26 February 2008

Now do you believe? -Morpheus

He's the One. And by he, I mean 0.999...

When I first encountered the seeming paradox, in 9th grade, that 0.999...=1, my mind was "blown." I kept wanting to say, "Yeah, it is close to one... but it is not identical." I mean, if it were true, my whole world view would be shaken. On Judgement Day, I could justify zoning out during a looong Sunday School class on Isiah with the argument... "I really wanted to pay attention. Really really wanted to. So technically I was paying attention... 0.999...=1, as you well know Mr. God."

Of course, I really just didn't understand infinity... or Isiah for that matter. For lost souls, like me, Wikipedia comes to the rescue. I like the simplest definitions best:

\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}

An even easier version of the same proof is based on the following equations:

\begin{align} \frac{9}{9} &= 1 \\ \frac{9}{9} &= 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots \end{align}

But, behold, Wikipedia lists many others as well that range from infinite series (something that I know something about) to Cauchy sequences (of which I know nothing).

Note: The Wikipedia article also has a good section on why students often reject the equality of 0.999... and 1. My favorite:
Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".

1 comment:

Chris said...

Here's some more cool food for thought about infinity! The Cantor Set was one of my math professors' favorite topics: take a finite part of the real line, such as [0,1] (meaning zero and one are included). Remove the middle third. Now you have two line segments, and you remove the middle thirds of those too. Carry this on to infinity, and you end up removing the whole length of the segment, yet there are uncountably many points there (i.e. not only is it an infinite set, but it is uncountably infinite).