Of late I have been trying to figure out Pi and there are some two observations that I would like to hear your input:-
1) from wiki it is defined as "When a circle's diameter is 1, its circumference is pi" - question what is the reference point for measuring 1??
2) Is there an end to the decimal points
I am missing out something
~Sig coming soon~
Tyberius Prime
Maniac (V) Mad Scientist with Finglongers
From: Germany Insane since: Sep 2001
posted 05-25-2009 11:08
1) there is none. Pi is the ratio of circumfence to diameter. I.e. you can choose your unit arbitrarily, as long as it's '1 unit' Pi will be circumfence / diameter = 3.14...
2) No. To quote wikipedia:
quote:
π is an irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers. Consequently, its decimal representation never ends or repeats.
"When a circle's diameter is 1, its circumference is pi"
I find it kind of interesting that they would say that. When working with code and trig, the working circle has a radius of 1, so the circumference is 2pi. It makes sense why this route is used, but it can be confusing.
I once wrote a Mickey Mouse prog that used an inscribed polygon to estimate pi. Fairly simple formula. The more sides the inscribed polygon has, the more accurate the calculation (sort of considering irrationality).
It was a fun exercise.
I know that estimating pi is kind of an informal benchmark. How fast can your computer figure out pi to 1 billion decimal places?
Now-a-days I rarely find that I need to go past 3.14159 for what I use it for.
quote:1) from wiki it is defined as "When a circle's diameter is 1, its circumference is pi" - question what is the reference point for measuring 1??
To be a little more accurate, the 1 *is* the reference point. It's defined as 1. If it were anything else, you would just multiply the 1 and the PI by the relevant units and get your answer.
Also, unless you're doing something crazy, you generally don't need much precision. For napkin math, 3.14 or 22/7 work fine. When writing physics code or something, I tend to just use whatever abstraction in built into the language I'm using (M_PI, Math.PI) since if you have something revolving 1000 times, 3.14 starts to become very different from 3.1415927...
TP: ..so from what u r saying is that the grammar @ wiki is wrong?.and as long as you have a circle and measure the ratio of its circumference to the diameter using the same units it will always give you Pi?.i will need to run a few test on this to confirm this?..
Warjournal:?here is one for u?so at what exact point would the polyon became a circle?.hmmm
Iron_wallaby:?"its defined by 1" takes me back to my question?from the defn statement?what is the distance 1 ? probably you would give me values like 1 cm, 1 millimetre, 1 nanometre?.but again this are standards agreed upon?.so in short without the standards, measuring 1 for instance by using our forefingers the distance 1 in this case would be relative
...whats all this ranting abt.....we shld do away with Pi...as it appears not to be an absolute...n mayb work with values such as Phi....n probably we would get optimal results of whatever it is we r developing...
The polygon will become a circle as soon as it meets following definition:
You can define a circle as an infinite amount of points all the same distance to its center.
so your polygon would have infinite amount of edges.
quote:Iron_wallaby:?"its defined by 1" takes me back to my question?from the defn statement?what is the distance 1 ? probably you would give me values like 1 cm, 1 millimetre, 1 nanometre?.but again this are standards agreed upon?.so in short without the standards, measuring 1 for instance by using our forefingers the distance 1 in this case would be relative
I don't think you understand how the universe works. Everything is relative. Everything. No exceptions. So yes, while for general use people agree upon standard distances to make life convenient and make it possible to communicate properly, it isn't required mathematically. In math, you can define whatever arbitrary units you desire, since distances only mean something relative to each other. So defining a distance of 1 (without units) is perfectly reasonable. Remember, all distances are relative: PI is the distance around the circle RELATIVE to a diameter that, by comparison, has a distance of 1.
quote:...whats all this ranting abt.....we shld do away with Pi...as it appears not to be an absolute...n mayb work with values such as Phi....n probably we would get optimal results of whatever it is we r developing...
PI is just as absolute as PHI. They are both well-defined ratios.
I think you should perhaps study mathematics a bit more before you are qualified to make such statements. It may be confusing now, but after you've studied for enough time, it will click, and you'll suddenly understand. Trust me, PI is a very simple, reasonable, and useful concept.
quote:The numerical value of Pi; truncated to 50 decimal places is:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
See the links below and those at sequence A000796 in OEIS for more digits.
While the value of Pi; has been computed to more than a trillion (10^12) digits, elementary applications, such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of a circle the size of the earth (Petskull: !!!) with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom.
The polygon will become a circle as soon as it meets following definition:You can define a circle as an infinite amount of points all the same distance to its center. so your polygon would have infinite amount of edges.
Points that form a circle r relative to its circumfrence....and they are therefore not infinite...coz a certain number of points form a circle
On the other hand can u define a point....am sure in ur mind u picture it to be a small dot.....but if u were to zoom into the dot itself u will find that it agains becames a circle again taking u back to where u started from....
Logically speaking then there should that one final edge that converts the polygon to a circle
quote: iron_wallaby said:
quote:
PI is just as absolute as PHI. They are both well-defined ratios.I think you should perhaps study mathematics a bit more before you are qualified to make such statements. It may be confusing now, but after you've studied for enough time, it will click, and you'll suddenly understand. Trust me, PI is a very simple, reasonable, and useful concept.(Edited by iron_wallaby on 05-25-2009 21:12)
What is it that clicks that u cant expalin in a few statements
Peterskull:...do u agree the observerable universe is getting smaller and there fore the decimal places,,,r therefore relative to what we observe..
Someone once said u can never have a perfect circle coz by the time u finish drawing it
- the earth has moved
- the universe has moved
- space time has changed
Guys lets think outside the box....taking this deeper...I suspect circles may actually not be real
Points that form a circle r relative to its circumfrence....and they are therefore not infinite...coz a certain number of points form a circle
On the other hand can u define a point....am sure in ur mind u picture it to be a small dot.....but if u were to zoom into the dot itself u will find that it agains becames a circle again taking u back to where u started from....
This is incorrect. A point is a location in space.
Points are dimensionless. That is, a point has no width, length, or height.
quote:
In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue. Thus, a point is a 0-dimensional object. Because of their nature as one of the simplest geometric concepts, they are often used in one form or another as the fundamental constituents of geometry, physics, vector graphics, and many other fields.
There are times, my fellow headcases, when I think you like to make things seem far more difficult than they really are. You all have the makings of engineers in that respect. :P
EDIT: Binary, I like your thinking, but the while the Universe may be expanding, it is generally the space between objects that is getting larger, and not the dimensions of the objects themselves; conventionally, local space is not expanding, so the distance between points (on say, a measuring tape) is unchanging. That's the general rule, anyway.
I think perfect circles are imaginary, too. They are a two-dimensional concept or absract that can only be crudely represented in the real world; as an 'infinite number of points' cannot exist, even the finest resolution will reveal any physical representation of a circle to have a very finite number of 'points'.
:PEDIT: Binary, I like your thinking, but the while the Universe may be expanding, it is generally the space between objects that is getting larger, and not the dimensions of the objects themselves; conventionally, local space is not expanding, so the distance between points (on say, a measuring tape) is unchanging. (Edited by White Hawk on 05-26-2009 13:38)
I see where you r coming from....but consider this what if the objects themselves were expanding all objects ...this would also include even the measuring devices...therefore it would be impossible to measure or even be aware of the expanding objects
An infinite number of points is NOT impossible. In Continuous math you have infinite number of points on a curve. You do so in ALL real world.
Our RW is not describable by descerte numbers, but by Continuous funktions (sin, cos ..). Thus it is not possible to count the points creating a circle, but it does not mean the circle does not exist.
And a perfect circle exists in nature: A rainbow (even if not a complete Circle)
Following directly on the heels of a rather poignant explanation that a "point" is a construct holding mental placement but without physical dimensionality aside from position, that statement rings blatantly false. Since a point occupies no physical space, it then cannot be defined to have a follow-circumference (just to continue the sphero-centric theme), therefore another point can exist infinitely close to the first, just not in the exact same space (which...like PI can be defined to any incremental unit...microns...atomic...proton widths).
Sorry guys, the Tao is dead on with this one...infinitum is discernable simply through the logic of understanding that any measurable distance can be dealt with in decreasing units merely by the power of the observer and/or his observational tools.
To respond to Blacknight and DarkGarden (and agree with WhiteHawk), an infinite number of points is impossible in the real world. Space is discrete. This was proved by Zeno a few thousand years ago. (Well, sort of. Spacetime being noninfinite is the best interpretation we have so far, though. http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#The_dichotomy_paradox ) That said, though, mathematics isn't constrained by physics, and so you can define things mathematically that are impossible in the real world. Like, for example, a perfect circle.
quote:Points that form a circle r relative to its circumfrence....and they are therefore not infinite...coz a certain number of points form a circle
I don't know where you came up with this, binary, but it's not true. A circle is defined as the locus of points equidistant from some other point. This is, by definition, an infinite number of points. Therefore, a circle is defined as being composed of an infinite number of points.
quote:On the other hand can u define a point....am sure in ur mind u picture it to be a small dot.....but if u were to zoom into the dot itself u will find that it agains becames a circle again taking u back to where u started from....
This is also not true. You are confusing mathematics with physics again. In mathematics, if you zoom in on that point, it stays a point. It's always infinitely small.
quote:What is it that clicks that u cant expalin in a few statements?
You'll find as your grow older that there is much that can't be expressed in words. This is why poets are awesome.
quote:Someone once said u can never have a perfect circle coz by the time u finish drawing it
- the earth has moved
- the universe has moved
- space time has changed
This is more or less true. But drawing a perfect circle in the real world has absolutely nothing to do with how a circle is defined or what the number PI means.
iron_wallaby:
- I dont know why u keep throwing the word Real World in ur statements.... if u r follwn the latest discoveries in science n they r now on a day 2 day basis....u shld know that the RW is subject to the observer.....Zeno's paradoxes were also intended to prove that all is ONE.....n therefore stating that i am confusing physics and mathematics doesnt hold ground...n therefore emphasis shld be set upon establishng the point where the various sciences seem to separate....
- Blacknight stated: "Our RW is not describable by descerte numbers, but by Continuous funktions (sin, cos ..). "...honestly this feels very accurate.....n the current research in computing is how to incorporate the current quantum theories....n from this we will get faster processing speeds n who knows what
-again dont forget the OBJECTIVE: which is to ensure that we obtain optimum rslts of whatever systems we have be it mechanical or biological....coz they are based on principles that are not unified n this is not how our reality/nature operates
"grow older" ...note this is also relative to the observer
quote:I dont know why u keep throwing the word Real World in ur statements.... if u r follwn the latest discoveries in science n they r now on a day 2 day basis
Right, physics changes all the time. What I'm saying is that it has nothing to do (directly) with mathematics, which is subject only to its own definitions. The two systems are totally and completely different. Mathematics defines its own reality. Physics uses mathematics to describe our reality.
quote:u shld know that the RW is subject to the observer
Yes, but unless you're a solipsist, I'm going to assume that you live on planet Earth.
quote:Zeno's paradoxes were also intended to prove that all is ONE
Who cares how something is intended? What's important is how it's used. Michelangelo made religious artifacts. I look as the Sistene Chapel and see art.
quote:n therefore stating that i am confusing physics and mathematics doesnt hold ground
You're using a failed argument as a premise. Physics and mathematics aren't the same.
quote:"Our RW is not describable by descerte numbers, but by Continuous funktions (sin, cos ..). "...honestly this feels very accurate.....n the current research in computing is how to incorporate the current quantum theories....n from this we will get faster processing speeds n who knows what
You'll note that continuous equations are only an approximation to the real world, not a description of it. Quantum physics, as we currently understand it, is unpredictable. It only follows patterns on average.
So, to recap: we started out talking about what PI was. The thing to remember here is that PI is a mathematical definition and is, as such, only useful in a mathematical perspective. Basing PI on the physical world therefore doesn't make sense, which is what you were saying earlier. I agree with that, but I think that the conclusion you came to (that PI should be done away with) is a bad one, because mathematics does not depend on the physical world.
quote:"grow older" ...note this is also relative to the observer
Heh! Please don't take it as an insult, either. You're asking good questions, which already makes you smarter that most.
iron_wallaby:
....there those who discover theorems and formulas...and there are those who are experts in applying them.... (imo) pretty sure you fall in the latter...
Life = harmony = mathematics.....am not sure wether you have researched on the origins of mathematics....it all starts from very simple definations...
conclusion
....circles dont exist in our observable universe n they r just illusions.....notice that rotate any object n u end up hvng a circle within its outline...
a Diskrete Funktion by definition is a funktion that has holes/jumps. So you can not draw the funktion in one go.(Definition for primary school but it will do)
a continuous Funktion has no holes/jumps...
So if timespace where discrete it would have holes in between the points. wich can not be true.
If you are talking about quantums the the only thing that could be described as discrete is the energy output wich is quantumized, but not discrete.
To go even smaler. Waves(Micro, Makro what ever) Are Continuous Funktions even if they are sometimes pulsed. Each puls is continuous.
And if i where a physicist i could explain to you how subatomic partilces are nothing but a continuous probability funktion until they are observed and turn in to a particle(Wave?particle duality)
Math is simply a set of facts, logically derived from one another based off of some core axioms. Whether it describes the physical world is a different question (which tends to lead to some silly discussions).
Binary: Pi is the ratio of a circle's circumference to it's diameter. That is, a circle's circumference is always roughly 3.14 times the length of it's diameter. In the case of a circle 1 "unit" in diameter (where the unit is whatever you want to plug in - inches, feet, meters, lightyears), the circumference is always pi "units" around. In the case of a circle with a 1 unit radius, the circumference is 2*pi units. Both of these can be easily seen by plugging the values into Circumference = 2 * pi * radius. Mathematically speaking, there's no room for interpretation.
Math is 100% abstract and has a very weak relation to reality/the physical world surrounding us.
Circle is a concept, diameter is a concept, circumference is a concept. They all relate to each other in the same abstract, theoretical level. In the 'real world' there are no absolute circles, spheres, lines etc. Only things that we may choose to call circles or spheres because we feel that these concepts describe these physical objects.
We may use math to generalize, classify and describe the physical world but we can never use the physical world to describe math. It's a very one way relationship. http://xkcd.com/435/
quote: binary said:
Life = harmony = mathematics
I won't say that this is entirely wrong since I don't believe in randomness, I only believe in the incomprehensible illusion of randomness but I think that life and mathematics find a much better connection through chaos and not harmony. At least that's what scientists have chosen to call it.
Well, I suppose technically it isn't true, but neither is it false. So far, we just don't know enough about quantum mechanics to be sure. Zeno's argument is a pretty good one, though there are other ways to resolve the dilemma. Though, with concepts of Planck length ( http://en.wikipedia.org/wiki/Planck_length ) and similar things, there is definitely a point at which we can no longer measure space, so it seems likely that there is a point at which space is no longer divisible. Like pixels on a monitor.
quote:So if timespace where discrete it would have holes in between the points. wich can not be true.
The definition of discrete: http://en.wiktionary.org/wiki/discrete (Separate, distinct, individual.) Don't jump to discrete functions, those are a separate concept. There doesn't need to be space between points -- they can be nestled up together. Furthermore, and the quantum level, almost everything is empty space anyway, so why would it matter anyway?
Bear in mind that quantum mechanics is weird. Really weird. Things can exist in multiple places at once. Quanta can be entangled such that meddling with one produces the same change in the other, regardless of how far apart they are. Both of those are much stranger than space having, essentially, a resolution.
Also, I'm kinda sick of defending this tangential discussion. So, unless the train gravitates back to it's rails, I probably won't contribute any further to this spiral of insanity.
I've read and re-read all the posts above with a lot of concentration and none of you, I repeat none of you has any clue on Pi except NoJive's second last post.
quote:Pi is always better with ice cream!
This has been proven by so many taste buds, mine included.
Enuff said.
" heads back to his cell, with a Pi and spiked ice cream"
quote:consider this what if the objects themselves were expanding all objects ...this would also include even the measuring devices...therefore it would be impossible to measure or even be aware of the expanding objects
If all objects in the universe are expanding, as space itself is expanding, the object has not changed in size (relatively) which is why you wouldn't be able to measure the difference - the net result concurs somewhat with your observation, and is more-or-less the truth. The following article is quite handy, and the explanation sounds blindingly simple... yet I marvel at the intelligence of those who first wondered at, and developed an empirical methodology for the most profound aspects of our Universe.
On a side note, if you've ever wondered why the night sky isn't full of light (effectively infinite space and all the matter in it, filling the sky an' all), it all seems to tie-in nicely with the expanding universe, too...
quote:An infinite number of points is NOT impossible. In Continuous math you have infinite number of points on a curve.
...and an infinite number of imaginary pink bunnies can exist too, but in the real world, there is no way to physically represent it. Infinity itself is as abstract and imaginary as 'zero', or a two-dimensional object (such as a circle). A rainbow may appear to be a perfect circle, but that is only as you perceive it - one imperceptible photon out of place and your crude human eye may still perceive what appears to be a perfect curve, but it's still a bombardment of the retina with a finite (though stonkingly vast) number of photons.
_____
Thinking out loud: I remember reading once that a perfect crystal cannot exist as, by definition, it would be boundless, and that would leave no room for anything else in the universe. I wonder how profoundly impossible 'perfect' is; it's another relative or abstract concept, isn't it? Nothing is perfect.
From: The Happy Hunting Grounds... Insane since: Mar 2001
posted 05-31-2009 21:31
quote:...and an infinite number of imaginary pink bunnies can exist too, but in the real world, there is no way to physically represent it. Infinity itself is as abstract and imaginary as 'zero', or a two-dimensional object (such as a circle). A rainbow may appear to be a perfect circle, but that is only as you perceive it - one imperceptible photon out of place and your crude human eye may still perceive what appears to be a perfect curve, but it's still a bombardment of the retina with a finite (though stonkingly vast) number of photons.
That is unfortunately incorrect - for in RL we do have things like trajectories.
And given that a curve can have an infinitie number of points, it can therefore also have an infinite number of trajectories.
It is also true that a sine wave is infinite (and this also holds true in RL).
Also, do not consider the photon to be the smallest thing in existence - to be honest, we do not know if there is such a thing as the smallest thing in existence. We only know about those things that we can currently observe or postulate about.
I would consider a point to be the smallest thing in existence - it has no dimensions. And since one can select a point (for example, when doing a trajectory), therefore I would call such the smallest thing in existence. Though, of course, a point has never been seen, measured, nor has it been quantified in RL as such. Still, it is a term that we use to describe an exact place (well, as exact as the math, that is) and it is defined.
WebShaman | The keenest sorrow (and greatest truth) is to recognize ourselves as the sole cause of all our adversities.
- Sophocles
quote: White Hawk said:
If all objects in the universe are expanding, as space itself is expanding, the object has not changed in size (relatively)
Not true...the object has changed in size if you were to relatively compare it with it's previous state....also ever considered that the expansion could also be happening inwardly too....
quote:
White Hawk said:
quote:
...and an infinite number of imaginary pink bunnies can exist too,
......unless in this case we assume zero = infinity
The example given of a perfect circle was the curve of a rainbow. You perceive the curve of a rainbow by the light (photons?) striking your eye. While you may perceive a perfect curve, if you could take a snapshot and zoom in to the finest resolution, you would see that the curve is in fact rather fuzzy. If you could map individual photons striking receptors, the best you could say is that those which correspond to the various colours in the curve are falling roughly within the shape of the perceived curve.
It's not just a perfect circle that's imaginary, but a circle, full stop. As it is a two-dimensional object, attempting to represent it in the real world would be impractical. Make it out of matter, and it will have a thickness that renders a third dimension, thereby making it a very short cylinder. Even if we ignore the third dimension and attempt to make it appear as a perfect circle from the appropriate perspective, on a small enough scale, we eventually find the very atoms that make up the edge to be finite, and therefore (without imagining curves between these atoms) describing a polygon with an incredible number of sides (or a finite number of points determined by the positions of those atoms).
Even if we do describe a perfect curve between the very finite points that make up the edge of the circle, I would wager that the circle either changes size constantly (as relevant particles fall inside/outside of the curve) or could only be an average description of the bounding curve.
I suppose a circle of light could be used instead (though the resulting circle would be a cross section of a projected/three-dimensional cone/cylinder of light), but while the resolution is finer, and the thickness is arguably nil, the edges will still never form a perfect circle - what forms it is still finite, and given an appropriate method for observing it, one should still be able to see that the edges of the circle are a number of 'points' of light between which straight lines might be drawn to form a large, uneven polygon that roughly approximates a circle. Also, unless the surface upon which the circle is projected is perfect, various points will be at different depths, so rendering a third dimension anyway.
I simply referred to photons as a rainbow is something perceived by the eye (or a camera) as a result of photon interactions, and so the curve or circle only exists as a perception based upon the interaction of light with a receptive medium (the retina, or camera film). This means that something the size of a photon is redundant in my example, as the effective resolution is on a par with the smallest active particle of the receptive medium (a photo-receptor, or a particle of the photosensitive chemical in the film). It really doesn't matter how small the smallest thing in the universe is, as it isn't what's describing the circle in the real world. Even if it were, and we had no way of measuring it, ?finity? could still be presumed, and the circle is still not perfect.
I agree, points are the smallest things in the universe. Infinity is possibly the largest thing. Neither is real.
___
quote:Not true...the object has changed in size if you were to relatively compare it with it's previous state....also ever considered that the expansion could also be happening inwardly too....
No, that's backwards. The object may have changed size absolutely, but if space itself has expanded proportionately, it has not changed size relatively. The only difference in size that could be measured is that which has occurred out of proportion to the expansion of space (say, thermal expansion), and that would be a relative change of size.
If you draw a nine-by-nine pixel square, then change to a lower resolution (scaled according to screen size) it will have changed size (in this example, technically, scale), but relative to the constituent pixels no change has taken place. If space is expanding, the screen, the pixels on the screen, and anything described by the pixels is expanding too, but nothing is changing size relative to the expanding space. If the screen and the images upon it do not expand with space, they would be perceived as 'shrinking' relatively.
From: The Happy Hunting Grounds... Insane since: Mar 2001
posted 06-02-2009 12:56
Hmmm...I am wodering about a wave (like a radio wave, for example) - is the frequency perfect? If so, then it would make a perfect circle (being that the wave is 360°.
Anyone know?
WebShaman | The keenest sorrow (and greatest truth) is to recognize ourselves as the sole cause of all our adversities.
- Sophocles
Hmmm...I am wodering about a wave (like a radio wave, for example) - is the frequency perfect? If so, then it would make a perfect circle (being that the wave is 360°.Anyone know?WebShaman | The keenest sorrow (and greatest truth) is to recognize ourselves as the sole cause of all our adversities.- Sophocles
....this question doesnt make sense at all.....if consider how the radio waves are propagated from source you will see what I mean...
posted 05-25-2009 10:08
...whats all this ranting abt.....we shld do away with Pi...as it appears not to be an absolute...n mayb work with values such as Phi....n probably we would get optimal results of whatever it is we r developing...
