How Can Pi Be Represented as a Continued Fraction?

[Math Amateur]

Iam reading Julian Havil's book, The Irrationals: A Story of the Numbers You Can't Count On.

In Chapter 3 Havil is writing about progress in the eighteenth century in determining the nature of \(\displaystyle \pi\) and \(\displaystyle e\) through the use of continued fractions. He writes (pages 92 - 93):View attachment 2851
View attachment 2852

Can someone please explain how one can determine/show \(\displaystyle \pi\) as a continued fraction: that is how is the expression for \(\displaystyle \pi\) given at the bottom of the page derived - what is the method of derivation?

Peter

 

[chisigma]

Iam reading Julian Havil's book, The Irrationals: A Story of the Numbers You Can't Count On.

In Chapter 3 Havil is writing about progress in the eighteenth century in determining the nature of \(\displaystyle \pi\) and \(\displaystyle e\) through the use of continued fractions. He writes (pages 92 - 93):View attachment 2851
View attachment 2852

Can someone please explain how one can determine/show \(\displaystyle \pi\) as a continued fraction: that is how is the expression for \(\displaystyle \pi\) given at the bottom of the page derived - what is the method of derivation?

Peter

Trivially that is a numerical trick. Let's suppose You want to approximate $\displaystyle \pi \sim 3.1416$.One possibility among others is...

$\displaystyle \pi = 3 + \frac{1416}{10000}= 3 + \frac{177}{1250} = 3 + \frac{1}{\frac{1250}{177}}= 3 + \frac{1}{7 + \frac{11}{177}} = 3 + \frac{1}{7 + \frac{1}{\frac{177}{11}}} = 3 + \frac{1}{7 + \frac{1}{16 + \frac{1}{11}}}$

Of course we have approximated pi greek with a few decimals so that even the continued fraction is approximated ...

Kind regards

$\chi$ $\sigma$

 

[Math Amateur]

Trivially that is a numerical trick. Let's suppose You want to approximate $\displaystyle \pi \sim 3.1416$.One possibility among others is...

$\displaystyle \pi = 3 + \frac{1416}{10000}= 3 + \frac{177}{1250} = 3 + \frac{1}{\frac{1250}{177}}= 3 + \frac{1}{7 + \frac{11}{177}} = 3 + \frac{1}{7 + \frac{1}{\frac{177}{11}}} = 3 + \frac{1}{7 + \frac{1}{16 + \frac{1}{11}}}$

Of course we have approximated pi greek with a few decimals so that even the continued fraction is approximated ...

Kind regards

$\chi$ $\sigma$

Thanks chisigma ... so it is not a way of determining \(\displaystyle \pi \)... or even indeed of accurately representing \(\displaystyle \pi\) ... seems that all that is going on is that a decimal approximation for \(\displaystyle \pi\) is found in some way and continued fractions used to represent the decimal approximation ... is that correct?

Peter

 

[chisigma]

Thanks chisigma ... so it is not a way of determining \(\displaystyle \pi \)... or even indeed of accurately representing \(\displaystyle \pi\) ... seems that all that is going on is that a decimal approximation for \(\displaystyle \pi\) is found in some way and continued fractions used to represent the decimal approximation ... is that correct?

Peter

Yes, it is!... pratically You derive the continuous fraction starting from a known approximation of $\pi$ and not vice-versa... note that a different approach, using the sign '-' instead of the sign '+' would be...

$\displaystyle \pi \sim 4 - \frac{8584}{10000} = 4 - \frac{1073}{1250} = 4 - \frac{1}{\frac{1250}{1073}} = 4 - \frac{1}{2 - \frac{896}{1073}} = ... $

It is clear that it is a procedure of no practical usefulness ...

Kind regards

$\chi$ $\sigma$

 

[Deveno]

Unfortunately, chisigma is vastly understating the difficulty of what is going on, here.

The proof of this is a bit involved, and was arrived at by estimating a "quadrature" of 1/4 of a circle.

In modern terms:

$\displaystyle \pi = 4\cdot \int_0^1 \sqrt{1 - x^2}\ dx$

You may find the following references "somewhat" useful (there's some heavy lifting involved):

The world of Pi - Brounker

http://www.raco.cat/index.php/PublicacionsMatematiques/article/download/34548/216366

https://www.youtube.com/watch?v=rOaMumPwj4I

Anytime you see $\pi$ in a mathematical formula, you should suspect trigonometric functions (or their inverses) are lurking somewhere in the background. Often we have some formula involving $\sin x$, for example, and the formula simplifies to something that "appears" mysterious, when we choose some special value for $x$, like: $\pi/2$.

The Brouncker formula, which he derived from Wallis' formula, in its simplicity hides some deep mathematics (the video may be the most accessible way for you to see this).

Another "pithy" formula involving $\pi$ and the "natural base" $e$ is:

$e^{i\pi} + 1 = 0$

which, again, disguises some deep ideas involving analytic functions on the complex plane, which shows that trigonometry functions and the exponential function are "two slices of the same beast", ultimately, due to the "rotational" nature of complex-multiplication (where there's circle-arcs, there's subtended angles, and hence trigonometric relationships).

The proof of the above formula (often known as "Euler's formula") depends in an essential way on the theory of differential equations and/or complex power-series. In turns out that the "interlaced" behavior of the sine and cosine functions with respect to differentiation, along with certain "initial values" completely characterize these functions.

The circle is a profound mathematical object: one could study it for years, and not exhaust what it has to say. $\pi$ is only one of its mysteries.

 

[chisigma]

Unfortunately, chisigma is vastly understating the difficulty of what is going on, here...

It is hardly necessary to point out that Peter had brought to our attention precisely the following continued fraction ...

$\displaystyle \pi = 3 + \frac{1}{7 + \frac{1}{15 + \frac{1}{292 + ...}}}\ (1)$

...and not to other better known continued fractions as that due to Lord William Brounker...

$\displaystyle \frac{\pi}{4}= \frac{1}{1 + \frac{1^{2}}{2 + \frac{3^{2}}{2 + \frac{5^{2}}{2 +...}}}}\ (2)$

Even a pharmacist comes to understand that (2) is the result of a well-defined logical framework that allows the extension to a larger number of terms while (1) is not. From a purely practical point of view to use for the calculation of $\pi$ type the continued fraction (2) is less efficient than using a series expansion of the type ...

$\displaystyle \frac{\pi}{6} = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(\sqrt{3})^{2 n+1}\ (2 n + 1)}\ (3)$

Using (3) In fact, I can always increase the accuracy achieved by calculating a more term and having a precise criterion of accuracy achieved. Using (2) instead of adding a term means more repeat accounts from the beginning and You do not have a precise estimate of the accuracy achieved. For this reason I consider the use of the continued fraction for the calculation of the functions a topic of academic interest more than anything else ...

Kind regards

$\chi$ $\sigma$

 

[Deveno]

In all fairness, chisigma, I did not realize your reply was directed at the "last" continued fraction in Peter's post.

It is possible to make a continued fraction for $\pi$ from any cauchy sequence of rational numbers converging to $\pi$. These may not have any particular "rhyme or reason" to them, in terms of the numbers appearing in the "denominators" of said continued fraction.

Continued fractions for $\pi$ that DO have a predictable pattern to them, are another story. There are usually good reasons for this, ones that are often "not-so-easy" to explain.

Indeed, continued fractions are certainly (in most cases, at any rate), "computationally expensive", as compared to power series expansions. Whether such a thing only accords them "academic interest" is, itself, an "academic" question, unless one is tasked with computing an irrational number to an extreme degree of precision, with limited computational resources (such as our brethren were in the 17th century, for example).

I do apologize for my oversight, I understand I can sound more critical than my heart intends. (Angel)

 

[Math Amateur]

In all fairness, chisigma, I did not realize your reply was directed at the "last" continued fraction in Peter's post.

It is possible to make a continued fraction for $\pi$ from any cauchy sequence of rational numbers converging to $\pi$. These may not have any particular "rhyme or reason" to them, in terms of the numbers appearing in the "denominators" of said continued fraction.

Continued fractions for $\pi$ that DO have a predictable pattern to them, are another story. There are usually good reasons for this, ones that are often "not-so-easy" to explain.

Indeed, continued fractions are certainly (in most cases, at any rate), "computationally expensive", as compared to power series expansions. Whether such a thing only accords them "academic interest" is, itself, an "academic" question, unless one is tasked with computing an irrational number to an extreme degree of precision, with limited computational resources (such as our brethren were in the 17th century, for example).

I do apologize for my oversight, I understand I can sound more critical than my heart intends. (Angel)

Thanks to both Chisigma and Deveno for help and insights regarding my post above

Peter

 

[Math Amateur]

Thanks to both Chisigma and Deveno for help and insights regarding my post above

Peter

For those interested in the history of mathematics, I thought I would mention, in the context of continued fractions and \(\displaystyle \pi \), that Julian Havil mentions that the first person to publish a proof of the irrationality of \(\displaystyle \pi \) was Johann Heinrich Lambert (1728 - 1777) in 1761 using a continued fractions proof.

Some information on this is in Havil's book (pages 104 - 108) - page 104 reads as follows:View attachment 2870

Peter
***EDIT*** For those MHB members interested, I have attached pages 104 - 108 from Havil's fascinating book: The Irrationals: A Story of the Numbers YOu Can't Count On.

 

[chisigma]

For those interested in the history of mathematics, I thought I would mention, in the context of continued fractions and \(\displaystyle \pi \), that Julian Havil mentions that the first person to publish a proof of the irrationality of \(\displaystyle \pi \) was Johann Heinrich Lambert (1728 - 1777) in 1761 using a continued fractions proof.

Some information on this is in Havil's book (pages 104 - 108) - page 104 reads as follows:View attachment 2870

Peter
***EDIT*** For those MHB members interested, I have attached pages 104 - 108 from Havil's fascinating book: The Irrationals: A Story of the Numbers YOu Can't Count On.

Very interesting post!... congratulations to Peter! (Yes)... truly ingenious solution to the problem found by Lambert: if $\displaystyle x= \tan \frac{p}{q}$ is an irrational number and if $\displaystyle \tan \frac{\pi}{4} = 1$, 1 being rational, then $\displaystyle \frac{\pi}{4}$ can not be rational! (Rofl)...

Regarding the continued fraction head on page 104 and the 'mysterious phrase': '... the nice pattern with which Euler juggled is absent...', it can be that some investigation is helpful...

Kind regards

$\chi$ $\sigma$

 

[chisigma]

Regarding the infamous continued fraction a first result I found is the following ...

http://www.math.uiuc.edu/~hildebr/453.spring11/pi-cf.pdf

Here is associated the approximated...

π =3.14159265358979323846264338327950288419716939937510582097494459230781
6406286208998628034825342117067982148086513282306647093844609550582231
7253594081284811174502841027019385211055596446229489549303820... (1)

... with the set of coefficients ...

π = [3, 7, 15, 1,292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1,15, 3, 13, 1,
4, 2, 6, 6, 99, 1,2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1,
3, 1, 1, 8, 1, 1, 2,1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4, 16, 1, 161, 45, 1, 22, 1, 2, 2,1, . . .] (2)

The following note... ...there is no obvious pattern to CF digits of π ...

... clearly shows that (2) is derived from (1) and not vice versa, so that the importance of (2) in the calculation of $\pi$ is very small, as I have argued from the beginning ...

Kind regards

$\chi$ $\sigma$

 

[Deveno]

At the risk of sounding like an idiot (OK, I AM an idiot, but I don't like to SOUND like one), some questions worth investigating are:

1. On what basis do we take $\pi$ to be a number?

(a) For one, how do we KNOW the ratio of a circle's circumference to it's diameter is constant?

(b) How does one even begin to measure arc-length, or, for that matter, even straight lines?

2. What might a proof that $\pi$ is irrational entail?

3. Having established (2), what might a proof that $\pi$ is not algebraic entail?

I am not suggesting, of course, that these questions do not have answers, they do, and perfectly sound ones. I am suggesting that seeking to answer them might lead one to a good deal of interesting mathematics. The answer to (1), for example, took several centuries to answer properly, even though everyone "knew" it was SOME kind of number.