Here is my understanding:
It has to do with the fact that we can never measure a circumference to an infinite number of decimal places -- so whatever we do measure is a finite decimal expansion, and therefore rational. So calculating pi as a ratio of circumference to diameter is just a rational approximation to pi. The actual irrational number pi can be defined as various infinite sums or products.

There is a good Dr Math article about this:
http://mathforum.org/library/drmath/view/54660.html

Devantè · 2006-10-02 03:51:22

nhmathteacher wrote:

a circle has no beginning or end and therefore cannot be measured to any exactness.

I disagree.

Patrick · 2006-10-02 04:10:57

I have to disagree with the fact that a circle can't have a beginning/end or an exact circumference.. I might be wrong, since I havn't done much on this, though..

Last edited by Patrick (2006-10-02 04:11:58)

Dross · 2006-10-02 04:34:57

nhmathteacher wrote:

I have been told all of my life that pi is the ratio of the circumference to the diamer, and also that pi is irrational.
WELL...how can pi be a ratio and be irrational at the same time? It makes no sense.

The ratio of a rational number to a irrational number is itself irrational. i.e. the circle with unit diameter has a circumference that is irrational. You then have pi being the ratio of the (irrational) circumference to the (rational) diameter. All this means is that there exists no circle such that both it's diameter and circumference have integer lengths.

Devantè · 2006-10-02 07:24:35

If a circle has no end, then how come I can draw one?

The fact that a circle has no end is arguable.

Ricky · 2006-10-02 08:11:32

Remember that once you set the circumference of a circle, you aren't allowed to choose a radius. So if you set a circumference to be rational, the radius will be irrational. If you set the radius to be rational, the circumference will be irrational.

Being a ratio does not imply a ratio of integers.

I also have a hard time believing you have a BA in math without knowing this. Did you ever study elementary real analysis? Normally that includes at least some study of irrational numbers.

All_Is_Number · 2006-10-02 09:34:43

Dross wrote:

All this means is that there exists no circle such that both it's diameter and circumference have integer lengths.

And, there exists no circle such that both it's diameter and circumference have rational lengths.

nhmathteacher · 2006-10-03 01:55:25

Ricky wrote:

Remember that once you set the circumference of a circle, you aren't allowed to choose a radius. So if you set a circumference to be rational, the radius will be irrational. If you set the radius to be rational, the circumference will be irrational.
Being a ratio does not imply a ratio of integers.
I also have a hard time believing you have a BA in math without knowing this. Did you ever study elementary real analysis? Normally that includes at least some study of irrational numbers.

Thanks, I think that I had forgotten to think about the idea of a ratio not necessarily being integers.

As for your comment about having a hrd time believing that I have BA in math without kowing that...well, all I can say to that, beside that I find it relatively offensive, is that this is a question that I started thinking about in the past few months and remember very little Real Analysis. Also, all of my core class were taken in two semesters because I switched my major last minutes, so I think it is fair that I have forgotten some concepts.

MathsIsFun · 2006-10-03 09:59:52

I can believe the BA.

Because the courses focus on the complicated issues and assume the basics. In fact I met an Engineer who learned more from reading a book called something like "Why we don't fall through the floor" than his entire University course.

Anyway, π is simply amazing. It turns up in so many unexpected areas.

mathsyperson · 2006-10-03 10:08:31

π² is even more fun. It's a lot rarer than π, but the things that it does turn up in are really nice. Like donuts.

Ricky · 2006-10-03 12:48:47

mathsyperson wrote:

π² is even more fun. It's a lot rarer than π, but the things that it does turn up in are really nice. Like donuts.

Don't you mean coffee cups?

MathsIsFun · 2006-10-03 13:39:39

Tricky Ricky! A coffee cup IS a donut, just a little more crunchy.

George,Y · 2006-10-03 16:49:59

Good question.

First they assume a polygone with infinite sides and call it circle. Then they calculate polygones with finite sides and try to approach the circle, and state that they can write out as many digits of Pi as long as they've measured enough polygons. At last they say it's a state of Pi with infinite digits, which is the ratio for a circle.

I doubt such state do exists, and their logic from the 2nd sentence to the 3rd.

In real life, your cup is not round but a polylinder with many sides. But Pi is a good approximation for a multi-polygon's length and area.

Another example might be we use e to approximate reproduction process.
1 -> 2 -> 4 -> 8... =2^t=e^(tln2)
We write the way on the end with e and ln cancelling each other out simply for the simplification reason.

Devantè · 2006-10-05 21:32:02

mathsyperson wrote:

π² is even more fun. It's a lot rarer than π, but the things that it does turn up in are really nice. Like donuts.

A fun thing to do is to measure the outside, visible part of your eye socket, then use π.

Toky0Drift404 · 2006-10-07 17:24:33

Well to fully understand it you have to extrapolate from the data you have to work with and learn a little about the works of the mathematician Geog Cantor. His seminal work was his supposed continuum hypothesis. While it has yet to be fully ingratiated into our conventional formal system of mathematics we can use it for basic day to day applications. It is true that there are an infinite number of digits in Pi, but with simple encapsulation we can encompass all of it within one framework. You have to do this to work with it practically, as we obviously can not deal with a truly infinite magnituter in a pragmatic sense. This all may sound like a lot, but you can get the basics with some elementary googling (tm).

If you really want to get technical and downshift it, you might want to read up on some Godel as well. His observations on formal systems play in to all of this.

This must all surely be a mouthful, my personal way of dealing with this is sitting back on a fine evening with some David Bowie and taking it at my own pace.

Jai Ganesh · 2006-10-07 17:35:22

Toky, thanks for your lovely post.
But I'd suggest never ever read the work of Kurt Godel or George Cantor.
Save youself from insanity.
They were outstanding mathematicians, without doubt,
but works in number theory take you nowhere.
I am not putting off young mathematicians,
but one of the most outstanding mathematicians in
number theory, George Hardy, the person who
discovered the potential in Srinivasa Ramanujuan, a young Indian mathematicians,
said sometime in his life, all he had done in his life
wouldn't help mankind in any way, or something like that.

Karl_Hungus323 · 2006-10-07 18:15:07

collegehumor.db22c981028ef5f85f144119c0fb9368.jpg
*In response to Toky0Drift404*

I sincerely hoped you meant Georg Cantor (as opposed to Geog Cantor, who is a Nihilist of notable pedigree). Rather than focusing on Georg Cantor's mythical continuum hypothesis, we should focus on Carl Friedrich Gauss' theories of Flux, specifically Cylindrical Flux Capacitance. It is primarily through this pivotal theory that we truly ascertain the various variables regarding the enigmatic number pi. While it may lull at times, the reading becomes suprisingly exhillirating while listening to the 80's supergroup Foreigner... not David "I was That Guy in the movie Labyrinth" Bowie.

Toky0Drift404 · 2006-10-07 18:26:57

Karl_Hungus323 wrote:

*in response to Toky0Drift404*
I sincerely hoped you meant Georg Cantor (as opposed to Geog Cantor, who is a Nihilist of notable pedigree). Rather than focusing on Georg Cantor's mythical continuum hypothesis, we should focus on Carl Friedrich Gauss' theories of Flux, specifically Cylindrical Flux Capacitance. It is primarily through this pivotal theory that we truly ascertain the various variables regarding the enigmatic number pi. While it may lull at times, the reading becomes suprisingly exhillirating while listening to the 80's supergroup Foreigner... not David "I was That Guy in the movie Labyrinth" Bowie.

I would hardly call the continuum hypothesis "mythical". While it is true that our conventional system of maths does not allow for a concrete discernation of it, I think that it is fair to say that it has its legitimate uses. If you really want to split hairs here, for example, the orbital gear box in modern automatic cars could not have been accomplished without at least a rudimentary implementation of an understanding of discrete infinities.

MathsIsFun · 2006-10-07 18:29:00

Hi Toky0 and Karl, and Welcome!

What illuminating discussions ... musically I feel Ulrich Scnauss or perhaps Yello (Oh Yeah!) would work

George,Y · 2006-10-08 00:16:45

A Mathematical Analysis book would introduce George Cantor's theory. Is Kurt Godel real different? I heard that there is a trinity of the defination of Reals, which means the three are equivalent.

Thank you Toky0 anyway.

By the way, is your avator R2? Star Wars..Interesting.

MathsIsFun · 2006-10-08 19:01:28

I think the avatar is Johnny 5.

Devantè · 2006-10-08 19:04:38

The avatar looks nothing like R2D2.

Dross · 2006-10-09 02:12:22

Toky0Drift404 wrote:

I would hardly call the continuum hypothesis "mythical". While it is true that our conventional system of maths does not allow for a concrete discernation of it, I think that it is fair to say that it has its legitimate uses. If you really want to split hairs here, for example, the orbital gear box in modern automatic cars could not have been accomplished without at least a rudimentary implementation of an understanding of discrete infinities.

The continuum hypothesis itself may not be mythical as such, in the fact that the hypothesis itself is a valid one to make, but I think the poster was reffering to it's "mythical" truth-value.

Math Is Fun Forum

#1 2006-10-02 02:29:29

Pi

#2 2006-10-02 02:56:09

Re: Pi

#3 2006-10-02 03:27:48

Re: Pi

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Re: Pi

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Re: Pi

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Re: Pi

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Re: Pi

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Re: Pi

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Re: Pi

#10 2006-10-03 01:55:25

Re: Pi

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Re: Pi

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Re: Pi

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Re: Pi

#16 2006-10-05 21:32:02

Re: Pi

#17 2006-10-07 17:24:33

Re: Pi

#18 2006-10-07 17:35:22

Re: Pi

#19 2006-10-07 18:15:07

Re: Pi

#20 2006-10-07 18:26:57

Re: Pi

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Re: Pi

#23 2006-10-08 19:01:28

Re: Pi

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Re: Pi

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Re: Pi

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