A junior high school student I am tutoring asked me a question that stumped me – I was wondering if anyone could shed some light on it here.
We were talking about how the square root of 2 is an irrational number, and that means you can't write that value as the ratio of two integers. The decimal form of this value goes on forever, without repeating. And then we tried visualizing what it means when a decimal number "goes on forever."
I tried explaining it as imagining drawing a line, but you keep adding smaller and smaller pieces to its length. The pieces end up being so small, you effectively get a finite length. It was here that I could hear my own explanation breaking down. I realized I truly don't know how to visualize this concept.
The diagram I was using to describe this line was the diagonal of a square with a side length of 1. The length of this diagonal is the square root of 2, and clearly, it has a finite length (it fits inside the box, after all). Yet looking at that length in decimal form, apparently it isn't really finite. Sure those additional values that keep getting added to it as you go out from the right of the decimal point keep getting smaller and smaller, but they each have substance, adding to the overall length of the line.
What am I missing here? Or, is there a better way to explain this concept?
Best Answer
Try explaining it with a number line:
Divide the interval between 1 and 2 into ten equal parts and number them 0-9. Note that these are tenths, and so numbers of the form 1.4... fall into division four.
Now divide the interval between 1.4 and 1.5 into ten equal parts and note that these are hundredths, so numbers of the form 1.41... fall into division one.
Continuing this way demonstrates that at each step, adding more digits gives us a smaller and smaller interval that the number can be in and so increases accuracy rather than just making the number grow.