Questions on Pi

[JanisChambers]

Alright, I know this board isn't for simple science questions but I trust this place to have some rather intelligent minds here. I was just waking up when a very random thought crossed my mind from a Discovery channel program I watched the other day. The problem with Pi and why it never seems to end. I could never even start to do the math, but something crossed my mind in those few moments of waking. Pi is the ratio of a circles diameter to it's circumference. For me, Pi seems to be the relation of a one dimensional object (The line of the diameter) to a two dimensional object (The circle). It's almost like Pi is a type of a 1D to 2D conversion factor. I imagine that isn't a unique thought, but I just wonder how close to the Bulls eye I am.

 

[nimzov]

For me, Pi seems to be the relation of a one dimensional object (The line of the diameter) to a two dimensional object (The circle).

In my opinion PI is not the ratio of two objects but of two lengths (the diameter and the circumference) of a single mathematical object in two dimensional space.

Geometrically speaking of course, because PI can also be defined without reference to the circle.

nimzo

 

[shadron]

It is certainly the ratio of the two distances, and it is certainly transcendental (meaning it cannot be exactly stated as a fraction with finite numerator and denominator). There are, however, an whole slough of ways it can be computed besides trying to compute the ratio of a circle's measurements. For example, it is the sum of the infinite series

pi/4 = 1 - 1/3 + 1/5 - 1/7...

an easy formula, but it converges on the value very slowly, while the Indian mathematics wonder Ramanujan gave us the very useful equation:

1/pi = (sqr(8)/9801) (sum 0 to inf ((4n)!/(n!)^4) ((1103 + 26390n)/(396^4n)))

(see http://www.jstor.org/pss/2325206 for a much nicer representation; I'm not really up on my T_EK scripting). A remarkably "ugly" but elegant formula that converges very rapidly, making it excellent for piling up billions of pi's digits.

By the way, I recently have re-read Sagan's Contact. In the book the heroine, Ellie Arroway, director of the SETI project, receives a message from Vega that directs the world to create a machine in which 5 people can be conducted to Vega. It directly discusses the nature of PI:

They meet with both known and unknown emissaries on Vega, but it happens in such a way that the results are not different from the passengers having hallucinated the results. However, one of the insights caught on the trip is that pi has a secret. When she returns, to relative disgrace at the hands of an old enemy, she programs her computer to explore PI, and in the last pages of the book it discovers that the expansion of PI, in base eleven, way down the row of digits begins drawing a picture of a circle in 1's on a field of 0's.

The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by the diameter, measure closely enough and you find a miracle - anther circle, drawn kilometers downstream iof the decimal point. It doesn't matter what you look like, or what you're made of, or where you come from. As long as you live in he universe and have a modest talent for math, sooner or later you'll find it. It's already there. It is inside everything. You don't have to leave your planet to find it. In the fabric of space and the nature of matter, as in a great work of art, there is, written small, the artists's signature. Standing over humans, gods and demons, subsuming Caretakers and Tunnel Builders [Note: the aliens she actually met], there is an intelligence that antedates the universe.

The circle had closed.

She found what she had been searching for.

I told one of our Christian posters that that was my definition of a omnipotent god. That is a god I could unconditionally believe in, if not worship. But, it is just a story.

 

[NobbyNobbs]

Alright, I know this board isn't for simple science questions but I trust this place to have some rather intelligent minds here. I was just waking up when a very random thought crossed my mind from a Discovery channel program I watched the other day. The problem with Pi and why it never seems to end. I could never even start to do the math, but something crossed my mind in those few moments of waking. Pi is the ratio of a circles diameter to it's circumference. For me, Pi seems to be the relation of a one dimensional object (The line of the diameter) to a two dimensional object (The circle). It's almost like Pi is a type of a 1D to 2D conversion factor. I imagine that isn't a unique thought, but I just wonder how close to the Bulls eye I am.

The circle is a two dimensional object. Its circumference, however, is a measurement of length, in one dimension. If it helps, think of the circumference as a loop of string. Now cut the loop, stretch it out straight, and measure the length of the now one-dimensional item. That distance is defined as the circumference of the circle the loop of string had bounded.

 

[sol invictus]

It is certainly the ratio of the two distances, and it is certainly transcendental (meaning it cannot be exactly stated as a fraction with finite numerator and denominator).

Actually that's the definition of an irrational number. Transcendental numbers like pi also have the property that they are not the roots of any polynomial of finite order with rational coefficients (unlike e.g. the square root of 2, which is the root of x^2 - 2).

 

[shadron]

Actually that's the definition of an irrational number. Transcendental numbers like pi also have the property that they are not the roots of any polynomial of finite order with rational coefficients (unlike e.g. the square root of 2, which is the root of x^2 - 2).

Ouch! Thank you sir, can I have another?

 

[Diagoras]

What I can tell you is that if you memorize too many digits of pi, people will just become irritated. They will decline your offers to recite the number for them, and promptly tell you to stop by the time you get to digit 40 or 50.

Oh, and in case you were wondering, π ≈ 3.14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679
82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196.

 

[Jeff Corey]

"How I want a drink, alcoholic, of course, after the heavy chapters involving quantum mechanics."
Mnemonic, from who?
And this:
3.14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679
82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196
44288109756659334461284756482337867831652712019091
45648566923460348610454326648213393607260249141273
72458700660631558817488152092096282925409171536436
78925903600113305305488204665213841469519415116094
33057270365759591953092186117381932611793105118548
07446237996274956735188575272489122793818301194912
98336733624406566430860213949463952247371907021798
60943702770539217176293176752384674818467669405132
00056812714526356082778577134275778960917363717872
14684409012249534301465495853710507922796892589235
42019956112129021960864034418159813629774771309960
51870721134999999837297804995105973173281609631859
50244594553469083026425223082533446850352619311881
71010003137838752886587533208381420617177669147303
59825349042875546873115956286388235378759375195778
18577805321712268066130019278766111959092164201989
38095257201065485863278865936153381827968230301952
03530185296899577362259941389124972177528347913151
55748572424541506959508295331168617278558890750983
81754637464939319255060400927701671139009848824012
85836160356370766010471018194295559619894676783744
94482553797747268471040475346462080466842590694912
93313677028989152104752162056966024058038150193511
25338243003558764024749647326391419927260426992279
67823547816360093417216412199245863150302861829745
55706749838505494588586926995690927210797509302955
32116534498720275596023648066549911988183479775356
63698074265425278625518184175746728909777727938000
81647060016145249192173217214772350141441973568548
16136115735255213347574184946843852332390739414333
45477624168625189835694855620992192221842725502542
56887671790494601653466804988627232791786085784383
82796797668145410095388378636095068006422512520511
73929848960841284886269456042419652850222106611863
06744278622039194945047123713786960956364371917287
46776465757396241389086583264599581339047802759010

 

[arthwollipot]

Show off.

 

[69dodge]

3.141592653589793238462643
38327950288419716939937510
58209749445923078164062862
0899862803482 9 3421170679
82148086513282306647093844
60955058223172535940812848
11174502841027019385211055
5964462294895493038196

9?

Jeff says 5.

I think he's right.

 

[Diagoras]

No, actually the value of pi changed yesterday. Mathematicians are still trying to figure out why, not to mention the philosophers, for whom this sudden change in the seemingly fixed laws of mathematics has caused quite an existential and epistemological conundrum.

 

[learner]

All I know about Pi is that it cost me a valuable point in my local pub quiz night.

Quizmaster; "State the formula for finding the circumference of a circle".
Me; "There isn't one I know of. Pi is close"
Quizmaster; "Are you saying Pi?"
Me; "No".
Quizmaster; "No answer then?"
Me; "Nope".

No answer, no point. I lost the night by a point and 80 quid.

 

[nimzov]

I think there are 2 Pi.

1) The Pi from geometry (or "physical" Pi) which is not constant. This Pi is the ratio of the diameter to the circumference and as you get closer to a black hole, this ratio varies because the curvature of space varies. So this geometrical Pi has not the same value depending of where you are in the universe.

2) The Pi from number theory is a constant :

and has the same value everywhere in the universe.

nimzo

 

[I Ratant]

All I know about Pi is that it cost me a valuable point in my local pub quiz night.

Quizmaster; "State the formula for finding the circumference of a circle".
Me; "There isn't one I know of. Pi is close"
Quizmaster; "Are you saying Pi?"
Me; "No".
Quizmaster; "No answer then?"
Me; "Nope".

No answer, no point. I lost the night by a point and 80 quid.

.
circumference: PiD.
Area; pi are square (but look round in the pan)

 

[ravdin]

Pi is the ratio of a circles diameter to it's circumference. For me, Pi seems to be the relation of a one dimensional object (The line of the diameter) to a two dimensional object (The circle). It's almost like Pi is a type of a 1D to 2D conversion factor.

This shows up in many geometrical objects, not just circles. The ratio of a square's diagonal to the length of a side is the square root of 2, which also can't be expressed as the ratio of two integers.*

*Math geeks- yes, I know that pi is transcendental, unlike the square root of 2.

 

[TrueSceptic]

"How I want a drink, alcoholic, of course, after the heavy chapters involving quantum mechanics."
Mnemonic, from who?
And this:
3.14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679
82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196
44288109756659334461284756482337867831652712019091
45648566923460348610454326648213393607260249141273
72458700660631558817488152092096282925409171536436
78925903600113305305488204665213841469519415116094
33057270365759591953092186117381932611793105118548
07446237996274956735188575272489122793818301194912
98336733624406566430860213949463952247371907021798
60943702770539217176293176752384674818467669405132
00056812714526356082778577134275778960917363717872
14684409012249534301465495853710507922796892589235
42019956112129021960864034418159813629774771309960
51870721134999999837297804995105973173281609631859
50244594553469083026425223082533446850352619311881
71010003137838752886587533208381420617177669147303
59825349042875546873115956286388235378759375195778
18577805321712268066130019278766111959092164201989
38095257201065485863278865936153381827968230301952
03530185296899577362259941389124972177528347913151
55748572424541506959508295331168617278558890750983
81754637464939319255060400927701671139009848824012
85836160356370766010471018194295559619894676783744
94482553797747268471040475346462080466842590694912
93313677028989152104752162056966024058038150193511
25338243003558764024749647326391419927260426992279
67823547816360093417216412199245863150302861829745
55706749838505494588586926995690927210797509302955
32116534498720275596023648066549911988183479775356
63698074265425278625518184175746728909777727938000
81647060016145249192173217214772350141441973568548
16136115735255213347574184946843852332390739414333
45477624168625189835694855620992192221842725502542
56887671790494601653466804988627232791786085784383
82796797668145410095388378636095068006422512520511
73929848960841284886269456042419652850222106611863
06744278622039194945047123713786960956364371917287
46776465757396241389086583264599581339047802759010

Pah! Someone has memorised Pi to over 22,000 digits!

 

[TrueSceptic]

I think there are 2 Pi.

1) The Pi from geometry (or "physical" Pi) which is not constant. This Pi is the ratio of the diameter to the circumference and as you get closer to a black hole, this ratio varies because the curvature of space varies. So this geometrical Pi has not the same value depending of where you are in the universe.

2) The Pi from number theory is a constant :

[qimg]http://googolplex.net/temp/euler.gif%3C/a%3E%3Ca%20href=http://googolplex.net/temp/euler.gif%20target=_blank%3E[/qimg][qimg]http://googolplex.net/temp/euler.gif[/qimg]

and has the same value everywhere in the universe.

nimzo

Is not (1) defined as existing strictly in a single plane?

 

[nimzov]

Is not (1) defined as existing strictly in a single plane?

Maybe. But my point is that if you do geometry near a black hole then that "geometrical Pi" is not equal to the "number theory Pi". So I speculate that math may not be the same everywhere in the universe.

But of course on earth both Pi are equal.

nimzo

 

[RationalVetMed]

You can't have a pi without coolwhip.

Yuri

 

[Beerina]

For example, it is the sum of the infinite series

pi/4 = 1 - 1/3 + 1/5 - 1/7...

an easy formula, but it converges on the value very slowly, while the Indian mathematics wonder Ramanujan gave us the very useful equation:

1/pi = (sqr(8)/9801) (sum 0 to inf ((4n)!/(n!)^4) ((1103 + 26390n)/(396^4n)))

(see http://www.jstor.org/pss/2325206 for a much nicer representation; I'm not really up on my T_EK scripting).

Never done that kind of formula formatting, but I always wanted to learn it. Let's see now...

​

By the way, I take sqr(8) to be sqrt(8) = 2 sqrt(2), which seems to be the way it's usually shown in this formula.

BTW, this uses one of many online utilities to generate an embeddable picture. Is there some kind of direct input to the BB code here? BTW, the above formulae link to more info at the web site, but the links are broken, even though I cut and paste the "Here's your BB code version!" text directly. BTW, I love to use BTW.

By the way, I recently have re-read Sagan's Contact. In the book the heroine, Ellie Arroway, director of the SETI project, receives a message from Vega that directs the world to create a machine in which 5 people can be conducted to Vega. It directly discusses the nature of PI:

They meet with both known and unknown emissaries on Vega, but it happens in such a way that the results are not different from the passengers having hallucinated the results. However, one of the insights caught on the trip is that pi has a secret. When she returns, to relative disgrace at the hands of an old enemy, she programs her computer to explore PI, and in the last pages of the book it discovers that the expansion of PI, in base eleven, way down the row of digits begins drawing a picture of a circle in 1's on a field of 0's.

I told one of our Christian posters that that was my definition of a omnipotent god. That is a god I could unconditionally believe in, if not worship. But, it is just a story.

That was, more or less, Sagan's point, that only the true creator of logic itself could do such a thing. And such a thing would be higher than the mere creator of the universe and physics.

And also, of course, that it would be a message directed at intelligence, not some apocryphal ancient writings suggesting gods with savage emitions and petty controls via threats and promises of gifts, addressed to the ignorant.