Not Enough Information about ... Ellipses
Date: 12/01/2010 at 13:17:46 From: Jane Subject: Would you consider 3.666.... a rational number? Would you consider this a rational number even though you do not really know what the next digit is? 3.666... It could be 3.666789.... I would say that if the digit repeats, it should have a line over the last six. Thank you
Date: 12/01/2010 at 13:40:00 From: Doctor Peterson Subject: Re: Would you consider 3.666.... a rational number? Hi, Jane. If I saw this number on a calculator that showed only the digits 3.666 and implied that there are more digits after it, I would have no idea what the number actually is, or whether it is rational. The ellipsis "..." is used in two slightly different ways. Sometimes it just means "there's more here that I left out" (as in the calculator example). Other times it means "continue the pattern you see." The latter is a little ambiguous, since you really don't know what the pattern is; but it's generally used only in contexts where it is understood what kind of pattern to expect. One of those is to indicate a repeating decimal. So you're right that, in general, you can't tell whether 3.666... is rational or not; but in some contexts it would be somewhat of a quibble to insist that it can't be understood as a repeating decimal, even though the explicit notation is a far better way to say it. I suspect that often the ellipsis is used for a repeating decimal just because (a) the bar notation may not be understood by the reader, or (b) it is too hard to typeset the bar, so they use the next best thing. I like the notation 3.(6), which overcomes the second issue, though few people would understand it. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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