[lcrz]

So the authors tries to be rigorous, but again falls into the same traps that the people who claim 0.9… != 1 fall.

“0.999… = 1 - infinitesimal”

But this is simply not true. Only then they get back to a true statement:

“Inequality between two reals can be stated this way: if you subtract a from b, the result must be a nonzero real number c”.

This post doesn’t clear things up, nor is it mathematically rigorous.

Pointing towards hyperreals is another red herring, because again there 0.999… equals 1.

[hinkley]

I don’t like any of his examples at the top. Look, it’s not that hard:

    x = 0.999…

    2x = 1.999…

    2x - x = 1

    x = 1

Multiplying by ten just confused things and the result doesn’t follow for most people.

[derbaum]

Whether you multiply by 10 or 2, the same "counter" argument from the article stands. Only now you don't have a trailing zero after infinite nines, you have a trailing 8.

[ndsipa_pomu]

I don't understand how you can even have a trailing zero after an infinite number of nines. Surely any place that someone would want to put the zero can be refuted by correctly stating that a nine goes there (it's an infinite number of them, after all) and there is literally no "last" place.

[hinkley]

I’ve seen videos of actual mathematicians complaining to each other about how the general public thinks like GP. There is no last digit. Every time you reach the horizon there’s another horizon.

[anthk]

Technically you don't have an '8', you keep doing a carried sum forever, think about it. The last eight will be set to 9 forever and appended a new one to it. Thus, you are getting a periodical 1.9_ in practice.

[hinkley]

There is no eight. This is something I’ve heard actual mathematicians complain about to other actual mathematicians: the non math public misunderstands infinite series as “imagine a number so big you can’t fathom it and add 1 more number to it. That’s not how things work.

Going as far as you can imagine and a little farther is an infinitesimal of the real infinite.