What is Pi?

[silverpig]

Originally posted by: Jeff7

Originally posted by: jhu
what if we lived in a universe where pi was different number?

We'd have pi equaling 100.3423, and we'd be asking "what if we lived in a universe where pi was a different number?"

Laws of physics and geometry would have to be radically different.

Pi isn't always constant!

General relativity allows for the bending of space. For example: If you measured out 100 million km (~2/3 of the earth's distance from the sun) from the sun, and then drew a perfect circle around the sun, you'd have a circle with diameter 200 million km, but the circumference of this circle would be LESS THAN pi * 200 million km. This is because the sun curves space around it so it'd be like measuring the diameter of a bowl along it's bowled surface as the diameter, and then measuring the circumference around the bowl's lip as you'd expect.

This curvature is only very prominent in large gravitational fields; we'd be hard pressed to find any examples without the use of lasers and interferometers here on earth, but it is still measurable.

This situation where the circumference of a circle is less than pi*d happens in a positively curved space, or when space looks like a bowl. You can in fact have a situation where c > pi*d as well, and this occurs in negatively curved space, where space looks sort of like a pringle. Like this.

As for the overall curvature of the universe, well that's still yet to be fully determined, but despite local fluctuations due to massive objects, it appears that the universe is nearly perfectly flat, with a very very very very slight negative curvature to it.

 

[Vee]

Originally posted by: silverpig
[Pi isn't always constant!

General relativity allows for the bending of space. For example: If you measured out 100 million km (~2/3 of the earth's distance from the sun) from the sun, and then drew a perfect circle around the sun, you'd have a circle with diameter 200 million km, but the circumference of this circle would be LESS THAN pi * 200 million km. This is because the sun curves space around it so it'd be like measuring the diameter of a bowl along it's bowled surface as the diameter, and then measuring the circumference around the bowl's lip as you'd expect.

This curvature is only very prominent in large gravitational fields; we'd be hard pressed to find any examples without the use of lasers and interferometers here on earth, but it is still measurable.

This situation where the circumference of a circle is less than pi*d happens in a positively curved space, or when space looks like a bowl. You can in fact have a situation where c > pi*d as well, and this occurs in negatively curved space, where space looks sort of like a pringle. Like this.

As for the overall curvature of the universe, well that's still yet to be fully determined, but despite local fluctuations due to massive objects, it appears that the universe is nearly perfectly flat, with a very very very very slight negative curvature to it.

Well, I think you're wrong. Pi refers to c/d in Euclidian geometry. It is a mathematical entity. Not a physical. Your example refers to a physical situation where c'/d' is not Pi. Pi doesn't change.

 

[silverpig]

Originally posted by: Vee

Originally posted by: silverpig
[Pi isn't always constant!

General relativity allows for the bending of space. For example: If you measured out 100 million km (~2/3 of the earth's distance from the sun) from the sun, and then drew a perfect circle around the sun, you'd have a circle with diameter 200 million km, but the circumference of this circle would be LESS THAN pi * 200 million km. This is because the sun curves space around it so it'd be like measuring the diameter of a bowl along it's bowled surface as the diameter, and then measuring the circumference around the bowl's lip as you'd expect.

This curvature is only very prominent in large gravitational fields; we'd be hard pressed to find any examples without the use of lasers and interferometers here on earth, but it is still measurable.

This situation where the circumference of a circle is less than pi*d happens in a positively curved space, or when space looks like a bowl. You can in fact have a situation where c > pi*d as well, and this occurs in negatively curved space, where space looks sort of like a pringle. Like this.

As for the overall curvature of the universe, well that's still yet to be fully determined, but despite local fluctuations due to massive objects, it appears that the universe is nearly perfectly flat, with a very very very very slight negative curvature to it.

Well, I think you're wrong. Pi refers to c/d in Euclidian geometry. It is a mathematical entity. Not a physical. Your example refers to a physical situation where c'/d' is not Pi. Pi doesn't change.

I guess I was a bit sloppy yeah, but I haven't found a definitive definition as to what pi is. Some sources just quote it as 3.14159... others say c/d. Regardless, I should have said that c/d != pi always.

 

[Ryoga]

Indeed, once you start dealing with large scale gravity wells, Euclidian geometry breaks down. This is simply one result.

The premise "Given a straight line and a point not on it, there exists exactly one line perpendicular to the first line that passes through the point" is no longer true (as it turns out, the proof for that premise assumes itself to be true). Any geometery where this premise isn't assumed correct is "non-Euclidian", and gravit happens to be more easily explained in some kinds of non-Euclidian geometry.

Interesting note I recently read about: It turns out that knowing pi to more than 35 decimal places is irrelevant for describing the natural world. Assume that estimates on the age of the universe are correct, assume that the big-bang theory is correct, and assume that the speed of light is and has always been constant. That gives you the approximate size of the universe in light-years. Even if there should be a circle of atoms the size of the whole universe, more precision than 35 digits would be below the threshold for the size of an atom. That is, you'd stop being accurate because of Plank's length, not because of pi.

 

[Mday]

Well, it's safe to say that Euclidean geometry breaks down when you apply it to non Euclidean space... Hence "EUCLIDEAN geometry" ...

 

[BespinReactorShaft]

Originally posted by: sao123
I've always thought were probably only 1 transcendental number away from having some real scientific breakthrough, like warp fields or time travel.

This reminds me of a corny dubbed Japanese TV series in which a brilliant designer "explained" the workings of his time machine by fiddling around with a Moebius strip.

Anyway
I personally like the recursiveness of the golden ratio, but too bad it isn't transcendental.

BTW, anyone watched the film "Pi" lately?

 

[Calin]

Pi is a number non-rational and transcendent - it means that it can not be written as m/n with n and m natural numbers (integers) nor is the result of a polinomial type equation with integer coefficients.
As an example, square root of 2 is non-rational but not transcendent (is the solution of the x^2-2=0 equation).

so there is no polinomial equation with integer coefficients which has Pi as solution. But Pi can be obtained with any required precision thru the means of several recursive methods

Calin

EDIT: to late

 

[DrPizza]

Originally posted by: ming2020

Originally posted by: sao123
I've always thought were probably only 1 transcendental number away from having some real scientific breakthrough, like warp fields or time travel.

This reminds me of a corny dubbed Japanese TV series in which a brilliant designer "explained" the workings of his time machine by fiddling around with a Moebius strip.

Anyway
I personally like the recursiveness of the golden ratio, but too bad it isn't transcendental.

BTW, anyone watched the film "Pi" lately?

Yeah, I loved that movie... I was going to show it to my students at the end of the year (after we watched Beautiful Mind) but we didn't have quite enough time. I think I'll watch it again tonight though.

 

[CSMR]

Originally posted by: silverpig
I guess I was a bit sloppy yeah, but I haven't found a definitive definition as to what pi is. Some sources just quote it as 3.14159... others say c/d. Regardless, I should have said that c/d != pi always.

Pi is the smallest positive number satisfying e^(i Pi)=-1
(If you don't know what that means, define it as the smallest strictly positive such that Sin(Pi)=0. (Sin can be defined by Sin''+Sin=0, Sin(0)=0, Sin'(0)=1, where ' is differentiation.))

It is not a physical constant, and it is not possible to imagine a world in which Pi is a different number.

 

[Fandu]

As others have already mentioned, Pi can be described by a Taylor Series using rational numbers. However the series is infinitly long

 

[swatoa]

Yeah, I loved that movie... I was going to show it to my students at the end of the year (after we watched Beautiful Mind) but we didn't have quite enough time. I think I'll watch it again tonight though.

haha, do your students call you "Dr Pizza" as well?

"I really don't understand how to solve this math problem, could you help me, Dr. Pizza?"

edit: an interesting note is during the apollo moon missions, engineers only had to approximate pi to like 6 or 7 digits when programming on-board guidance computers on the CSM. Of course, more precision would be required for objects that are a greater distance away, but nevertheless it gives insight into the degree to which the number can be approximated to solve real world problems.

 

[unipidity]

Im not really sure wha the limit of use is for pi.

A circle with radius 14 bly? Needs accuracy of better than +/- 1 am? Still probably only need a few tens of digits...

 

[oneshot47]

PI IS EXACTLY 3!

I'm sorry it had to come to that.

--Who knows what that comes from?

 

[DrPizza]

Originally posted by: oneshot47
PI IS EXACTLY 3!

I'm sorry it had to come to that.

--Who knows what that comes from?

From the Bible, which doesn't describe it quite that way, and from an urban legend about some legislature in the South.

Originally posted by: sao123
I wonder...
How many other "important" transcendental numbers are out there that we havent discovered?

I've always thought were probably only 1 transcendental number away from having some real scientific breakthrough, like warp fields or time travel.

What a wonderful thought!

 

[oneshot47]

Originally posted by: DrPizza

Originally posted by: oneshot47
PI IS EXACTLY 3!

I'm sorry it had to come to that.

--Who knows what that comes from?

From the Bible, which doesn't describe it quite that way, and from an urban legend about some legislature in the South.

The simpsons idiot.

 

[unipidity]

I thought it was 3 1/3rd in the bible? Besides which, the happy literalist camp has an explnation for that anyway...

 

[Chaotic42]

In all seriousness goto your library and pick up a copy of a book called "A History of Pi". It's by Petr Beckmann. It's a small book, you can read it in about two hours.

 

[Chaotic42]

Originally posted by: DrPizza
From the Bible, which doesn't describe it quite that way, and from an urban legend about some legislature in the South.

Dr. Edwin J. Goodman of Indiana introduced a bill to the Indiana house in 1897 to make pi=16/?3.

See why I left Indiana?

 

[mickles]

Pi is also used in trigonometry to denote angles. For example, pi radians is equivalent to 180 degrees, 2 pi = 360 etc...

 

[NichowA]

Back when I said I thought it was interesting that pi was such an impossible number to express sucha simple relationship I was refering to it more on terms of principle than mathematics. Of course I know what pi is and that it is irrational and transcendental... if I didn't know the different types of numbers and their classifications I wouldn't post on a thread like this. I was simply stating that it's interesting that we derive such seemingly arbitrary numbers from such common situations (e is the other prominent example). It makes me wonder if things would be easier to solve in the natural world if we used a number system of base e or base pi. Hrm.

 

[BespinReactorShaft]

Originally posted by: swatoa
an interesting note is during the apollo moon missions, engineers only had to approximate pi to like 6 or 7 digits when programming on-board guidance computers on the CSM. Of course, more precision would be required for objects that are a greater distance away, but nevertheless it gives insight into the degree to which the number can be approximated to solve real world problems.

Brings to mind the film "Apollo 13" showing engineers performing mission critical calculations using slide rules. Or were there also logarithmic tables? The high school horror of it all...

 

[CSMR]

Originally posted by: NichowAIt makes me wonder if things would be easier to solve in the natural world if we used a number system of base e or base pi. Hrm.

The world would be interested to hear of a number system of non-integer base.

 

[Vee]

Originally posted by: NichowA
Back when I said I thought it was interesting that pi was such an impossible number to express sucha simple relationship I was refering to it more on terms of principle than mathematics. Of course I know what pi is and that it is irrational and transcendental... if I didn't know the different types of numbers and their classifications I wouldn't post on a thread like this. I was simply stating that it's interesting that we derive such seemingly arbitrary numbers from such common situations (e is the other prominent example). It makes me wonder if things would be easier to solve in the natural world if we used a number system of base e or base pi. Hrm.

I find this post rather confusing
But just to repeat what I've already said. In the realm of 'real' numbers, nonrational vastly outnumbers rational. Even though there is an infinite number (K) of rational numbers , N/K = infinite, where N is number of real numbers.

So if you would just pick any real number, the chance for it to be rational, is infinitesimal. The reason we sometimes (education) see so many rational numbers, is because we choose them, or they are derived by simple arithmetics from other chosen rational numbers. For Pi to be a rational number, would have been "impossibly" "lucky".

Secondly, while I can't even begin to fathom what you mean by "number system of base e or base pi" , in the case of the base for the positional system, it doesn't matter at all what base you're using. It is the same, any way you look at it.

Nor do I understand what the presumed benefits, from having Pi or e being rational numbers, would be?

 

[Vee]

Originally posted by: mickles
Pi is also used in trigonometry to denote angles. For example, pi radians is equivalent to 180 degrees, 2 pi = 360 etc...

No. radians and degrees are used to denote angles! 1 radian = 180/pi degrees.
Pi is a number. And it doesn't ever become anything else.

 

[CSMR]

1 radian =1. (This is an arbitrary definition but a convenient one.)

Originally posted by: Vee
In the realm of 'real' numbers, nonrational vastly outnumbers rational. Even though there is an infinite number (K) of rational numbers , N/K = infinite, where N is number of real numbers.

That's true. N/K=N.

So if you would just pick any real number, the chance for it to be rational, is infinitesimal.

This is not an accurate way of speaking. You need to specify a probability measure if you want to talk about chances. There are probability measures on the real numbers which will give you rational numbers with posative probability, and others which give rational numbers with probability 0. There is no pre-eminent proability measure on the real numbers.

The reason we sometimes (education) see so many rational numbers, is because we choose them, or they are derived by simple arithmetics from other chosen rational numbers. For Pi to be a rational number, would have been "impossibly" "lucky".

I think this is a somewhat mystical way of speaking about numbers which derives from the lack of being priecise earlier.