Hygro
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A hand of fingers!
awkwardness ensues.
awkwardness ensues.
1=.999999...?
- Thread starter Mouthwash
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onejayhawk
Afflicted with reason
Lohrenswald
世界的 bottom ranked physicist
No they're not
and most rational numbers you can't make with adding 1's. Like 1/2, for example
onejayhawk
Afflicted with reason
You define multiplication as repeated addition. Then subtraction and division are inverse functions of addition and multiplication. With integers and division you can define rational numbers.
We could go back to talking politics but that's even more irrational.
J
You're right, it was only 27 pages long yesterday ?!
Jay you don't think the four axioms of arithmetics define 1 and + well enough ? 1 is just defined as the successor of 0 = {}. Addition comes quickly after that
warpus
Sommerswerd asked me to change this
Did you completely ignore the link I posted right above? These things have definitions.
onejayhawk
Afflicted with reason
Did you ignore it? Try reading some of the discussion.
These are attempts at definition. Foundations of Mathematics is a weird and arcane place, but that's where this discussion lives.
J
1 is many things, but afaik it is mostly set as a meter. Eg the smallest non-empty set of integers.
The progression of natural numbers is just steps in adding 1, obviously, yes. It means that each step coincides with the numerical value it has.
There are attempts to unify all math fields, so as to be able to view any problem which is difficult in one field as something tied to another field where it might be easier to examine. I do know that as a result of work on Fermat the so-called Taniyama (spelling?) hypothesis has been proven, linking two such fields (elliptical-?- equations and modular forms).
The progression of natural numbers is just steps in adding 1, obviously, yes. It means that each step coincides with the numerical value it has.
There are attempts to unify all math fields, so as to be able to view any problem which is difficult in one field as something tied to another field where it might be easier to examine. I do know that as a result of work on Fermat the so-called Taniyama (spelling?) hypothesis has been proven, linking two such fields (elliptical-?- equations and modular forms).
warpus
Sommerswerd asked me to change this
So you are ignoring definitions of the things you are saying aren't defined.. because.. you call them "attempts"? Even though they are fully proved and derived from first principles?
I'm confused. Are you just ignoring the definitions because you don't understand them?
onejayhawk
Afflicted with reason
That is because it is confusing. The short answer is no. No, they are not fully proved from first principle and no, they are not definitions. As it happens, definition is defined. https://en.wikipedia.org/wiki/Well-defined These attempts do not meet the criteria.
To the point of the thread, 0.999... is not well defined. There are (at least) two competing versions, only one of which is equal to one.
J
The other one is not a real definition, which means there's only one. Good, now that we have that settled we can definitely say that the only definition is equal to 1, end of story
onejayhawk
Afflicted with reason
The point is that neither is a real definition. To be well-defined, a proposed definition must be both unambiguous and dispositive, also stated necessary and sufficient. The definitions I have seen for 0.999... are neither. If there is a proper definition, I cannot find it.
J
Regardless of used/current definitions, intuitively it seems to make much more sense to claim that 0.9999.... is ever approaching a limit, namely of a progression like 1/2+1/4+1/8+... Afaik the limit is to 1 but you never get 1 while still in the progression, and intuitively writing down 0.999... is a progression, not the bit right after it ends (ie 1).
Eg the spiral of fibonacci numbers has a limit to the golden ratio spiral, approaching it from two sides (larger, smaller, larger, smaller etc). It never will become the golden ratio spiral, which is its limit and not part of the fibonacci series in any position of it.
Eg the spiral of fibonacci numbers has a limit to the golden ratio spiral, approaching it from two sides (larger, smaller, larger, smaller etc). It never will become the golden ratio spiral, which is its limit and not part of the fibonacci series in any position of it.
onejayhawk
Afflicted with reason
This is the key phrase. Or, perhaps, approaching is the key word. One approaches a limit. One does not reach it. However, that is the claim if 0.999... = 1. It is all another way of saying close enough.
J
I think so too. Although on my part i am not dealing with what definitions there may be in current use in some part of math. I do know that parts of math have definitions which are ongoing bits of discussion, eg in (some?) issues with different types of infinite groups.
Surely, though, to approach something forever is distinct from getting there, which is why i noted i wrote about ideas and notions ^^
So is the problem here that the undefined bit is the meaning of ...? Would everyone agree that?
I do think that myself and J indeed just note whether the "..." is intuitively connoting indefinate continuation of the progression, or it connotes the actual limit (not part of the progression). Eg if you wrote another progression with limit to 1 (1/2+1/4+...) you wouldn't claim that the progression itself is the limit; it just has a limit to 1. So yes, from my part anyway i am talking about connotation and not definitions where a priori you have the "..." be in tautology to "limit".
PS: tied issue: in the above again i see "1" not as built by any progression up to it, but as something itself, and occasionally tied to some progressions as their limit. Ie i do not see 1 there as having to exist only as a limit (and thus tied to) of a progression. Again this may be not in the spirit of the math definitions formed there, but i am not discussing reasons why some definitions may have been formed or whether they are compatible with some linking structures in math.
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onejayhawk
Afflicted with reason
Yikes. That does not mean anything. If you mean to ask if limit(sum(9*10^-n) for all n>0)) = 1 then yes. The question is whether 0.999... = limit(sum(9*10^-n) for all n>0)). The alternative is that 0.999... = (sum(9*10^-n) for all n>0) =/= 1.So is the problem here that the undefined bit is the meaning of ...? Would everyone agree that?
There is no unambiguous definition of repeating decimals that I can find. However, there are many discussions which omit the limit, eg wiki. https://en.wikipedia.org/wiki/Repeating_decimal
I like Kyriakos phrasing. It has a limit does not mean it is the limit.
J
warpus
Sommerswerd asked me to change this
I mean, they are fully defined from first principles, and yes they are definitions. Just because you don't understand the math involved (I don't blame you, it's not easy to wrap your head around this stuff, I would be lying if I said I could see the big picture after looking through some of the proofs involved) doesn't mean that it's not a proper mathematical definition.
I just showed you a complete definition of addition, defined from the ground up. That's exactly what you were looking for. If you ignore it just means you have an agenda and aren't really after the truth here.
