Is 0,999999.... actually equal to 1?

[entropy1]

Is 0,999999... actually equal to 1, or does it approach 1?

 

[member 587159]

This is mathematics, so before we give an answer I ask you: what is your definition of $0,9999\dots$?

 

[fresh_42]

Is 0,999999... actually equal to 1, or does it approach 1?

This has been discussed a thousand times on this forum. I suggest to perform a forum search. The keyword 0.999 should do.

 

[entropy1]

This is mathematics, so before we give an answer I ask you: what is your definition of $0,9999\dots$?

I would say something like the limit of n to infinity of $\lim_{n \to \infty}\sum _{n}\frac{1}{9\cdot 10^n}$.

 

[entropy1]

This has been discussed a thousand times on this forum. I suggest to perform a forum search. The keyword 0.999 should do.

Ok. Figures. Will do.

 

[fresh_42]

$$
\lim_{n \to \infty} \sum_{k=1}^n \dfrac{9}{10^k}=9\cdot \sum_{k=1}^\infty \dfrac{1}{10^k}=9\cdot\left( \dfrac{1}{1-\dfrac{1}{10}}-1\right)=1
$$

 

[weirdoguy]

I would say something like the limit

And limits (if exist) are numbers, and numbers do not approach anything, they just are.
As an aside note: unfortunately quite a lot of students use that improper phrasing, that is "limit approaches something"... Teachers should emphasise that it is incorrect. I always do.

 

[trees and plants]

You can read other proofs of this too. When you subtract the infinite digits because those infinite digits do not end in a digit, they can be subtracted, so it is correct according to the definitions of infinite sums and their subtraction. If they ended in a digit 9 then 0.9 would remain but they do not end.

So the possible wrong doubt is about the correspondance of the digits subtracted, but the correspondance is for infinite digits of 9 not finite. When x=0.999... and 10x=9.999... the 10x contains as many infnine digits of 9 after the dot as x does. So does 100x or 100000000x or generally kx where x is a mulitple of 10.

 

[trees and plants]

kx, where k is a multiple of x contains the same infinite amount of 9s as x because that amount of 9s has the same cardinality as N, the set of natural numbers does i think. It is not a different kind of cardinality like Cantor showed with sets like R or I or Q i think. Is this correct?

 

[FactChecker]

Without getting into the mathematical technicalities of the infinite representation, here is one way of thinking about it.
If they are different, how large is abs(1 - 0.999...)? No matter what number you pick, it is easy to show that the true difference is smaller than that.

 

[fresh_42]

The question has been answered and there are already existing threads, too, so this one will be closed.