0.9 recurring is equal to 1, for the same reason that lim (n --> inf) 1/n is equal to 0. In my understanding the expression "0.9 recurring" describes a limit process that converges on 1.
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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branchandroot
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Date: 2006-06-20 10:18 pm (UTC)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!" :-)