or quarter spheres. Thing is, space between the spheres gives me no accurate measure of distance, to bring them together changes the angle, and I no longer have a square, but a rhombus...
I am making no claim that Pythagoras got it wrong, I certainly don't know much of anything myself. I am just making a suggestion to an alternative way of thinking. More of what if sort of thing. Maybe someone with a little more brain power than me could take this suggestion in some direction.
Another point is that this is all dealing with only 2 spacial dimensions.