I’m usually pretty good at pointing out the flaws in mathematical puzzles that give contradictory answers… the types that give you results like 1 equals 0, you get the idea.

I just found this one that has had me stumped:

First we set:

x=0.999999999…… (infinitely recurring)

Multiplying both sides by 10, we have,

10x=9.999999999….. (infinitely recurring)

subtracting the first equation from the second one,

10x – x = 9.999999999…… – 0.999999999…….

Therefore,

9x = 9

We divide both sides by 9 to get,

x = 1

so do we have, from the first statement,

1 = .999999999….. ?

Apparently, this is true! No kidding. Yeah, I was pretty surprise as well. I expected to find a hitch in the proof, an inconsistency of some sort. Nothing. Nada. Zilch. I looked it up online even. You’d be surprised how popular this issue is on the web. Amongst mathematicians at any rate. Wikipedia has a pretty exhaustive, and somewhat exhausting article about this here. The image you see at the beginning of the post is from there. So is the alternative proof that follows:

Oh, here is another interesting piece of information I found while looking up this puzzle. Though quite a few of you probably know about this: Any recurring (non-terminating repeating) decimal can be converted to a fraction. Use the method in the first proof.

Here is a related page with some other elegant examples.

(Update: I hit the publish button before I meant to post. Here is the ending)

This puzzle illustrates the somewhat philosophical issues in our interpretation of mathematics. While we inherently believe that the number .999999999….. has a last 9 at infinity, one must realize that there is no *last *9 and that the expansion of the number never ends. Stating that there is something *at *infinity is meaningless. We often treat infinity as if it were a number, or a location (a point on a number line). This is something we need to get past.

This entire discussion curiously reminds me of a particular strip from Calvin and Hobbes. This one: