Approximating pi

Task

1. Different cultures through history have used many different rational numbers to approximate $\pi$. For each of the historical approximations below, use a calculator to determine approximately how far the fraction is from $\pi$:

   1. $\frac{22}{7}\,$ 
   2. $\frac{25}{8}\,$ 
   3. $\left(\frac{9785}{5568}\right)^2\,$ 
   4. $\frac{355}{113}\,$

2. The fraction with denominator 1 which is closest to $\pi$ is $\frac{3}{1} = 3$. For each of the following whole numbers, find the fraction with this denominator which is closest to $\pi$:

   1. 3
   2. 10
   3. 20
   4. 100
   5. 113
   6. 10000
3. Look for patterns in the data you've collected thus far.  What effect does increasing the denominator have on the accuracy of the best approximation?

IM Commentary

The goal of this task is to explore some important aspects of approximating an irrational number with rational numbers. The irrational number chosen here is $\pi$ because it is one of the most interesting, well known, and grade appropriate irrational numbers. The fact that finite decimal representations are approximations by fractions, whose denominators are powers of $10$, is emphasized by parts ii, iv and vi of the second part of the task.

There is a rich and interesting history of finding approximations of $\pi$. Many details, including the historical approximations in the first part of the question, can be found at the Wikipedia page for $\pi$.

Some of the work done before calculators is truly astounding: Daniel Ferguson found 620 digits of $\pi$ in 1946 without the aid of a calculator. Earlier, in 1873, a British mathematician named Shanks calculated 707 digits but after the 528th place, the calculations were incorrect! For this task, students are highly encouraged to use a calculator. In doing so, it is important to realize that the $\pi$ button on a calculator only gives a (very good) approximation of $\pi$, not an exact value.

The theory of approximating irrational numbers with rational numbers is a rich and old one. The fact that $\pi$ can be extremely well approximated by rational numbers whose denominators are not that large is closely related to the fact that it is an irrational number. It was not known until the second half of the 19th century that $\pi$ is an irrational number and there is still no elementary proof of this fact today. Many of the historical estimates for the size of $\pi$ came from inscribing polygons inside a circle (or circumscribing a polygon around a circle) and then calculating the perimeter of those polygons: this technique, appropriate for high school geometry, is explored in https://www.illustrativemathematics.org/content-standards/tasks/1567.

The task can be extended by having students compare the errors of the approximations with sizes of unit fractions. Particularly good approximations have a small ratio of error to the size of unit fraction. Some such approximations are given by continued fraction convergents, which include the Archimedean and Chinese approximations. Continued fractions have not been part of popular mathematics culture for some time but the theory is fascinating and some information is available here: http://en.wikipedia.org/wiki/Continued_fraction.

Solution

1.  
   1. We have $\frac{22}{7} \approx 3.142857$ and $\pi \approx 3.14159265$. These differ by a little more than $\frac{1}{1000}$.
   2. Since $\frac{25}{8} = 3.125$ we can see that this approximation is not as good as $\frac{22}{7}$. It differs from $\pi$ by about $\frac{17}{1000}$.
   3. This approximation is certainly the most complex of the group. Using a calculator we find that $\left(\frac{9785}{5568}\right)^2 \approx 3.088$ and this differs from $\pi$ by a little more than $\frac{1}{20}$. It is the least accurate of all of these estimates.
   4. We have $\frac{355}{113} \approx 3.1415929$ which is extremely close to $\pi$. It differs by a few ten-millionths!
2. 1. With denominator 3, the two numbers closest to $\pi$ are $3 = \frac{9}{3}$ and $\frac{10}{3}$. The closer of these is $3$, differing by a little more than 14 hundredths.
   2. We have $3.1 \lt \pi \lt 3.2$ and $\pi$ is a little closer to 3.1 than to 3.2.
   3. With denominator 20, the closest fractions to $\pi$ are $\frac{62}{20} = 3.1$ and $\frac{63}{20} = 3.15$. The closest of these to $\pi$ is $\frac{63}{20}$.
   4. With denominator 100, we see that $\frac{314}{100} \lt \pi \lt \frac{315}{20}$ and $\frac{315}{100}$ is closer. This is the same approximation as $\frac{63}{20}$.
   5. In part iii of the first part, we saw that $\frac{355}{113} \approx 3.1415929$ is very close to $\pi$. It is a little bigger than $\pi$ so $\pi$ falls between $\frac{355}{113}$ and $\frac{354}{113} \approx 3.133$. So $\pi$ is closest to $\frac{355}{113}$. 
   6. With denominator 10000, $\pi$ lies between $\frac{31415}{10000}$ and $\frac{31416}{10000}$. It is closer to $\frac{31416}{10000}$.

3. One trend we can see from the answers to parts (ii), (iv), and (vi) to b is that if the denominator of the fraction is a power of 10 then we can find the closest fraction to $\pi$ by studying its decimal expansion. For example, since $\pi \approx 3.14159265$, we can see both that it lies between 3.1 and 3.2 and that it is closer to 3.1. We also gain an additional decimal of accuracy each time we increase the power of 10. This reasoning only applies when the denominator is a power of ten.

   Generally speaking, one would expect larger denominators to help get a better approximation for $\pi$, since there is a smaller distance between successive fractions with that denominator.  For example, since fractions of the form $\frac{n}{17}$ (with $n$ an integer) are $\frac{1}{17}$ apart, and since $\pi$ lies between two such numbers, the farthest it could be from the closest one is $\frac{1}{17}$ (in fact, half of this distance, $\frac{1}{34}$).
   
   On the other hand, it is not always true that increasing the denominator permits a more accurate approximation of $\pi$.  For example, as calculated above, the denominator 113 allows for a much more accurate estimate than a denominator of even 10000.  One thing we can say is that multiplying the denominator by another integer permits at least as good an approximation -- for example, we can get as least as good an approximation with denominator 51 as with 17, for the simple reason that every number of the form $\frac{n}{17}$ can also be written as $\frac{3n}{51}$.