math proof: 1=.999…?

There is a proof on Brainfood website that shows an interesting and intriguing proof for the following: 1 = .999…. This idea and its proof have been troubling me, causing me some minor cognitive dissonance. The proof itself is simple enough. It’s my inquiry into the nature of infinity and number representation that has been troubling. The proof for the statement is:

a)      Let x = .999…

b)      10x = 9.999…

c)      10x – x = 9.999… – x

d)     9x = 9

e)      x = 1

f)       1 = .999…

The trouble/cognitive dissonance for me comes in when considering the counterintuitive idea that 1 = .999… as the repeating number will simply never reach 1. I think of it in graphing terms, where a curve approaches but never reaches a point or one of the axis).  Even more troublesome and confusing (yet strangely engaging) is reading some of the debates and various proofs against the above statement which revolve around the idea of infinity (is it a finite number?  is it a mathematical concept or process only?). There’s even a rather long Wiki page dedicated to the proof.

What I found intriguing about this little Brainfood puzzler is the extension, through my personal curiosity, to follow the idea to deeper thinking. Admittedly, the introduction of analytical proof, infinite series and sequences, nested intervals and least upper bounds, Dedkind cuts, Cauchey sequences, etc.  left me scratching my head. I suppose I could make sense of it, if I slowed down enough and took them step-by-step. A puzzler of this kind, however, could be useful in a classroom to spark student curiosity, engage an appreciation for the logic of mathematical proofs, and possibly be used as an extension activity for those students who are able to handle the more complex concepts and proofs.

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One thought on “math proof: 1=.999…?

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