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© Copyright 1999, Jim Loy
.999999... is equal to 1, exactly. Here is the simple proof:
1/9=.111111... (easy to prove using Geometric Series)
This fact (.999999...=1, along with similar equations like .1249999...=.125) slightly complicates a few proofs in set theory. When dealing with all possible decimal representations, you may have to show that you are accounting for the ones that are redundant. We deal with the redundancy by outlawing any decimal which ends in infinitely many nines (replacing them with their equivalent non-repeating decimal).
I received email suggesting that the above proof may be easier with 1/3=.333333... as the first step. It may be more obvious, as more people know that 1/3=.333333... But the proof is virtually identical then because, even though we all know that 1/3=.333333..., we still need to use a geometric series to prove that. In my opinion, 1/9=.111111... is slightly more esthetically pleasing, for some reason.
Originally posted by Vanguard
im sticking with 0.9 recurring tends to 1 rather than equals 1
Originally posted by djohnsto77
Well my last statement is if you don't accept that .9 repeating is exactly, completely and utterly equal to 1, you must also accept that .1 repeating isn't equal to 1/9, .3 repeating isn't equal to 1/3, .6 repeating isn't equal to 2/3 etc....
Originally posted by BlackGuardXIII
If I divide 9 into 1 I get .1 repeating........Divide 1 into 1 = 1, not .9 repeating.
Pi is an irrational number