# What does "Pi" mean? **URL:** \ **Category:** Factual Questions\ **Created:** [September 30, 2003, 2:19am UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610 "2003-09-30T02:19:21Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![MonkeyMensch](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@MonkeyMensch](https://boards.straightdope.com/u/MonkeyMensch)\ **Post date:** [September 30, 2003, 7:45pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/21 "2003-09-30T19:45:35Z")
Thanks for the mention of _perimetron_ from which pi comes. I’ve always thought that pi is a fundamental aspect of mathematics. As a fun aside, though, it represents the ratio of circumference to diameter. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 30, 2003, 8:16pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/22 "2003-09-30T20:16:35Z")
If we’re getting into mnemnonics, you can also use “May I have a large container of coffee”, and count the letters in each word, for 31415926, better than **CurtC** ’s ratio. > [@](#): > > I’ve always thought that pi is a fundamental aspect of mathematics. You mean, define it by something like pi = 4/1 - 4/3 + 4/5 - 4/7 + … , and then have it as a theorem that pi times diameter equals circumference? I suppose you could do it that way, but it makes sense to use the most obvious application as the definition. ---
**Author:** ![N9IWP](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/n9iwp/32/3154_2.png) [@N9IWP](https://boards.straightdope.com/u/N9IWP)\ **Post date:** [September 30, 2003, 8:29pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/23 "2003-09-30T20:29:17Z")
Read “Contact” for one reason to figure out pi to a zillion digits. Bragging rights is probably the main reason. Brian ---
**Author:** ![MonkeyMensch](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@MonkeyMensch](https://boards.straightdope.com/u/MonkeyMensch)\ **Post date:** [September 30, 2003, 8:41pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/24 "2003-09-30T20:41:06Z")
> [@](#): > > \*Originally posted by Chronos \* > \*\*If we’re getting into mnemnonics, you can also use “May I have a large container of coffee”, and count the letters in each word, for 31415926, better than **CurtC** ’s ratio.You mean, define it by something like pi = 4/1 - 4/3 + 4/5 - 4/7 + … , and then have it as a theorem that pi times diameter equals circumference? I suppose you could do it that way, but it makes sense to use the most obvious application as the definition. \*\* Or my favorite, _How I wish I could recollect of circle round The exact relation Archimede unwound_ I know you have to cheat on Archmedes name, but hey: how else are you going to the number nine in there.😉 As to the application of pi, **Chronos** , I was thinking more along the lines of how it crops up in power series and Gaussian distributions and a number of other places. It is a fundamental number in math, as I’m sure you know, and a feature of that essential nature is the ratio of circumference to diameter. But it is not the definition of the the number. It is rather one of the key bricks at the bottom of the structure. By the way, love your posts, \*\*Chronos[/]. Getting cold up there yet? ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [September 30, 2003, 8:42pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/25 "2003-09-30T20:42:21Z")
Here’s an impressive…well, I don’t think _mnemonic_ is quite the right word, since I’m doubtful that it could serve much use as a memory aid…piece of constrained writing is a better description. It’s constrained by the first (nearly) 4000 digits of pi (there are other constraining devices in it as well, I believe): [http://users.aol.com/s6sj7gt/cadenza.htm](http://users.aol.com/s6sj7gt/cadenza.htm) ---
**Author:** ![biqu](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@biqu](https://boards.straightdope.com/u/biqu)\ **Post date:** [September 30, 2003, 9:54pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/26 "2003-09-30T21:54:15Z")
> [@](#): > > _Originally posted by Chronos_ > You mean, define it by something like pi = 4/1 - 4/3 + 4/5 - 4/7 + … , and then have it as a theorem that pi times diameter equals circumference? I suppose you could do it that way, but it makes sense to use the most obvious application as the definition. > [@](#): > > _from Goldberg’s Methods of Real Analysis_ > Before we begin our development of the trigonometric functions we would like to point out that, in most elementary calculus texts, the proof that the derivative of the sine function is the cosine function depends on the fact that \lim\_{x \rightarrow 0} ( \sin x ) / x = 1. This limit is usually established by a geometric argument that involves the formula for the total area of a circle. Hence, to find the derivative of the sine function by this approach we must know the formula for the area of a circle. However, one of the chief early applications of the trigonometric functions in a calculus course is the evaluation of 4 \int\_{0}^{r} \sqrt{ r^{2} - x^{2} } dx > – the integral that represents the area of a circle. Thus in this overall approach, the “proof” that the area of a circle is \pi r^{2} using trigonometric substitution is not valid, since it involves knowing that the derivative of the sine is the cosine, while this in turn involves knowing that the area of a circle is \pi r^{2}. > In our development of the trigonometric functions we assume no formulae from geometry. In fact, along with everything else, we define \pi itself (without geometry). ---
**Author:** ![The\_Ryan](https://avatars.discourse-cdn.com/v4/letter/t/7feea3/32.png) [@The\_Ryan](https://boards.straightdope.com/u/The_Ryan)\ **Post date:** [September 30, 2003, 11:30pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/27 "2003-09-30T23:30:05Z")
Another mnemonic is the Pi Round. It’s sung to the tune of Frere Jacques: Three point one four (three point one four) One five nine (one five nine) Two six five three five eight (two six five three five eight Nine seven nine (nine seven nine) Besides being defined as being equal to C/D, another way to define pi is to say that it is (ln -1)/sqrt(-1). In some ways, that is a more natural and useful definition. amore ac studio: Your quote does present an example of circular (heh) reasoning, but it can avoided by performing the integral in polar coordinates. ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [September 30, 2003, 11:38pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/28 "2003-09-30T23:38:46Z")
Is ln(-1)/sqrt(-1) really a more useful defintion as both numerator and denominator are imaginery? I prefer the sequence as you can obtain a value with arbitary error from it. ---
**Author:** ![biqu](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@biqu](https://boards.straightdope.com/u/biqu)\ **Post date:** [October 1, 2003, 1:18am UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/29 "2003-10-01T01:18:28Z")
> [@](#): > > _Originally posted by The Ryan_ > **amore ac studio** : > Your quote does present an example of circular (heh) reasoning, but it can be avoided by performing the integral in polar coordinates. Polar coordinates, eh? How would you know the value of the Jacobian without knowing the derivatives of the trig functions? Maybe with some handwaving you could convince yourself that the polar area element is r dr d(theta), but Goldberg explicitly wanted to avoid assuming formulae from geometry. His definition of \pi is in terms of a definite integral whose integrand has the antiderivative arcsine. He defines almost all the elementary functions in terms of integrals, but he is able to get the exponential function almost for free, as the sum of cosh and sinh. It’s not the most intuitive presentation, but it does avoid circular reasoning like the example quoted above. ---
**Author:** ![robby](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/robby/32/11048_2.png) [@robby](https://boards.straightdope.com/u/robby)\ **Post date:** [October 1, 2003, 2:02am UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/30 "2003-10-01T02:02:22Z")
> [@](#): > > \*Originally posted by quiltguy154 \* > \*\*I JUST GOTTA SAY IT, I HOPE YOU’LL ALL INDULGE ME. > > Pi r round, cornbread r square. > > Thank you, and good morning. \*\* Then at least do the complete joke! 😛 "Teacher: The area of a circle is equal to p_r[sup]2[/sup]_ [pi r squared]. Doofus student: Pie are not squared! Cake are squared! Pie are _round_." ---
**Author:** ![TaxGuy](https://avatars.discourse-cdn.com/v4/letter/t/b19c9b/32.png) [@TaxGuy](https://boards.straightdope.com/u/TaxGuy)\ **Post date:** [October 1, 2003, 2:43am UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/31 "2003-10-01T02:43:54Z")
> [@](#): > > Polar coordinates, eh? How would you know the value of the Jacobian without knowing the derivatives of the trig functions? Maybe with some handwaving you could convince yourself that the polar area element is r dr d(theta), but Goldberg explicitly wanted to avoid assuming formulae from geometry. Best. GQ rant. Ever. The value of the Jacobian indeed . . . ---
**Author:** ![The\_Ryan](https://avatars.discourse-cdn.com/v4/letter/t/7feea3/32.png) [@The\_Ryan](https://boards.straightdope.com/u/The_Ryan)\ **Post date:** [October 1, 2003, 5:04pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/32 "2003-10-01T17:04:32Z")
Jacobian? What do you need the Jacobian for? The Jacobian is used in double integrals. This is single integral. Sure, you could look at this as a special case of a double integral, in which the integrand is an indicator function, but why? Take any wedge wi. Let li be the largest triangle contained in wi, and ui be the smallest triangle that contains wi. Let lt=sum (li), wt=sum (wi), and ut= sum (ui). Since for each i, li\ [@](#): > > I prefer the sequence as you can obtain a value with arbitary error from it. Actually, it is possible, with a little work, to get a series expression for pi from the definition I gave. It’s not as nice as the normal one, though. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [October 1, 2003, 7:56pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/33 "2003-10-01T19:56:43Z")
> [@](#): > > Hence, to find the derivative of the sine function by this approach we must know the formula for the area of a circle. However, one of the chief early applications of the trigonometric functions in a calculus course is the evaluation of 4 \int\_{0}^{r} \sqrt{ r^{2} - x^{2} } dx > – the integral that represents the area of a circle. But of course, there are many other methods for finding the formula for area of a circle. Archimedes didn’t know trig, but he was still able to determine that a circle has the same area as a triangle with base equal to the circle’s circumference and height equal to the radius. He did so by showing that an inscribed polygon always has an area less than this amount, and a circumscribed polygon always has area greater than this amount (today we would do this using limits, but the Wizard of Syracuse didn’t (quite) have that tool at his disposal). And I would hardly say that ln(-1) / sqrt (-1) is superior as a definition; it depends either on the plane interpretation of complex numbers, in which case you are just measuring a circle (the circle of roots of unity), or on the power-series definition of the exponential function (and hence of the log function), which ends up just being the 4 - 4/3 + 4/5 … series I mentioned earlier, in an uglier form. ---
**Author:** ![biqu](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@biqu](https://boards.straightdope.com/u/biqu)\ **Post date:** [October 1, 2003, 9:34pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/34 "2003-10-01T21:34:05Z")
> [@](#): > > _Originally posted by The Ryan_ > Jacobian? What do you need the Jacobian for? The Jacobian is used in double integrals. If you wanted to do the integral in polar coordinates to avoid circular reasoning, it would have to be a double integral. I’ve never heard of a polar coordinate system for a one-dimensional space. ---
**Author:** ![The\_Ryan](https://avatars.discourse-cdn.com/v4/letter/t/7feea3/32.png) [@The\_Ryan](https://boards.straightdope.com/u/The_Ryan)\ **Post date:** [October 1, 2003, 9:38pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/35 "2003-10-01T21:38:33Z")
Seems to me that the polygon method is a good start, but one must still show that the inner and outer areas converge to the same amount. Did Archimedes do so? (Without limits, I don’t see how he could). I don’t know what you mean when you say that it depends on the plane interpretation of complex numbers. pi can be defined as the number such that e^pi\*sqrt(-1)=-1, regardless of how one interprets sqrt(-1). ---
**Author:** ![biqu](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@biqu](https://boards.straightdope.com/u/biqu)\ **Post date:** [October 1, 2003, 10:03pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/36 "2003-10-01T22:03:20Z")
> [@](#): > > _Originally posted by Chronos_ > Archimedes didn’t know trig, but he was still able to determine that a circle has the same area as a triangle with base equal to the circle’s circumference and height equal to the radius. He did so by showing that an inscribed polygon always has an area less than this amount, and a circumscribed polygon always has area greater than this amount (today we would do this using limits, but the Wizard of Syracuse didn’t (quite) have that tool at his disposal). I remember a historical note in one of the calculus textbooks of my father’s generation, which mentioned that if Archimedes had not been killed by the Romans, he probably would have gone on to invent the integral calculus. He had already established many area and volume formulas using techniques that we would classify today as calculus, but in order to convince his contemporaries who were trained in the Euclidean method of geometry, he had to derive his results again without the calculus. The same is true of Newton, who in his Principia did not employ any techniques of differential calculus, but instead proved his results with Euclidean geometry. It sounds too good to be true, almost like the notion that Aristotle set science back 2000 years by marginalizing the beliefs of forward-thinking men like Democritus. > [@](#): > > _Originally posted by amore ac studio_ > If you wanted to do the integral in polar coordinates to avoid circular reasoning, it would have to be a double integral. I’ve never heard of a polar coordinate system for a one-dimensional space. Perhaps you meant evaluating the one-dimensional integral \int\_{0}^{2\pi} \frac{1}{2} r^{2} d heta = \pi r^{2}. That will avoid circular reasoning, as you intended. ---
**Author:** ![bonzer](https://avatars.discourse-cdn.com/v4/letter/b/45deac/32.png) [@bonzer](https://boards.straightdope.com/u/bonzer)\ **Post date:** [October 1, 2003, 11:46pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/37 "2003-10-01T23:46:14Z")
> [@](#): > > \*Originally posted by amore ac studio \* > The same is true of Newton, who in his Principia did not employ any techniques of differential calculus, but instead proved his results with Euclidean geometry. The case with Newton is more that, having pretty much spectacularly self-educated himself in mathematics, he invents calculus, but that in then learning more about what other people had done, he’s impressed by the classical Greek style. So that when he comes to develop and write _Principia_ he thinks that the latter is the best way of doing things. In particular, he doesn’t rewrite most of the results into the more traditional style. Those are probably mainly worked out in the Greek style in the first place, rather than using calculus. There’s also the factor that he believed that everything in _Principia_ was a rediscovery of things known to the ancients. He didn’t think he was doing anything the Greeks hadn’t. Furthermore, there are parts of the book that do use techniques that can only be described as calculus. (And, goodness knows, Newton was prepared to fight down and dirty to defend these techniques as innovative in the 17th century.) It’s however striking that much of _Principia_ is written in such a way as to segregate these sections. Later propositions often include qualifications like “granting the quadratures”. He’s shown how you can work out areas at the start, but he’s thus careful to formulate the later sections in such a way that they don’t depend on the acceptance of this particular method. ---
**Author:** ![The\_Ryan](https://avatars.discourse-cdn.com/v4/letter/t/7feea3/32.png) [@The\_Ryan](https://boards.straightdope.com/u/The_Ryan)\ **Post date:** [October 3, 2003, 8:18pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/38 "2003-10-03T20:18:55Z")
“If you wanted to do the integral in polar coordinates to avoid circular reasoning, it would have to be a double integral. I’ve never heard of a polar coordinate system for a one-dimensional space.” No, you could integrate theta against r, or r against theta. Either way, it would be one integral. Yes, it’s two dimensional space, that’s why we only need one integral. Double integrals are for three dimensional spaces. In one integral, the integrand ranges over one dimensional, and the dummy variable varies over another. One integral, two dimensions. ---
**Author:** ![Malacandra](https://avatars.discourse-cdn.com/v4/letter/m/45deac/32.png) [@Malacandra](https://boards.straightdope.com/u/Malacandra)\ **Post date:** [October 3, 2003, 9:38pm UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/39 "2003-10-03T21:38:09Z")
Some more mnemonics: “How I need a drink - alcoholic of course! - after the heavy chapters involving quantum mechanics”, and: "Now I, even I, would celebrate "In rhymes unapt the great "Immortal Syracusan rivaled\* nevermore "Who in his wondrous lore passed on before “Gave men his guidances how to circles mensurate”. \*Is this the US spelling? In English we would spell it “rivalled”, but then the mnemonic doesn’t work - as can be seen by reference to the previous mnemonic. ---
**Author:** ![Seven](https://avatars.discourse-cdn.com/v4/letter/s/3d9bf3/32.png) [@Seven](https://boards.straightdope.com/u/Seven)\ **Post date:** [October 4, 2003, 12:25am UTC](https://boards.straightdope.com/t/what-does-pi-mean/204610/40 "2003-10-04T00:25:18Z")
> [@](#): > > \*Originally posted by elmwood \* > **It’s Greek for "1920s style death ray."** AAAAAAAAAGH! _hits head on desk_ WACK-WACK-WACK-WACK [Previous page](https://boards.straightdope.com/t/what-does-pi-mean/204610.md?page=1) [Next page](https://boards.straightdope.com/t/what-does-pi-mean/204610.md?page=3)