Last week I started the chapter on fractions in decimals in our *Introduction to Number Theory* book with my younger son. Yesterday we were talking about repeating decimals like 1/3 and found that he had a little misconception about decimals:

He thought that since 1/3 = 0.3333…., that 10/3 would be 0.3030303030….. because when you multiply each of the 3’s in 1/3 by 10 you’d get a 30. Interesting idea, and it took me a little bit to see what he was doing. We spent a little time in yesterday’s video talking about ways to see that 10/3 was not equal to 0.30303030…., but I wanted to dig in a little deeper today. In particular, I wanted to show him where his thinking was right.

We started by reviewing the decimal expansion for 1/3 and what it means to write “1/3 = 0.33333…..”

Next we checked to see if we could use our knowledge of decimal expansions to understand what the decimal expansion of 10/3 would be. Right away we saw a bunch of 30’s – that opened the door to explaining why my younger son’s initial thoughts about 10/3 were nearly right. For it to be right, you need a digit that represents the number 30. That number, which I write in the video, isn’t the same as .30303030…., though.

Next we looked more carefully at the series 10/3 = 30/10 + 30/100 + 30/1,000 + 30/10,000 . . . . We saw that the right hand side could be simplified to become: 3 + 3/10 + 3/100 + 3/1000 . . ., which is exactly 3 + 1/3.

Finally, a little bit of an unexpected ending to the project. After I turned off the camera after the first movie my older son asked what the number .3030303030… was actually equal to. Good question. We set out to answer that question here. The approach I took was similar to the “standard” proof that 0.99999… = 1, though the boys took the proof in a pretty clever direction:

So, a fun little morning project. My younger son’s thoughts about 10/3 were really interesting to me. Hopefully this bit of extra work today helped him get a better understanding of place value and arithmetic.