3 ms·
if we say that infinitesimals exist. that 1/3 != 0.33.. and 1 != 0.9999... and the probability of possible events is never 0. what are the properties that we
by tomtomtom1 6y ago
if we say that infinitesimals exist.
that 1/3 != 0.33.. and 1 != 0.9999... and the probability of possible events is never 0.
what are the properties that we would lose?
- MaxBarraclough 6y agoThe obvious one seems terrible enough, that division is no longer the inverse of multiplication: (1/3)*3 != 1
- mFixman 6y agoIf we say that infinitesimals exist, it still happens that 1 = 0.999…. It just happens that 0.999 ≠ 1 - 𝛚. 0.999… = 1 is a property of the way we write some rational numbers, not of the number system itself.
- tomtomtom1 6y agowouldn't 0.999.. be equal to 1 - 10^w since it's only a countably finite series of nines.
- mFixman 6y ago0.999...9 with a countable finite amount of nines is clearly less than 1. 0.999... with infinite nines is equal to 1.
- tomtomtom1 6y agosorry I typoed I meant, countably infinite. by w I meant the ordinal.
- ulucs 6y agoNonstandard analysis exists (with infinitesimal and infinite numbers) , but 1/3 and 9/9 is the same there. The problem is that the numbers 0.333... and 0.999... don't really exist.
- LudwigNagasena 6y agoCompleteness is one of the most important properties of real numbers. Basically, you will have to completely throw away real analysis.
- marcosdumay 6y agoI have never seen a number system with infinitesimals where the addition wasn't updated to ignore smaller classes if they come with larger ones. That is, for any number system I've seen, 1 = 1 + dx, and infinity = infinity + 100.
- 2OEH8eoCRo0 6y ago1/3 does equal 0.3.. though.