Can a Triangle Have Sides of 1m and Hypotenuse of √2m?

[Cheman]

Square root of 2...

Is it actually possible to produce a right angled triangle with sides exactly equal to 1m and 1m? Because this would produce a hypotenuse with length "square root of 2" m, which has no exact length.

Thanks in advance.

 

[Gale]

root two is as exact a number as one. if you can exactly draw one, you can as exactly draw the other.

 

[Crosson]

because this would produce a hypotenuse with length "square root of 2" m, which has no exact length.

Let me fix your sentence to be true:

"...because this would produce a hypotenuse with length "square root of 2" , which has no finite decimal representation."

 

[ramsey2879]

Mr Crosson. Does 1/3 have a "finite" decimal representation? I am not clear what is meant since I view 0.33333... as an infinite decimal representation. Maybe you consider 0.(3) wherein the parenthesis surrounding the 3 mean the standard notation, 3 bar, (the bar is placed over the portion of the decimal representation to denote the infinitely repeating series of digits on the righthand side of the decimal representation) as a finite decimal representation.
Regarding the question asked. Somewhere but not necessary in this forum, I read a related thread wherein someone tried to prove the imposibility of exactly dividing a crystalline object a certain way based upon the fact that the object divides along crystalline planes. Math is an exact science, but whether or not you can exactly measure or cut an object to an exact length depends upon what meaning is to be attached to exact. Anyone with knowledge of physics knows that a polished surface in never in fact "plane" in the mathematical sense since there will be hills and valleys among the atoms. Even so I know a plane surface when I see one. And diamonds can be polished so that the plane surface does not follow the crystalline planes. The arguments that there can not an exact right triangle with sides of 10cm cut from diamond is like saying that 1*10^-123456789 is different from zero. One can even more easily cut and polish a glass sheet to such a shape. And to cut paper to any shape is simple. The original question does not belong in this forum.

 

[pivoxa15]

This simply example/problem posted by Cheman is a perfect example of why mathematics does not exactly represent or describe physical reality.

First of all, it is imperative to remember that mathematics is not affected by time and the two should not be thought together.

Humans cannot comprehend let alone measure an irrational physical quantity such as root 2m or any othe "exact" infinite represented number. Our physical intuition suggest that a length such as root 2m is something that is forever growing (ever smaller growth). This is because of its infinite decimal expansion. But that is wrong. root 2m is not growing, it was and always will be root 2m. Just think about the unit triangle, the hypotenuse of root 2 is a fixed, finite line (in theory or in the mathematical/Platonic world). Our physical intuition failed us because we cannot comprehend the infinite and we also naturally factor in time. I.e. it takes some time to draw 1.1m and longer for 1.11m. Imagine drawing 1.11...m? It would physically take forever, in other words it is unfinishable. Meanwhile the line gets longer and longer if looking from a finite/physically intuitive perspective.

The moral of the story is that mathematics should only be used as a guide to modelling the world around us and should never be taken literlly. There could be a better system than mathematics to describe the world around us.

 

[Berislav]

Draw two lines of length 1 m intersecting at 90°. Connect their ends with another line.

 

[Hurkyl]

The moral of the story is that mathematics ... should never be taken literlly.

That's the exact opposite of the problem -- mathematics, quite literally, says absolutely nothing about the physical universe. But the point of your statement is correct: mathematics is not about modelling the physical universe with math. That's the physicist's job!


Not everyone intuits decimals as things that "grow". In fact, I didn't even realize anyone would think such a thing until I realized that this was the mistake made by some of the crackpots we've had in the past.

(Ack, I don't mean to make it sound like I'm calling you a crackpot -- these people were crackpots because they absolutely refused to believe their intuition could be incorrect)

 

[mathwonk]

the greeks believed that a length was measurable bya number. all well and good. the problem comes when they tried to compre two differewnt lengtha and hence two diffeent numbers.

their method of comparing numbers was to subdivide both until some subdivisons were the same. i.e. to compare 5 and 2, we subdivide 5 5 times, and subdivide 2 2 times and in both cases we get the same lengthm anmely one.

but then they found that the edge length and the hypotenus of a square could not be compared this way. that throws off the whole method of using decimals to measure numbers, as only numbers thatc an be compared to powers of 10 are writable as a finited ecimal.


so if you,pick your unit length to be the edge of a square, then you have trouble writing the length of the hypotenuse as a finite number, and vice versa, if you pick the hypotenuse to be your uynit loengthm then you have trouvble writing the edge length as a finite number.

i.e. a choice of numbers syetm involves a choice of what length shall be called "one". once this choice is made one ahs troublew writinf any other numbers that do not conmpare well to that unit length.

there is no difference in the lengths, one is as good as the other, but if you choose one, and then to use numbers to represent all thenothers, you have no way to use only finite numbers.


OK?


it turned out some pairs of num

 

[pivoxa15]

you have no way to use only finite numbers.

I understand what you said but wasn't it Hilbert who wanted to recreate mathematics making everything finite? Surely he would have known what you were saying. Or maybe by finite he meant something else? His passion for Cantor's infinities is highly bizarre and contradictory to his philiosophy as well?

When I said "There could be a better system than mathematics to describe the world around us." I was actually thinking about digital physics (i.e. Wolfram's theory) where everything is black and white and discrete. This system/framework on the surface (because this is as much as I know on the subject) seems to make much more sense because experimentally, we know that the fundalmentlal constituents of matter exist and is certaintly not infinitesimally (whatever that means) small as calculus (the main and most important mathematical tool in physics?) demands.

 

[master_coda]

In order to stop using math to describe the universe, you would need to create a theory of physics that does not use math. I do not see how this is even possible.

 

[Icebreaker]

It could be done, but it would be very qualitative and useless.

 

[robert Ihnot]

Berislav: Draw two lines of length 1 m intersecting at 90°. Connect their ends with another line.

Of course you can draw a line representing the square root of 2.

I found it interesting in Physics, that if you have a barrel with a small hole punched in it such that water flows straight out under pressure shooting 16 ft in a second, and the force of gravity after one second pulls it down the same distance, why then the vector is 16\sqrt{2}, if I remember this right. So, of course, the \sqrt2 can occur in a physics problem, as well as a triangle problem. You can watch the water flow out.

What is being argued about is the decimal representation of a number, which has certain limitations. But it does not prevent the existence of the square root of two as an absolute numeric value.

 

[Canute]

Is this relevant?

"Herein I see the genesis of the conflict between geometrical intuition, from which our physical concepts derive, and the logic of arithmetic. The harmony of the universe knows only one musical form - the legato; while the symphony of numbers knows only its opposite, - the staccato. All attempts to reconcile this discrepancy are based on the hope that an accelerated staccato may appear to our senses as legato. Yet our intellect will always brand such attempts as deceptions and reject such theories as an insult, as a metaphysics that purports to explain away a concept by resolving it into its opposite."

Tobias Dantzig
Number - The language of Science

 

[robert Ihnot]

Canute: an accelerated staccato may appear to our senses as legato.

A certain segment hypothesizes that space is not continuous; In fact, a friend in Berkeley says that he spoke to Hawkings, and Hawking felt that space might be discontinuous in small enough segments.?

Statement attributed to Hawking assistant: Essentially, what all of this (and he) said was "when things get that small, we can no longer measure them so we don't know what the hell is going on."
http://forumserver.twoplustwo.com/showflat.php?Cat=&Number=2839191&Main=2819438

 

[selfAdjoint]

The question is, can you construct two exactly equal perfectly straight line segments at exactly a right angle to each other. If you can do that then the ratio of the distance between their end points to their common length is \sqrt{2}. Conceptually this is easy, but physically I think it is impossible.

 

[katlpablo]

"paradoxically…"

…it is impossible to produce —on a flat surface— a circle whose circumference's ratio to its diameter is not ∏(Pi) which, —like the 'square root of 2'— has no finite decimal representation…

 

[Hurkyl]

"Herein I see the genesis of the conflict between geometrical intuition, ...

I've always thought such an attitude suffers from tunnel vision -- they focus specifically on the algebraic structure, and ignore that topology does a fine job capturing the notion of a "continuum".

 

[pivoxa15]

A lot of theoretical physics uses mathematics developed by pure mathematicians such as root 2, pie etc. They are nice to use in physical models in that it creates a sense of wonder and beauty to the theoretician while keeping the model simple/elegant but the big question is, does reality really exactly confine to these abstract symbols/concepts. Experiments may agree to some degree but does it agree exactly? Most likely not as Hawking suggests as well.

In the 'Elegant Universe', Greene said that in quantum mechanics, there is a probability (1/(close to inifnity) one may walk through a brick wall. I haven't studied quantum mechanics but I have a feeling that this is another example of theoretical physics using pure mathematics as a way to explain physical phenomena - and taking the mathematics literally which is potentially dangerous. In general, the theory may be successful to some degree (from the experiments) but with this specific case, one can see that just like the infinit number of decimals in root 2 meters being very likely to be physically impossible, so too is walking through a brick wall. The mathematics of quantum mechanics may suggest a non zero probability but in reality or physically, I believe the probability is 0.

 

[Hurkyl]

The microscopic analog of walking through a brick wall is well-documented: it's called quantum tunneling.

If anyone of our particles has a nonzero probability of appearing on the other side of the brick wall, there ought to be a nonzero probability that they all do at the same time.


By the way, scientists don't just say "Oh look, the math says this, it must be correct!"; when the math leads them to a new prediction, they test it, if they can. Quantum mechanics has survived all of the tests of its strange predictions that we've been able to try.

one can see that just like the infinit number of decimals in root 2 meters being very likely to be physically impossible

You'll have to be more clear on just exactly what you mean by that. Some other things to consider, by the way, are that 1 = 1.00000... has just as many decimal digits as √2 = 1.41421..., and that the decimals are just one way to represent real numbers.

 

[DaveC426913]

1] It is as impossible to physically construct a line that is exactly 1m in length as it is to physically construct a line that is exactly \sqrt2 in length.

2] Decimal numbers are an arbitrary and wholly human creation. A number that is infinitely repeating is merely a byproduct of that creation, and has no effect on reality.

 

[pivoxa15]

I guess what I am trying to get at is 'what is nature really like, exactly?' Without consideration of any mathematics or even pretending that mathematics was never invented. In other words I want to know, if you like, 'God's thoughts'. This is probably a good analogy because this kind of exactness of describing our Universe is only likely from a higher dimensional point of view.

It is may be true that 'Quantum mechanics has survived all of the tests of its strange predictions that we've been able to try'. But does nature exactly obey and work according to the rules of this theory? Or does nature work sort of like what QM suggests but not exactly? I am worried about the word 'exact' like the way the most rigorous pure mathematician would worry about it.

1] It is as impossible to physically construct a line that is exactly 1m in length as it is to physically construct a line that is exactly \sqrt2 in length.

I agree and this backs up the fact that theoretical physics which uses different types of ideal mathematical numbers will always be an approximation to reality. Although this approximation may get better as more insightful models are built but by in large, as long as one uses traditional mathematics involving the real number line than one will always be approximating reality.

 

[Gokul43201]

This thread is evolving from "gross misunderstanding of math" (which, it is perfectly fine to address here) to "pseudo-mathematical mumbo jumbo" (which makes it not belong in this subforum).

I agree and this backs up the fact that theoretical physics which uses different types of ideal mathematical numbers will always be an approximation to reality. Although this approximation may get better as more insightful models are built but by in large, as long as one uses traditional mathematics involving the real number line than one will always be approximating reality.

Theoretical physics says that if you take 2 apples and give an equal number to Bob and Tom, each of them has 1 apple. Physics uses the integers to describe the reality that is the quantity of apples. The physics is perfectly accurate because it is defined to be so. Physics defines a one-to-one relationship between the quantity of apples and the integers. So, by definition, the physics can not be inaccurate.

 

[master_coda]

I guess what I am trying to get at is 'what is nature really like, exactly?' Without consideration of any mathematics or even pretending that mathematics was never invented. In other words I want to know, if you like, 'God's thoughts'. This is probably a good analogy because this kind of exactness of describing our Universe is only likely from a higher dimensional point of view.

I don't think you realize how realize how much you would have to throw away to create a physics theory without mathematics. It doesn't just mean getting rid of calculus, or the real numbers. It also means you can't use the natural numbers. So your theory can't even depend on the ability to count, or do any basic arithmetic.

 

[pivoxa15]

Theoretical physics says that if you take 2 apples and give an equal number to Bob and Tom, each of them has 1 apple. Physics uses the integers to describe the reality that is the quantity of apples. The physics is perfectly accurate because it is defined to be so. Physics defines a one-to-one relationship between the quantity of apples and the integers. So, by definition, the physics can not be inaccurate.

If only theoretical physics used just integers. But there is a whole lot of more to it including real numbers. For me, that is where the trouble starts. As far as I know, we might say that all theoretical physics is mathematically true but physically true? Most likely, as empirical evidence suggets but exactly? Probably not as many distinguished physicists suggests. People like Hawkings, Einstein, Feymann, Schrödinger.

By using only integers to describe physical reality, we have assumed everything to be discrete. We therefore also get results that are discrete. It so happens that we see objects as large as apples discretely. Likewise, when using the real number system to model dynamical systems with calculus, we have assumed everything to be infinitesimally small hence the whole universe is some sort of a contiumm and other ideal properties (although we may not have chosen these properties but were forced on us because of their convinience to work with, mathematically). So we will also get answers with these ideal properties and may get an answer such as \sqrt2 meters. This time our senses cannot tell whether such a distance physically exists. Out of the infinitely different ways our Universe can be, why does it have to turn out to be exactly mathematical in the traditional sense?

I don't think you realize how realize how much you would have to throw away to create a physics theory without mathematics. It doesn't just mean getting rid of calculus, or the real numbers. It also means you can't use the natural numbers. So your theory can't even depend on the ability to count, or do any basic arithmetic.

It is highly likely that reality obey logical rules so maybe reality does obey some type of mathematical system that have not been created or even thought about yet. Einstein hinted this point and said 'one can give good reasons why reality cannot be represented as a contiuous field ... Quntum phenomena ... must lead to an attempt to find a purely algebraic theory for the description of reality.'

 

[master_coda]

It is highly likely that reality obey logical rules so maybe reality does obey some type of mathematical system that have not been created or even thought about yet. Einstein hinted this point and said 'one can give good reasons why reality cannot be represented as a contiuous field ... Quntum phenomena ... must lead to an attempt to find a purely algebraic theory for the description of reality.'

Highly likely that reality obeys logical rules? If reality doesn't obey logical rules, then there isn't anything that physics can say about it either, with or without math. The whole point of physics is to come up with logical rules to describe reality.

Any attempt to understand how the universe works would be pretty much meaningless if it didn't obey logical rules; there would be nothing to understand.

 

[bao_ho]

counting meters is like counting space. counting space is counting time and counting velocity. if you started counting exactly when you started and finish exactly when you finish, and counting steadily exactly as you go, you will be exactly on track.

 

[mepcotterell]

It is my opinion that all infinitly represented decimal values represent a definite region, length or space. It is this presumption that leads me to believe that it is possible to have a square root of two hypotenuse when compared approximatley, relativley and yes, even exactly to the length 1 legs.

 

[Antiphon]

You must not confuse the mathematical operation you perform with the reality you
want to model. This is how you fall into Zeno's paradox.

There is nothing mathematically special about the mapping between an irrational
or trancendental number and a physical quantity like a length.

To convince yourself, merely change your number base from 10 to 12. Then 1/3
is no longer an infinite repeating representation.

Of course root 2 is still infinite and non-repeating in any base, but
that's incidental.