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Hi. New member here. I have been discussing this problem with some students of mine and have reached a bit of a wall. I have a way of thinking about this and I hope someone can point me in right direction.
We started with the following "proof".
let x = 0.99999999.......
10x = 9.9999999........
10x - x = 9.999999....... - 0.999999......
9x = 9
x = 1
My problem is with the following line:
9.999999....... - 0.999999...... = 9
The kids are happy to accept this and gave the following response (among others).
"Consider 9.999... - 0.999....
If this is subtracted digit by digit it become clear that the answer is 9:
First we have 9-0 so we have 9 'units'.
Then 9-9 so 0 tenths; then 9-9 so 0 hundredeths....
As it is an infinte decimal this will continue forever so the answer will be:
9.00.... = 9"
Now my response to this was to write down each term for the subtraction as follows:
9.9999.... = 9 + 0.9 + 0.09 + 0.009 + ....
0.99999... = 0.9 + 0.09 + 0.009 + 0.0009 + ....
So far so good. If we then combine these then we get
9.9999... - 0.99999... = 9 + 0.9 - 0.9 + 0.09 - 0.09 + ....
It therefore appears that each term apart from the 9 will cancel giving the answer 9 (exactly). However, I have a problem with this and it is as follows:
If we group the terms together in a different way then I think there is a different solution.
9.999.... - 0.9999.... = 9 x [(1-0.1) + (0.1 -0.01) + (0.01 - 0.001) +....]
9.999.... - 0.9999.... = 9 x [(0.9) + (0.09) + (0.009) + (0.0009)....]
= 9 x 9 (0.1 + 0.01 + 0.001 + 0.001 + ...)
= 81 x (0.1 + 0.01 + 0.001 + 0.001 + ...)
= 8.1 + 0.81 + 0.081 +0.0081 + ...
= 8.999...
Therefore the alternative line in the original proof should be
9x = 9.999... - 0.9999.... = 8.9999...
which gives the trivial answer that x = 0.9999.... which is where we started. It has not been shown that 0.9999.... = 1.
My reason for grouping the terms in this way can be understood if we again consider the series for each value.
9.9999.... = 9 x sum (n=1 to n= ∞ )[10^-(n-1)]
0.99999.... = 9 x sum (n= 1 to n= ∞ )[10^-(n)]
Apologies for the notation but I hope that this is clear.
Now if these expressions are grouped together we get
9.999... - 0.9999... = 9 x sum (from n = 1 to n = ∞ ) [(10^ -(n-1)) - (10^ -n) ]
which indeed gives the expression that does not allow 0.9999... to equal 1.
( 9.999.... - 0.9999.... = 9 x [(1-0.1) + (0.1 -0.01) + (0.01 - 0.001) +....] )
My logic is that even though there are an infinite number of different terms in each original series the statement (from n = 1 to n = ∞ ) also suggests there should be the same number of terms in each expression. For the expression to be written in a form that allows all the non-9 terms to cancel the series for 9.9999... must contain one extra term than the series for 0.9999... If this is the case then the two series cannot be combined to give the expression
9.999... - 0.9999... = 9 x sum (from n = 1 to n = ∞ ) [(10^ -(n-1)) - (10^ -n) ]
and thus,
9.999... - 0.999... = 8.999
and not 1.
So finally here is my problem. If the symbol ∞ is defined in the system to mean infinite then this surely must have a definite meaning. My thinking is that the symbol ∞ cannot mean infinity and (infinity + 1) in the same axiomatic system. I am happy to accept that infinity + 1 is still infinite but is it possible that ∞ and ∞ + 1 are actually different numbers. I am aware that Cantor defined different classes of transfinite numbers but know very little about his methods and am wondering if anyone can elaborate. If these numbers are different then all the terms other than 9 will never cancel and it is surely true that trying to prove 0.9999.... = 1 (by this proof at least) is a fallacy.
Apologies for the overly long message and thanks in advance.
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