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Btw, the naive perspective on this issue is because people intuitively perceive the hyperreal numbers. In the hyperreal number system, .9999999999999999999999... and 1 are different. But in almost all phrasings of this question it is implied to be talking about the real number system.
Specifying it gets rid of all ambiguity. The real numbers .999... and 1 are equal. The hyperreal numbers .999... and 1 are not.
Earlier quoted context omitted.
(sorry to see you downvoted - because it is actually sometimes tought to grasp but...) it doesnt approach one- your thinking of a nice graph where the line does approach but never touch, however 0.9... is NOT a function, cannot be plotted. It exists for infinite length. Any distance it is from 1 is constant.
I interpreted repeating 9's as: the function that writes some number of repeating 9's (replacing the zero in the last decimal place with a 9). This function approaches 1, does it not? f(1) = 0.9 f(2) = 0.99 f(3) = 0.999 so f as defined above approaches 1 I interpreted the question in the context of numerical precision. Since we are usually not dealing with infinite precision, the two are not equal. I am familiar with…
Earlier quoted context omitted.
I interpreted repeating 9's as: the function that writes some number of repeating 9's (replacing the zero in the last decimal place with a 9). This function approaches 1, does it not? f(1) = 0.9 f(2) = 0.99 f(3) = 0.999 so f as defined above approaches 1 I interpreted the question in the context of numerical precision. Since we are usually not dealing with infinite precision, the two are not equal. I am familiar with…
ahhh No the OP was referring to the value at the limit as x->inf of your function f.
1/3 = .333333.... 1/3 + 1/3 + 1/3 = 1 So, yes
I've always used this variation 1/9 = 0.111... 0.111... + 0.111... + ... = 0.999.... 9 * 1/9 = 1 Or we could use base 9, instead of base 10 to get rid of that infinite sequence of digits that seem to confuse people so much. 1/10 = 0.1 10 * 1/10 = 1