4 ms·
Well, if I could choose I wouldn't personally accept 1/3 = 0.333... . But rather, I'd say it equals a limit: 1/3 = lim(N -> oo) 0.3{N} (3 is N times re
by captainmuon 7y ago
Well, if I could choose I wouldn't personally accept 1/3 = 0.333... . But rather, I'd say it equals a limit:
1/3 = lim(N -> oo) 0.3{N} (3 is N times repeated)
Especially I would distinguish between infinitely many threes, and N threes, where N goes to infinity. In the first case, you would still be missing an infinitisimal amount, in the latter case you have the usual situation and the sequence has the least upper bound of 1/3.
When you are calculating a limit, you can never just plug in the value for N (say if N is in the denominator and the limit goes to 0). Why should you be able to do this when N is infinity?
At least this is my personal justification why I find non-standard reals interesting. They also justify the nice calculation method where you can cancel out 'dx'es from fractions.
- baddox 7y agoBut you absolutely can evaluate that limit as N goes to infinity and correctly conclude that 1/3 does equal 0.333 repeating.
- ball_of_lint 7y agoRepeating decimals may be introduced in a mathematics education long before any other infinite series or the methods used to tame them, such ass limits.
- polotics 7y agoI see the introduction of a conceptually incomplete notion as one of the big hurdle of school-level math: without limits, children are asked to just adopt the axiom that there is such a thing as a concept of infinity with any form of practical usefulness, and this riles. It's only with limits and proper formalism that I was reconciled with maths that frankly were just tending towards approaching an equality with bullshit.