# Communicating to students the meaning of extremely large numbers

I'm planning on showing my students a "deep zoom" video of the Mandelbrot set. The video is about 15 minutes long and, at the end, shows an image that is zoomed by a factor of $10^{220}$.

I'd like to convey somehow just how staggeringly huge that scale is. The best I have do far is that if the image shown at the beginning of the video were the size of the Planck scale (many orders of magnitude smaller than a proton), then after scaling it up by a factor of $10^{62}$ it would be about $100$ billion light years across, which is the size of the observable universe.

I suppose I could now iterate: take a speck the size of the Planck scale in the enlarged video, and scale it up until it's the size of the observable universe. Now we are at a scale of about $10^{124}$. Do this two more times; now you have some idea of how large $10^{220}$ is.

I am wondering if anybody has any other ideas for how to convey the enormity of this scale.

• Maybe some of the ideas I posted in this 8 April 2002 sci.math post could be of help. – Dave L Renfro Apr 26 '16 at 19:08
• @DaveLRenfro That is incredibly helpful, and if you would like to post it as an answer I will be happy to accept it. – mweiss Apr 26 '16 at 23:56
• Go ahead and accept another answer, as I'm not up to trying to format that post now. For one thing, I don't think I could resist adding a lot of extra stuff I've come across or thought of since then, and I don't really have the time to do this now. Incidentally, that post was the first of three posts I made at essentially the same time, the other two being BIG NUMBERS #2 and BIG NUMBERS #3. – Dave L Renfro Apr 27 '16 at 14:31
• When I was in secondary school (±30 years ago), I had a physics teacher who once teached me about the size of atoms: he said "Consider a small cube of copper from one cubic centimeter, and a crane, moving 10,000 atoms per second. How long will it take for the whole cube to be moved?", and the answer was "100 billion of centuries!". (Please don't kill me if the numbers are not accurate, it's been quite a while) For explaining the 10^62, this example might be very useful. – Dominique May 16 '18 at 12:18

My favorite video for this is powers of ten from 1977. Though we can get a little smaller today, I think it still does an excellent job with getting the scale of things starting from what we know.

They should pick up on that these numbers are far far far bigger than the largest scale in the video.

I was looking for a link for the book "The Lore of Large Numbers" that I read years ago and I thought might give you some ideas that I came to this new book "Really Big Numbers" that seems fascinating too.

I also like to mention a story from the years that I was a middle school teacher. To give an idea of how the powers of a number get bigger and bigger so quickly we worked on the idea of folding a paper (that gives us powers of two) and supposed that we can do that physically as many as we like. It was amazing for my students to see that just with "a few" folds we could climb to the moon!

You might also find this fascinating video (viewed by about 42 million) useful.

One suggestion would be to take an approach similar to that used for describing the $52!\approx 8.063*10^{67}$ ways to arrange a standard deck of playing cards as outlined in Scott Czepiel's Blog and popularized in this vsauce youTube video @ 14:40. The example given in the blog post is to set a timer for $8.06*10^{67}$ seconds and pass the time by

...picking your favorite spot on the equator. You're going to walk around the world along the equator, but take a very leisurely pace of one step every billion years. Make sure to pack a deck of playing cards, so you can get in a few trillion hands of solitaire between steps. After you complete your round the world trip, remove one drop of water from the Pacific Ocean. Now do the same thing again: walk around the world at one billion years per step, removing one drop of water from the Pacific Ocean each time you circle the globe. Continue until the ocean is empty. When it is, take one sheet of paper and place it flat on the ground. Now, fill the ocean back up and start the entire process all over again, adding a sheet of paper to the stack each time you’ve emptied the ocean. Do this until the stack of paper reaches from the Earth to the Sun. Take a glance at the timer, you will see that the three left-most digits haven’t even changed. You still have 8.063e67 more seconds to go. So, take the stack of papers down and do it all over again. One thousand times more. Unfortunately, that still won’t do it. There are still more than 5.385e67 seconds remaining. You’re just about a third of the way done.

Given that you want to emphasize how large $10^{220}$ is, you can stack all sorts of other counters on top of those mentioned (the linked blog post / video also gives the example of dealing a 5-card hand, buying a single lottery ticket, putting a grain of sand in the grand canyon, etc.). The key to all this is that multiplying numbers with the same base results in the exponents adding (this can be a good exercise for students, verifying that the calculations are correct).

I use a beautiful little story narrated by Helmut Kracke in order to demonstrate the size of the largest three-digit-number 9^9^9.

Wotan, the God of the Teutons, had a very precious ring called Draupnir. Every ninth night nine rings of the same shape dropped from it. He could use them for paying maintenance to his several women. But these rings did not drop further rings.

Now assume, since in modern times everything goes faster, we have a ring that drops nine rings in every second, and every dropped ring will also do so. Then after 10 seconds we have 10 billion rings, after 20 seconds we have as many as there are combinations of the Rubik cube, after 82 seconds as many as there are atoms in the universe. How long do we have to wait for 9^9^9? More than 11 years.

For more detail see https://www.hs-augsburg.de/~mueckenh/HI/HI01.PPT, slides 32-45.

I suggest coupling your Mandelbrot set zoom with the classic Eames Powers of Ten movie.

http://www.eamesoffice.com/the-work/powers-of-ten/

https://en.wikipedia.org/wiki/Powers_of_Ten_(film)