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Math Puzzle

by Aditya Sengupta, on August 5th, 2007

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:

\begin{align} 0.333\dots &= \frac{1}{3} \\ 3 \times 0.333\dots &= 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\ 0.999\dots &= 1 \end{align}

    

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:

August 5th, 2007 | Category: Technology/Science | Tweet | Share
  • The subtle error in both puzzle examples is that the values given are only approximately equal to each other. An infinitely recurring decimal, by definition, never reaches an absolute value. So, though x may equal .9999...., x then is only an approximation of one (1), not the absolute value. In the second example, .333.... is not genuinely equal to 1/3, only infinitely close to it. Equal and infinitely close to equal are not the same things even though the difference may not be possible to measure.

    This means that in the strict, categorical sense the "solutions" to your mathematical puzzles are not true. They've already been fudged by the first propositions.

    (By the way, thanks for the visit to my site.)
  • Interesting. Heard of Zeno's paradox?
  • OMG... Have I heard of Zeno's paradox? Not in a very, very long while! Zeno's paradoxes used to flummox me back in school. There was a time between the 6th and 8th grades (I can't remember when exactly) where I found the one about the tortoise and the hare absolutely fascinating. Not so much now after 3+ years of calculus. I did indeed think of including a simplified version above but changed my mind in the end ;-)
  • There is a way to verify the "subtle error" pointed above, and since you've had 3+ years of calculus, I need only state that 0.9999999... is not a number per se, it is a representation of an infinite series, which converges to, guess what? 1! If you have a problem with 0.9999..., you should have a problem with every infinite series. I have cleared this problem out of my mind by reasoning that the pure mathematicians know what they're doing ;).
  • I always get afterthoughts after hitting the submit button, so forgive me while I elaborate on the rigourousness of the infinite series (which you may know, but Quixote doesn't, by the looks of it).

    You take the sum of 9/10^n till a finite n. Now, if I give you a positive real number e, and you can find an integer N such that the sum of 9/10, 9/10^2, 9/10^3, ... 9/10^N is within e of 1, and if you can always find such an integer N given a number e, then your series converges to 1, that is, the sum of infinite terms is 1. Infinity has no meaning in maths, aside from being defined as a limit.

    I'll give an example. Suppose I want to find an N such that N 9's written after the decimal point would get me to within 0.00034 of 1. Simple: 4 9's, since 1 - 0.9999 = 0.0001 < 0.00034 . You can see that if I want the sum of N terms to be within any range of 1, I can always write enough 9's to do the trick. "Convince yourself" (in the colourful language of textbook authors) that infinite 9's get to 1.
  • Anonick- Thanks for the response... I appreciate it when someone takes the effort of explaining subtleties like these.
  • Interesting post and the responses are even more interesting.
  • Thanks! Well, this is what blogs are all about eventually- Conversations.
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