> if you were doubtful that 0.999... = 1, then you should also be doubtful that 0.333.. = 1/3

I disagree. Any middle school student can calculate 1/3 to be 0.33333... using long division, but there's no immediately obvious way to go from 1 (or 1/1) to 0.9999...

> Any middle school student can calculate 1/3 to be 0.33333...

I can just do it backwards - is 1/3 equal to 0.33333...?

1 / 3 = 0.33333... 3 * 0.33333... = 0.99999... and my child brain "knows" that 1 != 0.99999...

In my child brain this proves that 1 / 3 is not equal to 0.33333..., it's just an approximation.

So I agree with larschdk, those problems are equivalent and one can't be used to prove the other ...

> Any middle school student can calculate 1/3 to be 0.33333... using long division, ...

...the same way That Chuck Norris can count to infinity... twice!

I smart middle-schooler is absolutely capable of understanding that dividing 1/3 results in an infinitely repeating sequence of 0.33333... Even without understanding the concept of infinity, they will quickly realize that there's no reason to believe the problem will stop adding a 3 to the end of the result with each iteration.

> I smart middle-schooler is absolutely capable of understanding that dividing 1/3 results in an infinitely repeating sequence of 0.33333..

And how will I smart middle-schooler know that the result of running the long division algorithm is exactly 1/3, rather than some approximation.

Recursively?

> Any argument that 0.999... is not quite 1 can also be used to argue that 0.333... is not quite 1/3.

Yes, but there aren’t good arguments for either of them, and that’s the point. The difference is that you have probably already learned how to divide 1 by 3 and have thus convinced yourself that 1/3 does indeed equal 0.333 repeating. It’s not so simple to come to the conclusion that 0.999 repeating equals 1 from simple long division that you would encounter in grade school.

See my other reply along the same lines. I was thinking about adding this caveat to my original message as well, but I think understanding that decimal numbers with infinite digits exist and that 0.999.. = 1 are separate things. The second being less intuitive.

I agree. The argument ignores the rule that infinity is a point which can never be reached; it can only be approached. So repeating 9s infinitely many times will still not reach 1, it will only approach it.