<P> Byers also presents the following argument . Let </P> <Dl> <Dd> x = 0.999...10 x = 9.999...by "multiplying" by 10 10 x = 9 + 0.999...by "splitting" off integer part 10 x = 9 + x by definition of x 9 x = 9 by subtracting x x = 1 by dividing by 9 (\ displaystyle (\ begin (aligned) x& = 0.999 \ ldots \ \ 10x& = 9.999 \ ldots && (\ text (by "multiplying" by)) 10 \ \ 10x& = 9 + 0.999 \ ldots && (\ text (by "splitting" off integer part)) \ \ 10x& = 9 + x&& (\ text (by definition of)) x \ \ 9x& = 9&& (\ text (by subtracting)) x \ \ x& = 1&& (\ text (by dividing by)) 9 \ end (aligned))) </Dd> </Dl> <Dd> x = 0.999...10 x = 9.999...by "multiplying" by 10 10 x = 9 + 0.999...by "splitting" off integer part 10 x = 9 + x by definition of x 9 x = 9 by subtracting x x = 1 by dividing by 9 (\ displaystyle (\ begin (aligned) x& = 0.999 \ ldots \ \ 10x& = 9.999 \ ldots && (\ text (by "multiplying" by)) 10 \ \ 10x& = 9 + 0.999 \ ldots && (\ text (by "splitting" off integer part)) \ \ 10x& = 9 + x&& (\ text (by definition of)) x \ \ 9x& = 9&& (\ text (by subtracting)) x \ \ x& = 1&& (\ text (by dividing by)) 9 \ end (aligned))) </Dd> <P> Students who did not accept the first argument sometimes accept the second argument, but, in Byers' opinion, still have not resolved the ambiguity, and therefore do not understand the representation for infinite decimals . Peressini & Peressini (2007), presenting the same argument, also state that it does not explain the equality, indicating that such an explanation would likely involve concepts of infinity and completeness . Baldwin & Norton (2012), citing Katz & Katz (2010a), also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature . </P>

What repeating decimal is equivalent to the fraction 1/9