0.999999…

I’m really, really enjoying my Real Analysis class. My inner math geek has been riding on a cloud of euphoria for the last three weeks, and I finally have a good excuse to play with and learn

LaTeX

, something I’ve wanted to do since I was first introduced to it in my undergraduate calculus classes several years back. I’ve already learned a handful of really cool little algorithms for manually calculating certain number sets that we take for granted because we have calculators that do these things for us now. It’s been a fun ride so far, and I still have about 3.5 months left to jones over all this math-y goodness.

On the very first day of class, our professor ((Bless his heart – he’s not the greatest teacher in the world. He definitely knows his stuff, but he does have some trouble presenting it in a clear manner.)) opened a can of worms that continues to pop up in our analytical operations. He had everyone break up into groups and then wrote a group activity on the board for us to discuss:

bq. Is

0.overline{9} = 1

?

In case you’re a bit rusty on your math notation, that’s a zero and decimal point, followed by 9s _ad infinitum_. My initial gut reaction was to say that no, 0.999999… is _not_ equal to 1, that it’s infinitely close to 1 without ever actually reaching 1. But of course, when you think about that for half a second, you have to wonder how you can ever get infinitely close to something without touching it. I was incorrectly thinking in terms of asymptotes, like the equation

y = frac{1}{x}

, which is a pair of curved lines that _do_ approach the x and y axes without ever touching. But 0.999999… is a _straight line_ when graphed, and not the curved line of the aforementioned equation. And if you graph 0.999999…, you’ll see no discernable difference between it and a graphed line of 1 – because the two are the exact same number. They’re just different ways of writing the same number.

Our prof pointed us at “this math blog”:http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html, where the author has a series of five entries with proofs showing clearing that 0.99999… is equal to 1 – and lots of discussion in the comments about the validity of said proofs. It’s very interesting to read down through everything, even through the trolling comments of naysayers, because the interactions and ensuing discussions turn up a lot of really good mathematical equations and proofs. If you’re interested in this sort of thing, I highly recommend that you check it out and even add Polymathematics to your feed reader. ((There haven’t been any updates since 24 December, but I’m hoping that he’ll come back soon with some more juicy math goodness.))

So, are your eyes all glazed over now? Anyone still with me?

Have anything to add to the conversation?