[tsimionescu]

I think it's worse than this. Even with hyperreals, 0.999... = 1, I believe, since they have to obey all laws of arithmetic that are true for the reals. At the very least, 3 × 0.333... = 1, and not 0.999... even for the hyperreals.

[qayxc]

IMHO the confusion arises, because the author failed to recognise that N cannot be a natural number if they go down the nonstandard analysis path. N would have to be elevated to a hyperinteger as well, which would eliminate the infinitesimal they end up with.

[Tistron]

You're saying that 0.999...=1, and simultaneously you are saying that 3 × 0.333... = 1 and not 0.999...

What? How can it be that a=b and a≠c when b=c?

[tsimionescu]

I'm saying that, in the hyperreals as well as the reals, I am 100% certain that 3 × 0.33... = 1. I am not as sure that 0.999 = 1 with the hyperreals, BUT, if it's true as the author claims that 0.99... ≠ 1 in the hyperreals, then it must follow that 3 × 0.33... ≠ 0.99... in the hyperreals.