Re:Now I’m calculating with constructive reals
by leephillips on Wednesday April 27, 2022 @12:23AM
(#31176267)
I wouldn’t say nonstandard analysis is way more complicated. It replaces the use of limits with a different (hyperreal) number system. That makes some things look simpler.
Re:Now I’m calculating with constructive reals
by hither_shores on Wednesday April 27, 2022 @03:00AM
(#31177105)
Which is exactly why nonstandard analysis is way more complicated: the usual construction of the hyperreals involves some somewhat technical set theory, and even a relatively fast and loose version is going to require at least a first course in abstract algebra to be intelligible.
The difference between 0.9 recurring and 1 would have to be infinitely small, as you note. So if it's a number, it can only be zero. It can't be any nonzero number because none of them are infinitely small.
You may not like it intuitively, but this way of thinking is perfectly rigorous and logical and works very well for all practical calculations. Admitting infinitely small numbers makes math way more complicated (nonstandard analysis) and under the hood it ultimately falls back on the standard model anyway.