A small puzzler

Here is a question a high school math teacher posed to my class one day: does the number 0.999…, where the 9s go on forever, equal 1? As I recall, we spent the entire class period arguing about this. Perhaps surprisingly, the answer is yes: it is equal to 1. Here are a few ways of thinking about this problem that will help you see why that is so.

Divide by 3

What do you get when you divide 1 by 3? You get \(1/3\), which is the same as 0.333…, with the 3s going on forever. What do you get when you divide 0.999… by 3? You get 0.333…, with the 3s going on forever. As you may recall from high school algebra, if \(x/3 = y/3\), then \(x = y\); in this case, 1 = 0.999….

Divide and subtract

What do you get when you divide 0.999… by 10? You get 0.0999…. Now notice that 0.999… - 0.0999… = 0.9. What do you get when you divide 1 by 10? You get 0.1. Now notice that 1 - 0.1 = 0.9. With a little high school algebra again, you can figure out that if \(x - x/10 = y - y/10\), then \(x = y\); in this case, again, 1 = 0.999….