The Last Conformist
Irresistibly Attractive
Not much.El_Machinae said:
Infinite complexity
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El_Machinae said:
Nope, because we don't work out more and more digits of pi by taking measurements. We work it out by working out by calculating from the various infinite series that equal pi. Basically pi=limit(n->inf) f(n), where n is the particular series you're using. Measurement error and the accuracy of your measurement don't come into it, the only thing stopping you calculating more digits is the amount of computing power you're willing to use.
pboily
fingerlickinmathematickin
But isn't the foundation of pi and trig based on measurements?
I know we almost have a formula for pi, but how do we know that formula is correct past the sig digs? Just like we have a formula for the acceleration of gravity, but we learned later that we needed a more precise formula to explain more experiments.
pboily
fingerlickinmathematickin
You're inverting the steps. The Aristotelian (naturalist/engineer) approach to mathematics via natural phenomena has beed discarded in the last century. Pi is defined abstractly as the ratio of any "circle" to its "diameter", but no claim is made to the effect that the "circle" represents the round physical entity that we commonly call a circle. If you must know an approximate value, than pi = 3.1415....., but it is more important that for any circle (defined as the locus of points in a Euclidean plane at a common distance from a given point), the ratio "circonference"/"diameter" is constant.El_Machinae said:
All the trig functions, for instance, are defined without any reference to measurements of any kind, as uniformly convergent (infinite) Power Series. That these series correspond to the classical definition of the trig functions is a happy coincidence, but there's no extra significance there. So to find a "formula" for pi, it is sufficient to find a Taylor series for Arc tan, say, and evaluate at arctan(x = 1)-arctan(x=0) to obtain pi/4.
Judging from pboily's post ... maybe I need to remember more than I do ...
(I'll go back to my neurology review, thanks)
Note to self: google a page that talks about this and spend an hour getting to a marginal level of knowledge ...
pboily
fingerlickinmathematickin
nah, I just like to make it sound more complicated than it really is...
http://en.wikipedia.org/wiki/Leibniz_formula_for_pi
http://en.wikipedia.org/wiki/Leibniz_formula_for_pi
Show off ...
A circle is a well-defined mathematical curve - it's not something that exists in the Universe like gravity, and isn't something we have to physically measure. It's defined mathematically, and we can work out pi mathematically.El_Machinae said:
To take an obvious case, do you need to start taking measurements in order to know that the ratio between a square's perimeter and its length is exactly 4?
El_Machinae said:
What Pboily said. We don't almost have a formula for pi, we do have many formulas for pi, and for trig functions, that are not measurement based. As mdwh said, getting an exact value for the ratio of a square's perimeter to its side length is easy, and can be done without resorting to measurements. Same for pi, it's just the maths is slightly more complicated.
You really don't need that much maths to know more about this though. One of the simplest way to work things out, that doesn't involve nasty looking calculations, is to draw a square inside and outside a circle.
2*pi*r is the circumference of the circle, that must be bigger than the inside square, smaller than the outside square. Basic pythagoras theorem will give you the perimeter of the squares in terms of r. Inside square is 4sqrt(2)r, outside square is 8r. Which makes pi between 2sqrt(2) (2.828) and 4.
Make the square an octagon, and you get a better approximation. Use the half-angle formula, and you get the new perimeters in terms of r. Double the side length two more times, to 32, and you have a reasonable approximation of pi, without even needing to resort to (or invent) a calculator or a measurement.
Actually, writing this down it's probably easier to start with a hexagon, double it three times to 48 sides, and you have pi to 3 sig figs. (3.13935<pi<3.14609)
With just yr 10 trig and a calculator, you can write a general formula. Use 2 regular polygons with n sides, one inside, one outside. The inside one can be split into n isosceles triangles, with side lengths of r, r, x, and an angle of 2pi/n at the centre (360/n if you use degrees). Call this angle a. Cos rule says x^2=r^2 + r^2 - 2*r*r*cos (a). So x = sqrt(2)*r*sqrt(1-cos(a)), the perimeter is x*n, and you have a lower bound for pi of x*n/2r.
Do the same to the outside n-gon, this time you have n triangles, with the same angle in the centre, a height of pi, a base of y. You have a right angled triangle with side length y/2 & r, angle of a/2, so basic trig says tan (a/2) = (y/2)/r. y=2*r*tan (a/2), the perimeter is y*n, and you have an upper bound for pi of y*n/2r.
Plug in the biggest value of n you want to calculate things for (48 or 96 with pen & paper, using 10 million on your calculator will give you 12 sig figs), and you can as accurate a value for pi as you want. The stuff above looks complicated, but that's only because I'm forced to type it out, I can't draw a diagram and write the equation out nicely for you. It's really easy to do, anyone who knows cos rule for working out a triangle's side length given 2 sides and the enclosed angle is capable of doing this and deriving their own equation for pi, which they can then calculate themselves. I'd happily give it to a year 10 student as an assignment.
Xenocrates
Deity
Heisenberg doesn't apply to abstract non-real entities such as circles.
It relates to the impossibility of being accurate on momentum at the same time as being accurate on position if I remember correctly. Delta p *Delta x = h bar?
The discreteness or continuity of free space is interesting as if space is continous thermodynamics is wrong; if space is discrete dynamics is wrong. These theories are both taught as if they are correct, which is philosophically hard.
It relates to the impossibility of being accurate on momentum at the same time as being accurate on position if I remember correctly. Delta p *Delta x = h bar?
The discreteness or continuity of free space is interesting as if space is continous thermodynamics is wrong; if space is discrete dynamics is wrong. These theories are both taught as if they are correct, which is philosophically hard.
Ayatollah So
the spoof'll set you free
Xenocrates said:
Cool! Got links or references, I want to get a clue on this stuff. Also if space is discrete, how do we need to change our mathematics of spatial measurements - if at all?
The Last Conformist
Irresistibly Attractive
No, it isn't. This idea appears to be the root of your issues in this thread, so it bears repeating; pi is not defined using measurements.El_Machinae said:
Xenocrates
Deity
It's been a while, but ... err no my brain's fried.
Try again: if space is discrete, things can only exist at certain allowed points. There must be certain allowed points unless all entropies = infinity (thermodynamics).
If there are only certain points that something can exist at, objects must zoom infinitely quickly between them, as nothing can exist in-between these nodes. This means that the speed of an object depends only upon the time for which it hesitates before zooming to the adjacent location. This means that acceleration is hogwash because at any specific time the object is either travelling at infinite speed or still. And therefore F=MA fails because F's can only be infinite or 0 or M's can only be infinite or zero.
This makes a mockery of relativity and a c (2.998 *10^8 m/s) max cap on velocities doesn't hold. Discrete space would monkey everything around, except maths, since that's theoretical only.
A fellow called Xenocrates (no relation) worked this out 2000 years ago, i think but I may be wrong.
Phew!!
Try again: if space is discrete, things can only exist at certain allowed points. There must be certain allowed points unless all entropies = infinity (thermodynamics).
If there are only certain points that something can exist at, objects must zoom infinitely quickly between them, as nothing can exist in-between these nodes. This means that the speed of an object depends only upon the time for which it hesitates before zooming to the adjacent location. This means that acceleration is hogwash because at any specific time the object is either travelling at infinite speed or still. And therefore F=MA fails because F's can only be infinite or 0 or M's can only be infinite or zero.
This makes a mockery of relativity and a c (2.998 *10^8 m/s) max cap on velocities doesn't hold. Discrete space would monkey everything around, except maths, since that's theoretical only.
A fellow called Xenocrates (no relation) worked this out 2000 years ago, i think but I may be wrong.
Phew!!
Pure gold. Thanks.
Sanabas - my thanks to you. I'll pull out a pencil and work things out with your post over lunch.
El_Machinae said:
No worries. If you can't follow it or get stuck because my explanation is no good, just yell. I can probably come up with some diagrams of what I mean using paint and attach them.
@Xenocrates: Yeah, your last post makes sense for the implications of space being discrete. But why does space being continuous mean that thermodynamics doesn't hold? Admittedly I haven't looked very hard, but I can't think of any contradictions to what thermodynamics has to say if space is continuous.
nihilistic
Intergalatic Delivery Boy
El_Machinae said:
Please refrain from using words, phrases, and especially technical terms you do not know in a manner that gives the illusion to uninformed readers that you do know. Heisenberg Uncertainty Principle deal with the fact that to measure anything in the physical world, you must interact with it, therefore altering it. A 'circle' is not a physical construct, but a mathematical one. Then you refer to the mysterious 'division by the two' as if you know what the hell you are talking about, which clearly is not the case. And what do you mean by "running into significant digits"? Don't just pull out random buzzwords and expect people to nod. You are not in your creative writing class anymore.
If you do not know, you can ask a question earnestly. You do not have to fake an alibi so as to appear credible to the unlearned. It's extremely annoying and denegrating.
Pi is a real number, a mathematical construct. Its value is known exactly, as Pi, or as this infinite series:
http://mathworld.wolfram.com/PiFormulas.html
There are many other ways to express Pi exactly. It just happens that the decimal system is ill-equipped to do that efficiently because Pi is irrational.
El_Machinae said:
No. Rather unimpressed.
Lambert Simnel
One across
Nihilistic, I think you're being a bit harsh there. El Mac certainly started from an incorrect premise, but he's clearly listened to what others have said, accepted that he didn't understand, and has thanked Sanabas for his explanations.
There are dozens of dorky threads started every day on OT (no offence meant, El Mac), and it is staggeringly rare to find the original poster being willing to listen and accept another's explanation or view.
to El Mac, IMHO.
There are dozens of dorky threads started every day on OT (no offence meant, El Mac), and it is staggeringly rare to find the original poster being willing to listen and accept another's explanation or view.
to El Mac, IMHO.
brennan
Argumentative Brit
No it doesn't. It states that the momentum and position of a particle can never be known together to within a certain limit. It is nothing to do with measuring them, more a statement of a consequence of quantum mechanics.nihilistic said:
Sort of a bump, but after rereading what I posted, you don't even need cos rule. I was making things needlessly complicated. You can get formulas for both the perimeters of both the inside and outside n-gons using just right angled triangles. So all you need is SOHCAHTOA, and I'd give this to an intelligent Yr 8 class as an assignment.
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