I’m having a bit of a roller coaster ride through square roots with my younger son. Sometimes things that I think will be hard about square roots are easy for him, and sometimes things I think will be easy are hard. Today was the latter case as some initial discussions about the square root of 2 led to more confusion than clarity. So I decided to scrap the overall plans for today and just talk about about . The new goal was to see why it was not rational.
So, we started off by discussion why it wasn’t an integer:
Next we tried to see if the square root of 2 was a fraction. I intended to talk about the standard proof by contradiction here, but about 30 seconds in my son remembered our continued fraction approximation for . That surprise memory led to a quick review of the first couple of convergents in the continued fraction expansion. We saw some fractions that were nearly equal to but none of them were exactly equal.
He understood that if we could write as a fraction, the continued fraction expansion would eventually stop (it may be a stretch to say that he understands this, but he at least has the intuition that this fact would be true). Since the continued fraction expansion goes on forever, there must be no rational number that is exactly equal to .
Finally, we covered what I intended to cover in the last little talk – the usual proof by contradiction that is not rational. We end the conversation by mentioning some other numbers that are not rational – some for the same reason as and some for other reasons.
So a fun and unplanned discussion about the square root of 2. Hopefully these little side discussions end up building up his number sense a little and help him gain a better understanding of square roots.