Revisiting 1/3 in binary

Last night we talked about writing in base 3.   A long long long time ago we talked about writing 1/3 in binary.  Here are those two projects:

Pi in base 3

Writing 1/3 in Binary

I suspected that the boys wouldn’t remember the project about writing 1/3 in binary, so I thought it would make a good follow up to last night’s project.

I started by just posing the question and seeing where things went. They boys had lots of ideas and we eventually got most of the way there:

At the end of the last video they boys figured out that if our number was indeed 1/3, if we multiplied it by 3 we should get 1. That reminded them of the proof that 0.9999…. (repeating forever) = 1.

We reviewed that proof and applied it to the situation we had now.

Just one little problem . . . what if we apply the idea in this proof to a different series, say 1 + 2 + 4 + 8 + 16 + . . . . ?

We’ve looked at the idea in this video before:

Jordan Ellenberg’s “Algebraic Intimidation”

We felt pretty comfortable believing that 0.9999…. = 1 and that we’d found the correct series for 1/3 in binary, but do we believe the results when we apply the exact same ideas to a new series?

I love projects like this one 🙂