3 ms·
Well, a limit of a series is just a number and doesn't approach anything either. If the series approaches something, we say the limit exists and is equal to tha
by ramshorns 6y ago
Well, a limit of a series is just a number and doesn't approach anything either. If the series approaches something, we say the limit exists and is equal to that thing.
Anyway, in the surreal numbers you could probably make up a notation where 0.999... actually denotes 1 - ε or something. But I daresay it might not be very useful because then how do you denote 1 - ε/2 or anything else.
- ramshorns 6y agoOh, I figured it out, kind of. [I don't really know what I'm talking about either.] 1.000...0 = 1 1.000...05 = 1 + ε/2 1.000...1 = 1 + ε 0.999...8 = 1 - 2ε 0.999...9 = 1 - ε 0.999...98 = 1 - ε/5 . . .
- saalweachter 6y ago> 0.999...8 = 1 - 2ε > 0.999...98 = 1 - ε/5 cough
- ramshorns 6y agoSure, why not? Different notations for different surreal numbers. But it's not actually a good notation; I wouldn't know how to write 1 + 10ε.
- tom-thistime 6y agoWe've defined the series we're talking about, and we've defined 0.99... as the limit of that series. I don't know whether repeating decimals are useful for testing equality of surreal numbers. All I'm trying to say is we're all talking about anything and everything except the definition "when are two real numbers equal?" And then we're saying people who don't understand the consequences of that definition are kind of dummies ... while we continue to not actually say what the definition is.
- ramshorns 6y agoFair. Maybe the definition is that two real numbers are equal iff there is no real number between them. Which is nonintuitive if you're used to things like natural numbers which have successors. But if you teach people this way of thinking about the real numbers, then arguments like "it's the last number riiight before 1" stop working.
- CogitoCogito 6y agoIt's smart of you to bring up the surreal numbers: https://thatsmaths.com/2019/01/10/really-0-999999-is-equal-to-1-surreally-this-is-not-so/ https://thatsmaths.com/2019/01/10/really-0-999999-is-equal-t...
- tom-thistime 6y agoStrictly speaking I brought up the surreal numbers.
- CogitoCogito 6y agoThen I guess you're the smart one. My apologies.
- tom-thistime 6y agoSorry, sorry. I've been trying to bring up nonstandard analysis, and repeatedly getting poked by people saying "what do you think 'calculus' is?" "have you ever heard of a limit?" and so forth. In the process I have apparently become even more ornery than usual.
- esmi 6y agoAs we have no idea who you are don’t you think it makes sense we try to figure it out? How one answers the question will vary based on background.