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4 solutions
But how can u say 0.99999999..... = 1 ?? Without this proof I wouldnt claim that 0.99999... =1 but we could say the number is tending to 1 but will never be equal to 1.
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True. I also find it hard to visualize that. But if we were aware of the formula that @Janardhanan Sivaramakrishnan wrote, anyone would be able to solve this. I agree, but if any of us understand logic of this sort, this problem is not too complicated.
What you have actually proved is lim n → ∞ 9 ∑ i = 1 ∞ 1 0 − i = 1
without a limit sign , it is false .
Saying that 0 . 9 9 9 9 9 9 9 9 . . . . = 1 is really the same as saying that 1 0 ∞ 1 = 0 , since 0 . 9 9 9 9 9 9 9 9 . . . . = 1 − 1 0 ∞ 1
So, the question is, is 0 an infinitely small number?
If you consider that 1/3=0.333333333.. and 1/3+1/3+1/3=0.33333..*3 = 0.9999 = 1 and infer therefore by the truth of the proof 0.9999..=1 , that any final number m with an ending 9999999.. decimal sequence is actually the m+1 applying this in an analogous way we should also infer that 0.0000..1 =0 or m.00.. = m-1 anyway.
The only way the curiosity appears is in the formulation that an infinitely small number is 0 (nothing) and if there is some sort of error it might be with the interpretation of the mathematical truth that 0.000000..1=0 (found in the formualtion) rather than with the mathematical fact itself therefore such an interpretation needing to be changed. 0.00000..1 means that the 0's repeat ad infinitum describing that the number becomes infinitely small and therefore that it doesn't "exist" because the repeated 0's nonetheless will never end the final digit or digits putted there to signify a quantity working rather as a way of showing that infinity works by never ending by not being "actual" say. Of course the argument of my solution uses the so called "potential infinity" and if we would anyway accept that there is an actual infinity spoke of in the problem , that is a great paradox because we should have the entire infinity of 0's and after that a 1 which ends "after" the infinity was completed so to say meaning that nothing is something , an absurdity. Therefore , unless someone comes with some proof that uses actual infinity and shows that 0.00001=0 is still sense and correct , we should accept just potential infinity. Otherwise we might just as well get crazy thinking absurdities , in fact might you happen to know what would Cantor say of this thing or how set theorists handle such a question ?
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Let me just ask you this and keep this simple... is 0 an infinitely small number? Or no? Remember what we were talking about, transfinite numbers?
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It mostly depends on the interpretation of the predication "infinitely small quantity or number because depending on what it means I can say either yes or no. Therefore if I don't know what is the way you thought of it I can't know how to answer but at least I can show the way the predication can be thought of anyway.
Because as it was said if 0 would be an infinitely small number or quantity we should accept that an infinitely small number as 0.00000..1296 which is a qauntity and therefore something would equal 0 which is a 0 quantity or nothing therefore we should accept a reasoning which concludes the very absurdity something = nothing. Therefore if the interpretation of "0 being an infinitely small quantity"means that 0.00000..1296 is a quantity completed and self-contained we would have a contradiction because it's said therefore that a magnitude greater than 0 is a 0 quantity violates the principle of identity so I would say "no". But ,if the interpretation of 0.00000..1296 means that 0 repeats indefinetely such that the 1296 at the end of the number will never be reached this is just a statement which says that the infinity of 0's excludes the possibility of any other number "after" the infinite sequence or in other words excludes the possibility of being something after as it is infinite therefore , potential infinity , making the statement true without contradictions because it just means that something will never happen as 0 continues without completion forever and in this interpretation being no problem admitting that 0.00..1296 is infinitely small quantity because that's jsut a way of speaking resulted from the linguistic situation at hand in which we can imagine such a situation which characterizes in what way we can think at the concept of infinity to which is added something , so I can say "yes". But , it seems though that by putting this question you rather find that infinity is not simply a thinking construct as such an artificial stuff as you seem to think of "concept" for which in any way we shouldn't be bothered when we don't understand it too well or maybe when the formualtion and way we arrive at inconsistencies in our direct apprehension of the concept truly imposes rethought and that such problems regarding the inconsistency of the concept in the understanding itself as it's for the case mentioned truly impose and authentic objection to be "bothered" right ?
another solution to the problem is to do 1-0.9999999999999999999999999999999999999999........ the answer equals 0.00000000000000000000000000000........1 which is equal to 0(...... means that 0 or 9 goes on for infinity) because the 1 never comes
Rounding off gives only approximation, not real value. By your logic 0.9 = 1 (By rounding off), 98 = 100 ( By rounding off).
These examples show that to find the real value of a number by rounding off is not a great thing.
Rounding off a number to itself is not great. By rounding off, we cannot say 0.9999999....... = 1.
Assume 0 . 9 9 9 9 . . . = x . 1 0 x = 9 . 9 9 9 9 . . . , 1 0 x − x = 9 x , 9 . 9 9 9 9 . . . − 0 . 9 9 9 9 . . . = 9 , 9 x = 9 , x = 1