Decimal Approximations of Logarithms: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on the decimal expansions of logarithms.

Part 1: Pedagogical motivation: how can students develop a better understanding for the apparently random jumble of digits in irrational logarithms?

Part 2: Idea: use large powers.

Part 3: Further idea: use very large powers.

Part 4: Connect to continued fractions and convergents.

Part 5: Tips for students to find these very large powers.

 

Continued fractions and pi

I suggest the following activity for bright middle-school students who think that they know everything that there is to know about fractions.

The approximation to \pi that is most commonly taught to students is \displaystyle \frac{22}{7}. As I’ll discuss, this is the closest rational number to \pi using a denominator less than 100. However, it is possible to obtain closer rational approximations to \pi using larger numbers. Indeed, the ancient Chinese mathematicians were superior to the ancient Greeks in this regard, as they developed the approximation

\pi \approx \displaystyle \frac{355}{133}

It turns out that this is the best rational approximation to \pi using a denominator less than 16,000. In other words, \displaystyle \frac{355}{133} is the best approximation to \pi using a reasonably simple rational number.

Step 1. To begin, let’s find \pi with a calculator. Then let’s now subtract 3 and then find the inverse.

TIpi1

This calculation has shown that

\pi = \displaystyle 3 + \frac{1}{7.0625133\dots}

If we ignore the 0.0625133, we obtain the usual approximation

\pi \approx \displaystyle 3 + \frac{1}{7} = \frac{22}{7}

Step 2. However, there’s no reason to stop with one reciprocal, and this might give us some even better approximations. Let’s subtract 7 from the current denominator and find the reciprocal of the difference.

TIpi2

At this point, we have shown that

\pi = \displaystyle 3 + \frac{1}{7 + \displaystyle\frac{1}{15.9965944\dots}}

If we round the final denominator down to 15, we obtain the approximation

\pi \approx \displaystyle 3 + \frac{1}{7 + \displaystyle\frac{1}{15}}

\pi \approx \displaystyle 3 + \frac{1}{~~~\displaystyle \frac{106}{15}~~~}

\pi \approx \displaystyle 3 + \frac{15}{106}

\pi \approx \displaystyle \frac{333}{106}

Step 3. Continuing with the next denominator, we subtract 15 and take the reciprocal again.

TIpi3

At this point, we have shown that

\pi = \displaystyle 3 + \frac{1}{7 + \displaystyle\frac{1}{15 + \displaystyle \frac{1}{1.00341723\dots}}}

If we round the final denominator down to 1, we obtain the approximation

\pi \approx \displaystyle 3 + \frac{1}{7 + \displaystyle\frac{1}{16}}

\pi \approx \displaystyle 3 + \frac{1}{~~~\displaystyle \frac{113}{16}~~~}

\pi \approx \displaystyle 3 + \frac{16}{113}

\pi \approx \displaystyle \frac{355}{113}

Step 4. Let me show one more step.TIpi4

At this point, we have shown that

\pi = \displaystyle 3 + \frac{1}{7 + \displaystyle\frac{1}{15 + \displaystyle \frac{1}{1 + \displaystyle \frac{1}{292.634598\dots}}}}

If we round the final denominator down to 292, we (eventually) obtain the approximation

\pi \approx \displaystyle \frac{52163}{16604}

green lineThe calculations above are the initial steps in finding the continued fraction representation of \pi. A full treatment of continued fractions is well outside the scope of a single blog post. Instead, I’ll refer the interested reader to the good write-ups at MathWorld (http://mathworld.wolfram.com/ContinuedFraction.html) and Wikipedia (http://en.wikipedia.org/wiki/Continued_fraction) as well as the references therein.

But I would like to point out one important property of the convergents that we found above, which were

\displaystyle \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, ~ \hbox{and} ~ \frac{52163}{16604}

All of these fractions are pretty close to \pi, as shown below. (The first decimal below is the result for 22/7.)

TIpi5

In fact, these are the first terms in a sequence of best possible rational approximations to $\pi$ up to the given denominator. In other words:

  • \displaystyle \frac{22}{7} is the best rational approximation to \pi using a denominator less than $106$. In other words, no integer over 8 will be any closer to \pi than \displaystyle \frac{22}{7}.  No integer over 9 will be any closer to \pi than \displaystyle \frac{22}{7}. And so on, all the way up to denominators of 105. Small wonder that we usually teach children the approximation \pi \approx \displaystyle \frac{22}{7}.
  • Once we reach 106, the fraction \displaystyle \frac{323}{106} is the best rational approximation to \pi using a denominator less than 113.
  • Then \displaystyle \frac{355}{113} is the best rational approximation to \pi using a denominator less than 16604.

As noted above, the ancient Chinese mathematicians were superior to the ancient Greeks in this regard, as they were able to develop the approximation \pi \approx \displaystyle \frac{355}{113}. For example, Archimedes was able to establish that

3\frac{10}{71} < \pi < 3\frac{1}{7}