# What Do Numbers Mean?

This week I wrote about experimenting with number base systems in my AP Calculus BC class. A question came into my head yesterday about repeating decimals in base ten and whether/how we could decide if that number is also repeating in different number bases. It was really hard and the calculations got pretty ugly. So, today I started class with the following idea. I wrote a repeating decimal in a different number base and then converted it to base ten. The calculations are clearly more manageable and I had a clear idea that this could link back to our conversations about infinite series. What excited me today was that my vision of the infinite series was different than that suggested by my student Megan AND it was entirely different than a suggestion by my student Elijah.

I started with the base 3 number 0.122122122… I saw this as three different infinite geometric series’ each with a ratio of 1/27 and I worked the problem this way. Megan saw this as one series made of the first three terms with a ratio of 1/27. We, of course, arrived at the same answer and I really liked the way that her techniques was only one series to calculate instead of calculating three different series’ the way I saw it. I will put a picture below here with a different example showing Megan’s technique.

The example above started with the base 5 number 3.021021021021…

Elijah had a completely different approach, one based on how we teach converting base ten repeating decimals into fractions. The picture below shows his approach.

A couple of notes here. First, Elijah is a terrific math mind and this is a really creative approach. Second, this approach models the approach that my students have already seen. Third, this tactic encourages you to actually live more thoughtfully in this different number base.

I just came away SO impressed by the thoughtfulness, the persistence, and the creativity of my students this morning.