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.
root two is as exact a number as one. if you can exactly draw one, you can as exactly draw the other.
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."
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.
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 inifite 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.
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)
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
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 bizzare 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.
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.
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 [tex]16\sqrt{2}[/tex], if I remember this right. So, of course, the [tex]\sqrt2[/tex] 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.
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
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 Hawkings felt that space might be discontinuous in small enough segments.??? Statement attributed to Hawkings 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
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 [tex]\sqrt{2}[/tex]. Conceptually this is easy, but physically I think it is impossible.
"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…
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".
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 probabilty (1/(close to inifnity) one may walk through a brick wall. I haven't studied quatum 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.
The microscopic analog of walking through a brick wall is well-documented: it's called quantum tunneling. If any one 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. 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.
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 [tex]\sqrt2[/tex] 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.