Too many significant digits (Part 1)

The following appeared on my Facebook feed a while back:


Just look at that: the Nike app claimed to measure the length of my friend’s run with twelve decimal places of accuracy.

Let’s have some fun with this. Just suppose that the app was able to measure distance to the nearest trillionth of a mile. One trillionth of a mile is…

5.28 billionths of a foot,

or about 63.4 billionths of an inch,

or about 161 billionths of a centimeter,

or about 1.61 billionths of a meter,

or about 1.61 nanometers.

By way of comparison, the fingernails on the average adult grow about 3 millimeters a month. A world-class runner could run 6.25 miles in about 30 minutes; in those 30 minutes, his/her fingernails would grow about 2 microns, or about 2000 nanometers. (Of course, they’ll grow longer for less athletic runners covering the same distance at a slower speed.)

So if the Nike app can measure my distance to the nearest trillionth of a mile, it would have absolutely no difficulty measuring how much my fingernails grew while running.

Or, it could be that the Nike app really isn’t measuring the distance all that precisely. Probably the app used double-precision arithmetic, and whoever programmed the app didn’t tell it to truncate after a reasonable number of digits.

Lessons from teaching gifted elementary school students (Part 3b)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received:


A \times A = B

B \times B \times B = C

C \times C \times C \times C= D

If the pattern goes on, and if A = 2, what is Z?

In yesterday’s post, we found that the answer was

Z =2^{26!} =  10^{26! \log_{10} 2} \approx 10^{1.214 \times 10^{26}},

a number with approximately 1.214 \times 10^{26} digits.

How can we express this number in scientific notation? We need to actually compute the integer and decimal parts of 26! \log_{10} 2, and most calculators are not capable of making this computation.

Fortunately, Mathematica is able to do this. We find that

Z \approx 10^{121,402,826,794,262,735,225,162,069.4418253767}

\approx 10^{0.4418253767} \times 10^{121,402,826,794,262,735,225,162,069}

\approx 2.765829324 \times 10^{121,402,826,794,262,735,225,162,069}

Here’s the Mathematica syntax to justify this calculation. In Mathematica, \hbox{Log} means natural logarithm:


Again, just how big is this number? As discussed yesterday, it would take about 12.14 quadrillion sheets of paper to print out all of the digits of this number, assuming that Z was printed in a microscopic font that uses 100,000 characters per line and 100,000 lines per page. Since 250 sheets of paper is about an inch thick, the volume of the 12.14 quadrillion sheets of paper would be

1.214 \times 10^{16} \times 8.5 \times 11 \times \displaystyle \frac{1}{250} \hbox{in}^3 \approx 1.129 \times 10^{17} \hbox{in}^3

By comparison, assuming that the Earth is a sphere with radius 4000 miles, the surface area of the world is

4 \pi (4000 \times 5280 \times 12) \hbox{in}^2 \approx 8.072 \times 10^{17} \hbox{in}^2.

Dividing, all of this paper would cover the entire world with a layer of paper about 0.14 inches thick, or about 35 sheets deep. In other words, the whole planet would look something like the top of my desk.

green lineWhat if we didn’t want to print out the answer but just store the answer in a computer’s memory? When written in binary, the number 2^{26!} requires…

26! bits of memory, or…

about 4.03 \times 10^{26} bits of memory, or…

about $latex 5.04 \times 10^{25} bytes of memory, or …

about 5.04 \times 10^{13} terabytes of memory, or…

about 50.4 trillion terabytes of memory.

Suppose that this information is stored on 3-terabyte external hard drives, so that about 50.4/3 = 16.8 trillion of them are required. The factory specs say that each hard drive measures 129 \hbox{mm} \times 42 \hbox{mm} \times 167 \hbox{mm}. So the total volume of the hard drives would be 1.52 \times 10^{19} \hbox{mm}^3, or 15.2 \hbox{km}^3.

By way of comparison, the most voluminous building in the world, the Boeing Everett Factory (used for making airplanes), has a volume of only 0.0133 \hbox{km}^3. So it would take about 1136 of these buildings to hold all of the necessary hard drives.

The cost of all of these hard drives, at $100 each, would be about $1.680 quadrillion. So it’d be considerably cheaper to print this out on paper, which would be about one-seventh the price at $242 trillion.

Of course, a lot of this storage space would be quite repetitive since 2^{26!}, in binary, would be a one followed by 26! zeroes.

The Scale of the Universe

My former student Matt Wolodzko tipped me off about this excellent website that shows the scale of the universe, from the very large to the very small: I recommend it highly for engaging students with the concept of scientific notation.

While I’m on the topic, here are two videos that describe the scale of the universe. The first was a childhood favorite of mine — I vividly remember watching it at the Smithsonian National Air and Space Museum when I was a boy — while the second is more modern.

Full lesson plan: Designing a model solar system

Over the summer, I occasionally teach a small summer math class for my daughter and her friends around my dining room table. Mostly to preserve the memory for future years… and to provide a resource to my friends who wonder what their children are learning… I’ll write up the best of these lesson plans in full detail.

This was a fun activity that took a couple of hours: designing a model Solar System. I chose the scale so that most of the planets would fit on a straight section of sidewalk near my house; of course, the scale could be changed to fit the available space.

For my particular audience of students, I also worked through the basics of the metric system as well as decimals.

This lesson plan is written in a 5E format — engage, explore, explain, elaborate, evaluate — which promotes inquiry-based learning and fosters student engagement.

Model Solar System Handout

Model Solar System Lesson

Post Assessment

P.S. For what it’s worth, the world’s largest model solar system can be found in Sweden.