6 ms·
Why not 0.00...1?
by _bxg1 7y ago
Why not 0.00...1?
- tom-thistime 7y agoThen the question is, is 0.00....1 equal to zero? We use the definition of equality above and we say yes. EDIT: The number above seems well defined. It's lim n->inf (10^-n). That's zero.
- DavidVoid 7y agoIt is not. The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.
- tom-thistime 7y agoOk. (a) I think you're saying that 0.00 ... 1 is not a real number. (b) Do we agree that lim n->inf 10^-n is a real number? (I.e. zero?) (c) Then you're saying that limit in "(b)" is not a reasonable definition of the string "0.00 ... 1". Is that correct?
- DavidVoid 7y agoThat is correct. The limit of lim n->inf 10^-n is exactly 0, it is not 0.00...1.
- tom-thistime 7y agoSince we have no common definition of what 0.00...1 might possibly mean, let's say we agree.
- andrewprock 7y agoThis is usually described mathematically by saying that the representation of a decimal has to be countable, whereas the number 0.000...1 is not a countable representation. I will say though that if your explanation of 0.999... = 1.0 requires that you explain the distinction between countable and uncountable infinities, that's a big ask for most lay people.
- tom-thistime 7y agoCould you please explain a little more what "not a countable representation" means?
- aw1621107 7y agoI think it's just another way to word DavidVoid's explanation a few comments up.
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- andrewprock 7y agoA countable set is one where you can reach any element in a finite amount of steps. See: https://en.wikipedia.org/wiki/Countable_set https://en.wikipedia.org/wiki/Countable_set
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- tom-thistime 7y agoIf you're saying we never get from the initial 0 to the trailing 1 in a finite number of steps, that's true and that's what DavidVoid is saying. But I haven't made the list of digits uncountably large by adding one element. Instead I put that element in a transfinite position in the ordering and I have to watch out for weird consequences. I'm no expert, but countability doesn't depend on how the set is ordered. It depends on whether the elements can be placed in 1:1 correspondence with the integers. 1,0,0,... has a countable number of elements, and so does 0,0,0,...,1. They can be put in 1:1 correspondence with each other. This definition of countability is described in your link. I meant to ask, does "countable representation" some kind of detailed definition that I can look at?
- empath75 7y ago0.0000... is the repeating decimal representation of zero.
- jonsen 7y agoThis is not an infinite decimal. The digit 1 is somewhere out there.
- oconnor663 7y agoThough you could treat it as an infinite series in a similar way: 0.9 -> 0.99 -> 0.999 -> ... -> ? 0.1 -> 0.01 -> 0.001 -> ... -> ?
- jonsen 7y agoThe first is an infinite series: 9/10 + 9/100 + 9/1000 + ... + 9/10^n + ... The second is not?!
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- oconnor663 7y agoI mean you could if you want to: 1/10 + -9/100 + -9/1000 + -9/10000 + ... But I just meant them as series, not necessarily as sums.
- jonsen 7y agoOr 1 - 9/10 - 9/100 - ... or 1 - ( 9/10 + 9/100 + ... ) So it becomes circular: 0.00...1 = 1 - 0.999... (In math a series is a sum: https://en.m.wikipedia.org/wiki/Series_(mathematics) https://en.m.wikipedia.org/wiki/Series_(mathematics)
- oconnor663 7y agoWoops, you're totally right, I think I should've said "sequence".
- UncleMeat 7y agoBut this is not a compelling argument to somebody in this situation. While correct, it feels identical to saying "it just is".
- greenshackle2 7y agoWhat is 0.00...1 times 34?
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- andrewprock 7y ago0.00...34
- greenshackle2 7y agoIs 0.00...1 times 34 equal to 0.00..1 times 3.4?
- andrewprock 7y agoI think the answer to that question depends on the axioms you are using.
- DavidVoid 7y agoBecause unlike 0.99..., 0.00...1 is not a Real number. The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.