Is 0.999... really equal to 1?

[Kaura]

Does 0.999... equals 1?
I know that this is a very basic well known concept but recently I stumbled across a video on Youtube in which the creator argues that the two are not equivalent
I posted a comment arguing that in the case of Infinite sum of Σ_n=0 9(1/10)ⁿ you can find the sum of the series using the equation a/(1-r) which gives 9/(1-(1/10)) which equals 10
Thus the sequence 9 + 0.9 + 0.09... which should equal 9.999... also equals 10 and subtracting 9 from both values gives 0.999... equals 1
The creator of the video responded to this by stating that the partial sum of the series approaches the value of 1 but never reaches it which I suppose is true for limits however the fact that it is an infinite sum should lend it to be equal to what it approaches should it not?
This seems to simply be an argument about terminology and I am by no means a mathematics expert so I was curious as the what some of you guys had to say in regard to this

 

[.Scott]

They are equal. The only time I know where it mattered is in certain proofs that the cardinality of real numbers is greater than that for rational numbers. The proof goes something like this: presume you have listed all real numbers between 0 and 1 - which you should be able to do if they are countable (as are the rationals). Then build a new number by making its first digit different from the first digit of the first number in the list, its 2nd digit different from the 2nd digit of the 2nd number in the list, and so on. The new number that you created is a real number that is different from all other number in the list. Which disproves the existence of any such list.

However, when you do the substitution, it's best to avoid the digits 0 or 9. Otherwise you could construct something like 0.12399999... even though 0.124 is already in the list.

 

[berkeman]

Does 0.999... equals 1?
I know that this is a very basic well known concept but recently I stumbled across a video on Youtube in which the creator argues that the two are not equivalent
I posted a comment arguing that in the case of Infinite sum of Σ_n=0 9(1/10)ⁿ you can find the sum of the series using the equation a/(1-r) which gives 9/(1-(1/10)) which equals 10
Thus the sequence 9 + 0.9 + 0.09... which should equal 9.999... also equals 10 and subtracting 9 from both values gives 0.999... equals 1
The creator of the video responded to this by stating that the partial sum of the series approaches the value of 1 but never reaches it which I suppose is true for limits however the fact that it is an infinite sum should lend it to be equal to what it approaches should it not?
This seems to simply be an argument about terminology and I am by no means a mathematics expert so I was curious as the what some of you guys had to say in regard to this

Please read through the "Similar Discussion" threads listed at the bottom of the page. This is a frequently asked question here at the PF.

 

[Mark44]

The creator of the video responded to this by stating that the partial sum of the series approaches the value of 1 but never reaches it

Then the creator of the video is ignorant of mathematics, and in particular limits and infinite series.

 

[Greg Bernhardt]

We have an FAQ on this
https://www.physicsforums.com/insights/why-do-people-say-that-1-and-999-are-equal/