3 ms·
The naive perspective on this issue is because people intuitively perceive the hyperreal numbers. In the hyperreal number system, .9999999999999999999999... and
by rms 17y ago
The naive perspective on this issue is because people intuitively perceive the hyperreal numbers. In the hyperreal number system, .9999999999999999999999... and 1 are different. But in almost all phrasings of this question it is implied to be talking about the real number system.
Specifying it gets rid of all ambiguity. The real numbers .999... and 1 are equal. The hyperreal numbers .999... and 1 are not.
http://en.wikipedia.org/wiki/Hyperreal_number http://en.wikipedia.org/wiki/Hyperreal_number
reply
- MaysonL 17y agoThis seems to be the only comment here written with a little mathematical sophistication. Whether 1 and .999… are equivalent depends on how you define 1 and .999…
- rms 17y agoI find these threads deeply ironic (the 400 page ones, anyways) because both sides arguing are ultimately wrong because both perspectives are correct. People with the naive perspective aren't wrong for not specifying they aren't talking about the hyper real numbers, because the other side doesn't specify they are talking about real numbers. At a philosophical level, the answer is that .999999999999999999999999999... is different from 1. The difference is real enough to let you rigorously define calculus. This is one of those subjects that makes an online community go insane (with 400 page threads), because it's possible for the math people and the philosophical people to go on and on without figuring out the right answer.
- deleted 17y ago[deleted]
- deleted 17y ago[deleted]