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X.9999... and X+1 
20030823 

From David: I have read your answers to the questions on rational numbers, esp. 6.9999... = ? and still have a question: The simple algebraic stunt of converting repeating decimals to rational numbers seems to work for all numbers except X.999999.... where X is any integer. The fact that the method yields the integer X+1 in each case seems to violate the completeness axiom of the real numbers, namely that there is no space on the number line which does not have an number and conversely that every geometric point on the number line is associated with a unique real number. In the case of 3.999... for example, it seems that both the number 4 and the number 3.9999.... occupy the same point on the number line. How is this possible??? Answered by Penny Nom. 


