CKDextHavn reply to 'is pi an exact number'

[robot-penguin]

In CKDextHavn's over-complicated explanation of the definition of pi it was neglected to mention that pi is not an exact number NOR an infinitum. Pi can be calculated with the simple expression of 22÷7 which, on simple calculators, gives the answer 3.1428571428571428571428571428571.

[zut]

You're referring to this Staff Report (http://www.straightdope.com/mailbag/mpi.html) by CKDextHavn.

I'm afraid you've misunderstood [/b]Dex[/b]. Pi most certainly can not be calculated as 22/7. True, it's not a bad estimate, since it's easy to remember, but the first few digits of pi are: 3.14159265358979323846264338327950.... and so forth. Close to 3.1428571428571428571428571428571..., but not the same number at all. This page (http://www.cecm.sfu.ca/projects/ISC/dataB/isc/C/pi10000.txt)'ll give you the first 10,000 digits of pi, and you can go here (http://www.cecm.sfu.ca/pi/pi.html) if you want to know more about the number. Bottom line: pi is irrational and, if you write it out, the digits do not repeat.

[Irishman]

Actually, 22/7 isn't even that good an approximation of pi. You'd be better off using 3.142.

pi = 3.14159... which, rounded to 3rd decimal place is 3.142

22/7 = 3.14285... which, rounded to 3rd decimal place is 3.143

Heck, it's not that hard to memorize 3.14159. Use that instead of 22/7.

By the way, robot-penguin, where did you learn that? I think perhaps you misheard, unless the person you learned it from was the mistaken one.

[C K Dexter Haven]

Well, there's always the argument that pi is exactly 3, because the Bible says so. There is a reference, I think somewhere in Kings, to a circular something or other that is 10 cubits wide and 30 cubits around.

Welcome to the Straight Dope Message Boards, penguin, we're glad to have you here... but you are misremembering. 22/7 is a estimate of pi that is usable for many purposes, but is not the same as pi, as already noted by zut and Irishman.

[bibliophage]

I'm working on yet another Staff Report on pi right now, and I find it quite amazing the number of reasonable and intelligent people who are absolutely convinced that p=22/7 exactly. I blame the schools. Then again, I blame the schools for pretty much everything.

[lel]

Did anyone else learn to approximate pi as 355/113? Just like any approximation, it's not exact, but it is closer than 22/7.

22/7: 3.1428571428...

355/113: 3.1415929203...

Pi: 3.1415926535...

Also, here (http://bible.gospelcom.net/cgi-bin/bible?passage=1KGS+7:23&language=english&version=KJV&showfn=on&showxref=on) is where the Bible mentions the round molten sea that is 10 cubits wide and 30 around.

[waterj2]

How would 355/113 be any easier than 3.141593? I guess you can save one digit of memory space, but it would take longer to use in most aspects, such as entering it into a calculator.

[Achernar]

I personally like to use 30001/10 + 75/82. I think that's pretty easy.

[Chronos]

There is a reference, I think somewhere in Kings, to a circular something or other that is 10 cubits wide and 30 cubits around.What's more, it's actually possible that those measurements are exactly and precisely correct. The object in question is bowl-shaped, so if measurements are restricted to the surface of the bowl, then we're in spherical geometry, and p is less than the Euclidian value. It is possible for a circle in spherical geometry to have diameter of exactly 10 cubits and circumference of exactly 30 cubits.

[MEBuckner]

Originally posted by bibliophage
I blame the schools. Then again, I blame the schools for pretty much everything.
Hmmm...You know, Oswald was in a school book depository.

[Irishman]

I do remember hearing somewhere in grade school that 22/7 was an approximate, but I always found it easier to just memorize 3.14 (way back in grade school, 3 sig digits was ample). Later I just increased that to 3.14159. For most calculations, I could always use the pi button on the calculator, anyway. ;)

[ultrafilter]

To answer another aspect of the OP, there is no such thing as an "inexact" number when one is talking about reals. There are numbers which have no finite decimal expansion, but they're still "exact" (I'm not sure that that's an actual mathematical term--it's not listed at mathworld, at least. Dex?).

[C K Dexter Haven]

I don't disagree, ultrafilter, there's technically no such thing as an "exact" number vs an "inexact" number. But I thought I knew what the OP meant, sort of, using common English rather than mathematics as the language of communication.

Thus, 3.14159 is an certainly an "exact" number -- it is different from 3.141592 or 3.14158, for instance. But it is not "exactly" pi. That is, pi is not "exact" in the sense that pi is an irrational, so that you can never write down all of its decimals, hence whenever you stop, you will have an "inexact" approximation.

There is another possible sense of "inexact." The Real Numbers are, after all, a theoretical construct. Try though you will, you cannot construct a line that is "exactly" 1 cm long. Measurement is all inexact or approximate, having a margin of error. Thus, the exact/inexact can be used to distinguish between the real world and the math-theoretic world.

[h.sapiens]

If inside a circle a line,
Hits the center and goes spine-to-spine,
If the line's length is d,
The circumference will be:
d x 3.14159

I didn't write this, and I can't remember where I read it, but it's nifty, eh?

[John W. Kennedy]

Now I -- even I -- would celebrate
In rhymes inept the great
Immortal Syracusan rivaled nevermore,
Who by his wondrous lore,
Untold us before,
Made the way straight
How to circles mensurate.
-- Edouard Prevost

[Chronos]

Am I the only one who just finds it easiest to approximate it as 3.14159265358979323846?

A simpler mnemnonic, by the way, is "May I have a large container of coffee?". It's fortunate for the mnemonicizers that it happens to take a while for a zero to show up.

[Achernar]

I say, why approximate? Just leave your answer in terms of pi. For crying out loud, we're not engineers.

[aerodave]

Originally posted by Achernar
I say, why approximate? Just leave your answer in terms of pi. For crying out loud, we're not engineers.

Pardon me...but some of us are engineers. ;)

[Achernar]

Well, geez, if you're an engineer, then pi = 3 (http://www.straightdope.com/classics/a3_341.html) should be good enough for you, no? ;)

What's worse, I notice that both John W. Kennedy's mnemonic and Chronos's mnemonic round the wrong way. Unless Chronos counts the question mark. :)

[Tom Arctus]

Originally posted by Chronos
What's more, it's actually possible that those measurements [from the Christian Bible] are exactly and precisely correct . . . It is possible for a circle in spherical geometry to have diameter of exactly 10 cubits and circumference of exactly 30 cubits.

I'm sorry, I don't follow you. Are we talking about a circle that also happens to lie on the surface of a sphere in a three-dimensional Euclidian space? Because if we are it seems to me it would have to also lie in some plane in that space and conform to the other requirement of a circle, that of having all its points equidistant from its center. So its circumference would still have to be pi times its diameter, not exactly three times its diameter. There's no way that I know of to deform a circle as defined above in Euclidian or any other space so that the ratio of its circumference to its diameter conforms to the Biblical passage you cite.

I admit that yours is a more sophisticated argument, if only on the surface, than the more common one that the Bible writers were citing measurements of the outside diameter of a thick wall and its inside circumference or suchlike. But the bottom line is that the Christian Bible is not a mathematics text; the value of pi or its deriviation was unknown to its authors; they apparently had no other earthly or divine source for the correct information; and so 3 was close enough to pi for them.

I suppose I could be wrong as I never really got into non-Euclidian geometries, but for now I'd have to bet that I'm not wrong.

[Chronos]

On the contrary, mine does not round the wrong way, as it does not round at all! It truncates!

[Tom Arctus]

One might define the "diameter" of a circle lying on a sphere as the length of an arc segment passing through its center and also lying on the sphere, which I suppose one might be able to manipulate in order to get the ratio to come out to exactly three. But there's no indication in the passage you cite that the authors meant one should measure the distance from one side of a concave vessel to the other by measuring along the closed bottom surface of said vessel rather than using the intuitive method of measuring across its open top. I'll stick with the above until a different conclusion can be demonstrated.

[ultrafilter]

Originally posted by Tom Arctus
I'm sorry, I don't follow you. Are we talking about a circle that also happens to lie on the surface of a sphere in a three-dimensional Euclidian space? Because if we are it seems to me it would have to also lie in some plane in that space and conform to the other requirement of a circle, that of having all its points equidistant from its center. So its circumference would still have to be pi times its diameter, not exactly three times its diameter. There's no way that I know of to deform a circle as defined above in Euclidian or any other space so that the ratio of its circumference to its diameter conforms to the Biblical passage you cite.

A circle in spherical geometry is not the same as a circle on the surface of a sphere. Rather, it resembles two circles of equal diameter running parallel to a great circle (I think--this is a semi-educated guess), and its radius is the distance from any point on either circle to the closest pole. I'm not gonna work out the area or circumference right now, but I suspect that Chronos is correct.

[Tom Arctus]

Originally posted by ultrafilter


A circle in spherical geometry is not the same as a circle on the surface of a sphere. Rather, it resembles two circles of equal diameter running parallel to a great circle . . . and its radius is the distance from any point on either circle to the closest pole.

In that case it seems to me we now have two questions:

1. Might a circle in spherical geometry as described above have a diameter exactly three times its radius?

2. Could there be a physical object such as a vessel for holding liquid in three-dimensional, non-spherical Euclidian space that could have a diameter exactly three times its radius?

Even if the answer to 1. is, "yes," I'm sticking to my guns that the answer to 2. would still be, "not unless we redefine some terms."

[Chronos]

I presume that you meant "circumference three times its diameter" there, Tom. Diameter is always twice radius, by definition.

For a circle in spherical geometry, the ratio of circumference to diameter can be anything greater than or equal to 2 and less than the Euclidian value of p, depending on the size of the circle, so 3 is a possibility.

As for you second question in normal, 3-d Euclidean space, it depends on how one defines "the distance across" something. If you find the distance across a bowl by stretching a string across, then you're in flat geometry, and p = 3.14159265358979323846... . If you do it by rolling a measuring wheel along the bottom of the bowl, then it's less.

By the way, I was not trying to imply that this was the True Explanation for the passage in Kings: Roundoff error is a far more plausible explanation. I was just saying that it could be an answer.

[Tom Arctus]

Yeah, whatever. BTW, whatever happened to:

`` . . . it's actually possible that those measurements are exactly and precisely correct. The object in question is bowl-shaped, so if measurements are restricted to the surface of the bowl, then we're in spherical geometry, and pi is less than the Euclidian value.''

[ultrafilter]

Originally posted by Tom Arctus
Yeah, whatever. BTW, whatever happened to:

`` . . . it's actually possible that those measurements are exactly and precisely correct. The object in question is bowl-shaped, so if measurements are restricted to the surface of the bowl, then we're in spherical geometry, and pi is less than the Euclidian value.''

Pi refers to exactly one number, the ratio of the circumference of a circle to its diameter in Euclidean space.

[h.sapiens]

Mmmm... pi.

[Jabba]

Incidentally, the appproximations for p come from its continued fraction (http://www.cut-the-knot.com/do_you_know/fraction.shtml) expansion
p = [3,7,15,1,292,...].

The 1st convergent is 3 + 1/7 = 22/7. This is the best possible approximation to p by a rational number with denominator no greater than 56.

The 2nd is 3 + 1/(7 + 1/15) = 333/106

The 3rd is 3 + 1/(7 + 1/(15 + 1/1)) = 355/113. This approximation differs from p by less than 5*10-7 and was discovered by the Chinese mathematician Chao Jung-Tze c. AD 500.

The 4th is 3 + 1/(7 + 1/(15 + 1/(1 + 1/292))) = 103993/33102.

After that it starts to get silly.

[jmonster]

I just remember e^(pi i) + 1 = 0 and solve for whichever term I need.

[Exapno Mapcase]

p doesn’t have to be represented by a continued fraction. There are books filled with various expansions for p, some of them extremely clever and some impossible (for me) to understand how anyone could come up with them.

Here’s a simple one:

p = 4(1 – 1/3 + 1/5 – 1/7 + …)

And it doesn’t even have to be a summation. You can find p through multiplication (and square roots), as Euler did:

p 2 = 6[22 / (22 – 1) x 32 / (32 – 1) x 52 / (52 – 1) x 72 / (72 – 1) x …]

[Jabba]

More about p (http://pauillac.inria.fr/algo/bsolve/constant/pi/pi.html)

[Chronos]

There are books filled with various expansions for p, some of them extremely clever and some impossible (for me) to understand how anyone could come up with them.Usually, you don't start by saying "I want to know what the value of p is". You start with the infinite series, or the infinite product, or a definite integral, or whatever it is, and you want to find the value of it. Eventually, your number crunching leads you to some trignometric function, and that function gives you an answer expressed in terms of p. If you want to publish a book, you then say "Aha! That means that I can also express p as (long complicated expression that was difficult to solve in the first place)".

[tracer]

I memorized pi to 200 digits when I was in high school. Do I get a cookie?

[h.sapiens]

No, you get a piece of pi.

[tracer]

:smack: