BlitzkriegKommt

So I'm sitting there thinking, Pi is an non-repeating, non-ending value. Now this must mean that we can't tell it's exact value, so what is it? Does it mean that it really doesn't have an exact value? Or to further the idea more, is Pi only an idea or a tool that we created, and the thing doesn't even exist on a number line(although we know it's approximate value)?
I really hope it makes sense. Looking forward for answers.

joeshoe

Pi is an irrational number, like many other irrational number (infinitely many, in fact), like square root of 2, square root of any non-square number (I think), and e, etc.

Irrational numbers do exist on the number line, so Pi would also be on the number line. Now you wouldn't be able to draw it at its exact location, but you wouldn't be able to draw the number 1 at its exact location either, because there's always inaccuracy in physical measurements (for example, you might use a ruler and draw number 1 at one inch, but then the ruler's lines have some thickness, so if you want to get even more accurate, you'll have to use an even smaller ruler, but that would also have some inaccuracy, and so on...)

BlitzkriegKommt

I understand what you are saying, but the question isn't primarily about pi being on numberline, its about pi being a theory just like infinity. Since we can't tell what pi exactly is.

flstf

Pi is the ratio of the circumference of a circle to it's diameter. However it's value is irrational. It is not an idea or tool that we created (we found/discovered it). But in 1897 the Indiana state legislature tried to pass a bill making pi equal to 3.2, 3.23 or 4, you could take your pick, LOL. I have often thought that if there is an intelligent designer of our universe that useful numbers like pi would be rational.

welshbyte

Theoretically you could create a whole number base, based on pi. Where pi is 1. So the ratio of the cirumference of a circle to its diameter (or whatever pi is) would always be 1. Numbers, and therefore mathematics ares tools created by humans. Sometimes the universe likes to reveal how complicated it really is and get us all philosophical about concepts that we cant reinvent without evolving massively more intelligent brains to understand the universe better so we're just stuck with them.

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stingc

The concept of a circle is an abstract idealization. Assuming that physical lengths have meaning to arbitrary precision is also an abstraction. Pi is a well-defined logical construct, but we created it. The "physical definition" of pi is only an approximation to the mathematical one (although a very good one).

BlitzkriegKommt

That was very well explained, comendable job.

flstf

I am not very well schooled in math, but isn't the ratio of a circle's circumference to it's diameter something that just is? And the logical construct comes about when we try to assign a number to it?

In simple terms (for my benefit, LOL) if we take a wagon wheel and place a mark on it's outer rim then roll it in the sand until we come to that mark again we will have a line that equals the wheel's circumference. Then if we lay the wheel down on its side (diameter) over the line, it will take up a certain amount of the line's length. The long line (circumference) and the short line (diameter) will always be the same relation to each other (ratio) no matter what size wheel we use.

Isn't this more like a discovery than a logical construct at least until we try to assign an exact number to it?

stingc

Yes, it certainly has a physical meaning. What I meant is that when you start talking about pi being irrational, then you're not really referring to physical measurements anymore. It is an abstraction, although one which is motivated by experiments like the one you mention with a wagon wheel.

To be a little more precise in what I'm talking about, we know that geometry is not quite Euclidean. Different circles would have different effective values of pi (although they can be related to the usual pi given some extra parameters). More seriously, it seems that distances cannot take on continuous values. Space itself probably becomes discretized in a sense below a certain scale. In all of this, it is still useful to keep the mathematical definition of pi, but it loses some of its simple physical interpretation.

CSflim

Pi is as real as any other number. Just because it cannot be written down as a decimal does not change this. I don't know what you mean when you say we cannot know it's value. You can draw a line of length pi quite easily. But perhaps your objection is that it could never be exactly pi. Certainly, that's true. But it could never be exactly two either.

Also, pi is not a "social construct". There is a reason why were are beaming the encoded value of pi, and not the complete works of shakespeare, out into space to alert any extra-terrestrial life of our presence.

EDIT:
I know what you are saying, but just so as not to confuse anyone:

pi has only one value. Non Eucleudian cirlces do not have a circumference/diameter ratio of pi.

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asaris

CSflim, there are just as many imaginary numbers as real numbers

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1010011010

CSfilm, I don't think I've ever seen it stated that it's the circumference:diameter ratio of a circle on a flat plane. Not saying this isn't the understood definition of 3.14159..., but that technically you could define a different surface, pick a point, draw a circle around it, and calculate different values of pi.

Why is it that the sum of 4/{n[(-1)^(1+n)]} approaches pi (of the 3.14159... variety)?
This is the 4/1-4/2+4/3-4/4+4/5-4/6+4/7... series. It doesn't seem like there's any reason that it should.

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Odd Pi Facts


The world cacluation record currently is 1.241 trillion decimal places. If you lined up all of these digits in about font 14, it would reach from the Earth to the moon.
The world PC calculation record currently is 25 billion decimal places.
If you lined up one billion digits, they would stretch from Washington D.C. to Las Vegas.
If you take the third, sixth, and nineth digits, they are, respectivly, 1,2,3.
If you take the 100th, 200th, and 300th digits, they are, resectivly, 9,6,3.
The sequence 333 is found three times in the first 7,000 digits.
The sequence 666 is found six times in the first 7,000 digits.
Only 20 decimal digits are neccessary to calculate the circumfrance of the Earth down to a fraction of an inch.
Only 39 decimal digits are neccessary to calculate the circumfrance of the known universe down to the electron.
"Pi" is the sixteenth character of the Greek alphabet, and "p" is the sixteenth of ours.
If you "multiply" pi times "e" you get "pie".... (get it?)
Savannah, Georgia has a zip code of 31415.

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filtherton

Conditionally convergent series can be made to add up to different numbers depending on the sequence and the order of its terms. That there is one that adds up to pi shouldn't surprise you anymore than one that sums to two.

Slavakion

You can take any circle, and divide its circumference by its diameter. You get pi every time. A circle that doesn't do this isn't a circle by all modern definitions. If you can prove this wrong, you just broke math.

stingc

Circles are usually defined by the 1 dimensional set of points equidistant from a center in a 2-dimensional space. If that space is a Euclidean plane, then circumference/diameter=pi. If it is not a plane, then you don't get pi.

CSflim

You can certainly define a different surface, pick a point, draw a circle around it, and divide the circumference by the diameter, and get a whole load of values, but these other values would not be pi.

Because mathematics is beautiful.

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Slavakion

Well... wait. Are you trying to suggest we draw a circle on a curved surface? Maybe it's technically a circle (I'm not sure) but I wouldn't call it one. Circles are two-dimensional, and what I think you're suggesting is three-dimensional. I can't really stand my ground there, because I haven't done anything with 3-D points, vectors, etc. Only volumes of various solids.

CSflim

The millionth digit is one.

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CSflim

No, he's not talking about three dimensional geometry, he's talking about two dimensional geometry on a curved surface (e.g. drawing a map of the world on a globe: don't forget, we are talking about the surface of the globe which is 2D, not the actual globe itself, which is 3D). So they would be two-dimensional, just non-euclidean. They are still referred to as "circles", they just don't have the same properties as circles in a euclidean geometry (most pertinently, the fact that c/d does not equal pi).



EDIT:
From another page:
Is pi constant in relativity?(Emphasis added)

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stingc

Just to amplify this for those unfamiliar with non-Euclidean geometry, surfaces can be defined without embedding them in a higher dimensional space. A sphere for example is a 2-dimensional object. You can define what it means to do geometry on it without ever having to refer to an external space that it might (or might not) be sitting in. The distinction is subtle, but it turns out to be a useful one.

Slavakion

Oh... okay. *backs away*

NegativeNine

Perhaps Pi is rational and we'rethe irrational ones...

flstf

If I'm not mistaken I think pi does exist in Euclidean geometry (as just the ratio between c and d). It becomes irrational when we try to fit it's value into our existing numbering system (by definition within that system). As welshbyte said in an earlier post we could have designed a new numbering system around it and made pi = 1.
Someone let me know if I'm all wet, I think I am way over my head here, LOL.

CSflim

Just to clarify for those who don't realise, in this context "rational" does not mean "logical" or "reasonable" but ratio-nal, as in to do with ratios. In other words an irrational number cannot be expressed as a ratio of two whole numbers. Take a half: that can be expressed as 1/2. Or 0.66666... can be expressed as 2/3. Pi and other irrationals (e.g. Root 2) cannot be expressed as a ratio of two whole numbers.
Now, it is important to realise that I have made no reference to any particular number system. Numbers have the property of rationality or irationality regardless of the number system that is used to express them.
So if you were to construct a number system with base pi (I have never seen non-whole number bases, but it is not immediately obvious why you could not do so), then you would have created a number system with an irational base and the symbol '1' would still refer to an irrational number.

Pi is always the same value, and that value is always irrational.

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Zeraph

W4K3_U?_!

Wow, If I stare at that long enough the numbers kinda blur together and mix around... I wonder if I would find a pattern before I saw a proton decay between the screen and myself?

Sen

I think that about sums it up. I've heard no clearer explanation.

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Furry

Scroll down really fast and you get a "matrix" effect. Ack, my poor brain.

belkins

Yes it is!

MojoRisin

It doesn't do much to explain your question but, you should check out the movie 'Pi' (directed by Darren Aaranofsky). I'm sure it will further your interest with the number.

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CSflim

Pi is a (very good) piece of fiction.

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Stiltzkin

If I knew how to derive Pi I could explain this better, but you can look at 1.5 is either 1.5 or 3/2, both meaning the exact same thing. 1.5 is no more or less exact than 3/2, and they are both infinitely precise. So if you look at the equation that derives Pi, that equation is the exact value of Pi, with infinite precision.

Yes it is. In fact, all of mathematics is only an idea. They're called axioms:
http://dictionary.reference.com/search?q=axiom

EDIT: Read 1010011010's post. This is the exact value of Pi: 4/{n[(-1)^(1+n)]}
It doesn't look like you always dreamed it would, does it?

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Bobbias

there are 906 occourances of 151 in the first million digits of pi, and 151 also happens to be the last 3 digits in the first million.

fckm

I think one of the problems people have with the hole diameter/circumference thing is that they don't realize what a circle is.
A circle is not a wheel. A circle is not a ring.
A circle is a mathematically defined shape. When mathematians talk about circles, there are specific definitions that they apply.
Coincidentally, when you go build a wheel, that wheel is a pretty good approximation to a mathematical circle.

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