# Digits

I’m about to start a unit on logarithms. Kids don’t technically know that yet. To prime them, today I gave both my Algebra 2 classes a warm up. I was super nervous about this, because I haven’t seen a crazy amount of endurance from many of my kids when they get stuck on something. And I was going to give them something totally open-ended! And without a calculator allowed! Gasp!

I asked them to do the following. Think about $2^{60}$. It’s going to be a long number when it is all written out. I wanted them to come up with a guesstimate about how many digits there are in the expansion. To scaffold, I asked them for three things:

a) What’s a guess (for the number of digits) that is too low? How do you know? (Can you come up with a larger low estimate?)

b) What’s a guess (for the number of digits) that is too high? How do you know? (Can you come up with a smaller higher estimate?)

c) Based on your work and your intuition, if you had to make a guess, how many digits are in the expansion of $2^{60}$?

Honestly, it was one of the best things I’ve done recently. Kids were showing grit and so much flexibility in their thinking! I had to correct a few misconceptions and nudge a little here and there, but it was all on them how they wanted to go about this. It was beautiful. (At one point, a kid said they wanted to give up, but I came back around a few minutes later and they were rapidly making progress and hadn’t given up.)

At first, kids didn’t know where to start. I told them they were going to get time to work on this, so they could take on strategies that might take a while. (Normally, we start class with something short and quick. I wanted to indicate this wasn’t that.) Initially, I gave 7 minutes, but since so many kids were on a roll, I expanded it to 14 or 20 minutes. I honestly don’t remember how long.

What I adored is that this problem was definitely in their wheelhouse. Most groups were gung ho, and just started writing stuff down — and eventually (sometimes with a little encouragement/prompting from me), they came up with SUPER awesome solutions. Seriously, things I had never thought of.

The main two approaches I saw were:

1. Kids noticing that $2^{10}=1024$. Which is close to $10^3$. So $2^{60}=(2^{10})^6 \approx (10^3)^6=(10^{18})$. So that puts us at around 19 digits.
2. Kids noticing this pattern:

So after going up about every 3 exponents, we add an additional digit to the number. (I say about 3 because all groups who did this method saw that a few times, you’d get 4 exponents in a row which keep the same number of digits instead of 3. But it was usually 3.)Assuming the number of digits increases after going up every 3 exponents, that means that exponent 12 has 4 digits, exponent 15 has 5 digits, exponent 18 has 6 digits, exponent 21 has 7 digits… etc. So exponent 60 has 20 digits.

So that puts us around 20 digits (or maybe a little lower because of those occasional 4 exponents in a row).

That’s about all I wanted to share. I was a little out of my comfort zone because I didn’t know if they would all just throw their hands up and give up. But they didn’t, and instead did some phenomenal thinking.

I just realized… you might want to see how this relates to logarithms. It turns out that the number of digits is equal to doing the following: take the log of the number, and then take the floor function of that result, and then you add one. I won’t spoil it by explaining why, though. See if you can figure it out!