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If 0.9... is an infinite number then why do mathematicians insist it is equal to 1 which is finite.
The classic proof that it is 1 is 3 x 1/3 = 1 and since 1/3 = 0.3... then 3 x 0.3... = 1. However, you can not complete the calculation of 1/3. It is infinite. So you can not say it is 1/3 completely. In fact 1/3 is just a symbolic way of representing an unfinished division. It is easy to make the mistake that if 3 x 1/3 = 1 then 3 x 0.3... = 1.
Another flawed proof is
Let X = 0.999...
Then 10X = 9.999...
Subtract X from each side to give us: 9X = 9.999... - X
but we know that X is 0.999..., so: 9X = 9.999... - 0.999...
or:
9X = 9
Divide both sides by 9 gives X = 1
The flaw in this proof is that to complete the calculations they MUST be finite values so 9 x X really in this case is 9.99 (not 9.999...).
So the complete calculation of 9X - X is 9.99 - 0.999 which is 8.991 which is clearly NOT 9.
Also, 1 is always < 0.n
Also, 1 is a different number to 0.9...
The other classic proof that 0.9...=1 is using limits.
Mathematicians declare that if something is approaching a limit then 'at ∞' it IS the value of the limit. This step jumps over the fact that the number approaching the limit is infinite and makes it a finite number (the limit). Its one small step for mathkind, but one giant leap for valid logic! If you want an example of why this approach doesn't work in every case then try n/ ∞. By the rule that at ∞ it is the limit we would expect therefore that n/ ∞ = 0 since 0 is the limit.
Please discuss.
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