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Here's an interesting article that postulates that .999 recurring = 1.
For those of you on my friends list who don't have a PhD in Mathematics, and the one of you who does but in an unrelated field.. the ideal is that since 1/3 = .333 recurring, and 2/3 = .666 recurring, and .333 + .666 = .999 and 1/3 + 2/3 = 3/3 = 1, .999 and 1 are the same number. The proof is further suggested by the fact that if .999 is less than 1, how much less? an infinate number of 0's, followed by a single one? :)
It's a cute read. For many purposes, .999 does equal one, but in other terms, it's like saying π = 4
For those of you on my friends list who don't have a PhD in Mathematics, and the one of you who does but in an unrelated field.. the ideal is that since 1/3 = .333 recurring, and 2/3 = .666 recurring, and .333 + .666 = .999 and 1/3 + 2/3 = 3/3 = 1, .999 and 1 are the same number. The proof is further suggested by the fact that if .999 is less than 1, how much less? an infinate number of 0's, followed by a single one? :)
It's a cute read. For many purposes, .999 does equal one, but in other terms, it's like saying π = 4
no subject
If you don't follow the idea of the geometric sum to infinity and so on, you are better off considering 0.9 recurring as 1-. That is, the non-descript instant before whatever happens at '1' happens. In electronics, this might be the very instant before the switch is flicked, in calculus this might be the value of a limit approaching the left-hand side of a discontinuous point.
You must also be aware that all measured data can only be read to a certain number of significant digits and further calculations on this data cannot have more significant digits than the measured answer. Thus, if you calculate 0.9999, but your most accurate measurement is only to 3 significant figures, the answer is 1.00 (you have to round up).
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You might be interested to know that the real numbers are defined this way anyway (i.e. one represents real numbers as the set of "Cauchy" rational sequences - sequences of rational numbers in which the terms get arbitrarily close together - and the irrational numbers are then those such sequences which are Cauchy but do not converge to a rational number).
Haven't read the link, but I'll go to my grave swearing "but Officer, 0.9 recurring IS 1!" :-)
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My reasoning went a number smaller than 1 is (1-1/n) so a number infinitesimly smaller than 1 is lim (n->inf) (1 - 1/n), break apart by the linearity property and you end up 1 - 0 (from the limit you gave above).
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