1=.999999...?

[onejayhawk]

No it's not. Are you going to continue to redefine even the most basic mathematics just because they prove you wrong and your ego can't handle this ?

Saying "0,999... != 1" is wrong but at least it can be understandable due to its counter-intuitiveness.
Saying "9 != 9" is just dumb.

You continue to insist that 0.9999... = 1, even though it is many times disproven.

∑.9+.09+.009+... =/= lim (∑.9+.09+.009+...) = 1

in fact, &#8721;.9+.09+.009+... < 1.

J

 

[Akka]

You continue to insist that 0.9999... = 1, even though it is many times disproven.

It's proven, by actual real mathematics and mathematicians for centuries, actually.

It's not because some random nobody from the Internet who doesn't even grasp basic calculus thinks himself as a Nobel Prize genius and claims the whole world is wrong and he's right that it suddendly changes reality.

Appeal to authority might be a fallacy when it's used as pure logical argument, but it should tell you something that everyone who is actually knowledgeable in mathematics say one thing, and a bunch of ignorant fools say the opposite.

 

[Lohrenswald]

No there aren't, not in real numbers. Some people just think that's an easier way to think integrals etc.

Pretty much all my professors use it like that. What does it really mean, then?

I explicitly said that they continue the 9's forever. If you do something forever, of course you never "reach" anything. That is the point.

You misunderstand. No matter how many times you do it (write another 9 at the end or add 9/10^n or whatever) you won't get 1 because you can't do it indefinetely. This is why when there are infinitely many 9's behind the comma, it equals 1

Yes. (5char)

Thanks for lying this entire thread, then.
As I've said (I think), that 1=0,999... is a consecuense of "the completeness principle", which is one of the most fundamental things math is based on
Or am I wrong about that, Atticus and Leif?

 

[brennan]

See my posts 451, 496 and 501. It isn't my fault if people don't bother to apprise themselves of what people's stated positions are before they jump in to pat each other on the back for calling others stupid.

 

[Akka]

See my posts 451, 496 and 501. It isn't my fault if people don't bother to apprise themselves of what people's stated positions are before they jump in to pat each other on the back for calling others stupid.

See Mise's post 631.

 

[Lohrenswald]

Alright then Brennan

Some people are asking "is there any a'priori reason for assuming that 1-0.999...=0?" and the answer appears to be "maths defines it that way". Which looks to me like a 'no'. Those saying 'no' seem not to realise what implications this has.

The previously mentioned completeness primciple, that any amount of numbers has an upper limit. For example, in the amount (0,1) or [0,1], there are no numbers larger than 1, so 1 is the upper limit.
I don't remember exactly how 1=0,999... follows, and I can't find my notes
Hopefully someone else with better math experience can complement this answer

The limit of that sequence as x tends to infinity is 1, that follows from the definition of what a limit is. The question is whether 0.999... is necessarily the same 'thing' as that limit. IOW is there an a'priori reason why mathematics uses axioms/definitions under which the two are said to be identical. The only answer so far appears to be ad-infinitum repetition of the fact that our system of mathematics defines it this way. Plus some insults.

I also adressed this too in my last post. The limit is the same thing as the number, yes.

I have suggested that it would be perfectly possible to use an alternative axiom system in mathematics that would be entirely self-consistent, functional (of real world use) and avoid the apparent paradox of 1=0.999... I have yet to see a refutation.

Going by that above, a system like that would have to allow that between say the numbers 0 and 10 there are numbers larger than 10

 

[brennan]

(c) he expressed his claim in a deliberately controversial way, in order to make a pretty trivial observation seem much more profound and meaningful than it really is.

Tldr: brennan is right. Plus some character assassination.

Personally I don't find it trivial to point out that the rules have been chosen and are not immutable.

 

[Hygro]

First of all, if that is his present claim, then either (a) his claim has changed (in substance) from the one he made at the beginning of the thread, (b) he expressed himself poorly at the beginning of the thread, or (c) he expressed his claim in a deliberately controversial way, in order to make a pretty trivial observation seem much more profound and meaningful than it really is. The fact that "nothing has to be so" is trivially true in an axiom-based field like maths. Secondly, literally all the "maths guys" have been saying that exact thing this entire time. Nobody said it was "some immutable law of the universe". They've all been saying (in their own language) that, sure, you can define 0.999... to be equal to whatever you want, but the way it's defined in standard, mainstream maths is that it is equal to one and the result of calculating a limit (rather than the limit itself). The reason we define it that way is that you can do waaaaay more maths this way; it's far more useful and far less problematic than defining it in another way. And for an axiomatic field like maths, this is what the concept of "truth" actually means.

P.S. 1+1=3 is a perfectly valid mathematical claim, one that Atticus (and brennan) made earlier. But, again, for various reasons, nobody would actually say that 1+1=3, unless they were making the trivial point that maths is based on a set of axioms. If this thread were about any other subject, then we would accuse brennan of equivocating on the word "0.999...". If the argument was about the War on Drugs, and brennan said "I don't see any tanks, therefore there is no war on drugs", then he'd be equivocating on the word "war", and we'd all have to tell him "no, actually, the word 'war' in this context doesn't mean a literal war in which tanks are involved, but a figurative war in which the government attempts to crack down on drug use, imports, supply chains, dealers, domestic production and so on". That's basically what's been happening for the past 32 pages. Then someone comes along and say "well actually the word 'war' could be used to mean a literal war involving tanks, therefore brennan is making a perfectly valid argument".

(Assuming of course that your interpretation of brennan's argument is correct.)

That's an interesting point. The war on drugs is a misnomer, a name designed for the purposes of propaganda to rally the people, who must be united when at war, in a basic policing infrastructure. It's a pretty good example of the misuse of a word to get the desired effect. If "1" means "war", and "0.999...." means "war" when used in a "war on drugs/poverty/Christmas" then any thread saying "the war on drugs is the same as a war against rival nations" should in fact bear the contention that, no, they are in fact not the same, and we should not fall victim to that equivication.

At the same time, there are sufficient parallels that the use of the equivication can hold discursive merit. And as such, because our decimal system is not absolutely perfect at communicating all real numbers eloquently we are stuck with awkward seams on the fabric of our mathematical communication. In that sense, one might argue that yes, of course awkward seams are a valid method to finish off a garment, but perhaps one can consider that we should not dogmatically, scornfully, ragefully attack anyone pointing out its inelegance.

 

[Lohrenswald]

That's an interesting point. The war on drugs is a misnomer, a name designed for the purposes of propaganda to rally the people, who must be united when at war, in a basic policing infrastructure. It's a pretty good example of the misuse of a word to get the desired effect. If "1" means "war", and "0.999...." means "war" when used in a "war on drugs/poverty/Christmas" then any thread saying "the war on drugs is the same as a war against rival nations" should in fact bear the contention that, no, they are in fact not the same, and we should not fall victim to that equivication.

At the same time, there are sufficient parallels that the use of the equivication can hold discursive merit. And as such, because our decimal system is not absolutely perfect at communicating all real numbers eloquently we are stuck with awkward seams on the fabric of our mathematical communication. In that sense, one might argue that yes, of course awkward seams are a valid method to finish off a garment, but perhaps one can consider that we should not dogmatically, scornfully, ragefully attack anyone pointing out its inelegance.

What?
First of all, your war-comparison is about different things with one name, while 1=0,999... is about the same thing with more names, so to say.
Then the second paragraph doesn't make sense. it's neither awkward or inelegant, really.
Also, my post 646 is important, okay

 

[Atticus]

See my posts 451, 496 and 501. It isn't my fault if people don't bother to apprise themselves of what people's stated positions are before they jump in to pat each other on the back for calling others stupid.

You entered the thread saying:

The problem here seems to be that 0.999... is being confused for a number when it is a procedure.
---
Then it pretty clear that 0.999... is not a number, if it was then 10x would be 9.990

In posts that followed you presented all kinds of fallacious and nonsensical things like:

0.9 is not equal to 1.
0.99 ditto
0.999 likewise

If 0.999... =1 then it is clearly not an equivalent entity.

If someone has mistaken that you weren't saying only that there are possible systems with 0.999... != 1, then the blame is on you.

At the same time, there are sufficient parallels that the use of the equivication can hold discursive merit. And as such, because our decimal system is not absolutely perfect at communicating all real numbers eloquently we are stuck with awkward seams on the fabric of our mathematical communication. In that sense, one might argue that yes, of course awkward seams are a valid method to finish off a garment, but perhaps one can consider that we should not dogmatically, scornfully, ragefully attack anyone pointing out its inelegance.

I hope you noticed that those who say 0.999... < 1 aren't pointing out inelegance in the decimal notation. Those who say that 0.999... = 1 however are saying that there's two different notations for the same number, and thus are pointing out an inelegance. It looks like we have been dogmatically, scornfully and ragefully been attacked for that too.

Pretty much all my professors use it like that. What does it really mean, then?
---
As I've said (I think), that 1=0,999... is a consecuense of "the completeness principle", which is one of the most fundamental things math is based on
Or am I wrong about that, Atticus and Leif?

They do use that in the maths courses for engineers, economists, physicists etc. It's not that bad in the sense that it usually gets the job done and makes calculations easier. However, people rarely think if their use of the infinitesimals is consistent.

It's not consequence of the completeness axiom in the sense that in rationals the sequence 9/10, 99/100, 999/1000... also tend to 1 (even though the axiom of completeness doesn't hold on rationals). But in similar kind of examples it would have something to do with the completeness of R, so you're not so far off.

 

[brennan]

I hope you noticed that those who say 0.999... < 1 aren't pointing out inelegance in the decimal notation.

Perhaps you should read post 301 again...?

As I already pointed out, decimal cannot elegantly represent these numbers, which is what leads to the obvious problem that 1 appears to have two identities.

 

[Atticus]

So you were claiming that 0.999... <1?

 

[TheMeInTeam]

So you were claiming that 0.999... <1?

For those attempting to assert that .999... isn't equal to 1, I can't imagine they'd claim it's *greater* . You're stuck with the assertion that it's less, or that there isn't such a thing in the first place and switching to some other conceptualization of numbers.

 

[Narz]

0.999... is caused by a fault in the real number system.

Hyperreal notation

Taking 1/3 * 3 for example:

1 divided by 3 = 0.333... this is actually inaccurate. There is no way, using our real number system, to properly notate 1/3 in decimal form. Thus, when you take 1/3 * 3, you will get an inaccurate answer.

Let's pretend that we have bad rounding skills and that 1/3 = 0.3. Based on this, 1/3 * 3 = 0.9. In actuality, our rounding is much better, but the error still exists. This discrepancy from 1 is not caused by anything except the mathematics language itself being unable to notate the proper answer.

Best answer IMO (I only read first 40 posts)

 

[warpus]

For those attempting to assert that .999... isn't equal to 1, I can't imagine they'd claim it's *greater* . You're stuck with the assertion that it's less, or that there isn't such a thing in the first place and switching to some other conceptualization of numbers.

If .999... < 1 then my question is

.999... < x < 1

find x

 

[brennan]

x=0.000...1

 

[Akka]

This HAS to be a troll.

 

[brennan]

It's an infinitesimally small number.

Calculus textbooks based on infinitesimals include the classicCalculus Made EasybySilvanus P. Thompson, and bearing the motto "What one fool can do another can".[13]Pioneering works based onAbraham Robinson's infinitesimals include texts byStroyan(dating from 1972) andHoward Jerome Keisler(Elementary Calculus: An Infinitesimal Approach). Students easily relate to the intuitive notion of an infinitesimal difference 1-"0.999...", where "0.999..." differs from its standard meaning as the real number 1, and is reinterpreted as an infinite terminating extended decimal that is strictly less than 1.[14][15]

https://en.m.wikipedia.org/wiki/Infinitesimals

My notation is bound to be wrong ofc.

 

[Borachio]

Well, there you go.

It would seem you couldn't do any calculus without infinitesimals.

 

[brennan]

I wouldn't say that, unless I completely missed the point about 10 pages ago its a non-standard method. Most people don't use them.