Robert Hansen (RH) posted Dec 27, 2012 9:41 PM: > > On Dec 27, 2012, at 7:28 AM, Domenico Rosa > <DRosa@post.edu> wrote: > > > On 15 Dec 2012, Robert hansen wrote: > > > >> One of his observations was that students will > accept > >> that 0.333... is 1/3 but have trouble accepting > that > >> 0.999... is 1. > > > > I wonder if "accepting" this would be a problem if > these students had been taught how to convert a > repeating decimal into a fraction, as we were when I > was in high school? > > > > Let x=0.999... > > then 10x=9.999... > > then 9x=9 > > and x=1 > > > Wow, it's been so long since I did that. I don't > think these students will be doing something like > this today. You can't do this with a picture.:( > > Bob Hansen > A couple of moments of thought on the issue can certainly lead one to a pretty useful and meaningful picture that 'does it'. True enough, to provide a picture using Word (or even Visio) would be difficult - but it is quite easy to do on a black- or white-board (using hand, appropriate writing implement and mind).
1. Draw a square (of 'adequate' size). This is the 'whole'
2. Divide the whole by vertical lines into 10 equal rectangular parts. Cross out one of the rectangles, then the remainder = 9/10 of the original whole.
3. Take another square (same size) [for the next step]. Divide it as before.
4. Take one of the rectangles. Divide that by vertical lines into 10 equal parts. The original 9/10 + the new 9/10 of the rectangle = 9/10 + 9/100 = 99/100 of the original square.
5. And so on...
Pictures (even informal sketches) are ALWAYS most useful to help us get our thinking clarified. Even more useful to try to help young people understand 'abstractions' that may be extremely difficult to convey in words.
Most useful of all is to learn to use the mind a bit before making claims.
True enough, the picture suggested above is by no means a 'rigorous proof' (as currently accepted by the 'traditionalists').
But see the works of John Sowa on 'Knowledge Representation' and 'Conceptual Graphics' - He has written a major book on the subject (which I've unfortunately not yet read). See: - -- http://en.wikipedia.org/wiki/John_F._Sowa - -- http://www.jfsowa.com/ - -- http://www.amazon.com/John-F.-Sowa/e/B001HCZ3QG He has been trying to create a rigorous science of 'knowledge representation' via 'conceptual graphs' ('conceptual graphics' is my term). Sowa is, I understand, the originator of the concept of 'conceptual graphs'. He and his followers have been working pretty extensively on various aspects of 'knowledge representation' and 'conceptual graphs' - the entire mathematics (and science) of the subject. To the best of my understanding, they have not yet 'got there' - but they are indeed pretty well advanced along the way 'there'.
See also the attachments to my message http://mathforum.org/kb/thread.jspa?threadID=2419536 at the thread "Democracy - how to achieve it?" in this forum for a description of tools that rise from the same philosophical basis (though they are not directly derived from the works of John Sowa - my tools are derived from the contributions of the late John N. Warfield to systems science). In contrast to Sowa's approach, mine is an relatively 'informal' approach.