1=.999999...?

[brennan]

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 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.

 

[Akka]

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.

Err... of course ?
As you said in the previous sentence, "that follows from the definition of what a limit is" ?

 

[brennan]

...and your post neatly illustrates my point.

 

[dutchfire]

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.

Real number are (can be) constructed as Cauchy sequences of rational numbers with an equivalence relation. You could also consider the same Cauchy sequences without this equivalence relation. Then the sequences x_n=1 and y_n = 1-10^-n are not the same. This set of Cauchy sequences is a perfectly fine mathematical object, but it will not have many of the properties you'd want from numbers.

 

[brennan]

Sorry. Knee jerk response there. Thanks Dutchfire.

What faults does this system have?

 

[Smote]

This is just as bad as the 0.333... = 1/3 "solution". And likewise, it says essentially: Here is why x is true. It is because x is true. Math wizardry people!
Just more symbols.

If you put it like that it would be a weird objection, in deed.
An objection that makes actually sense:
0.999... does not describe a quantity at all, for one. That is just a fact. Forget mathematical rules. If you have any idea what, in terms of quantities, the number 0.999... actually means, then you should be able to recognize the truth of that statement. I feel really silly if this has to be explained. Common sense, man.
1 does describe an exact quantity.

Now that is a rather akward divergence in traits for numbers which are merely supposed to be different styles of writing.

....

Not sure weather I will respond to the responses of my last post. I'd like to, but it looks like so much tedious work Maybe

Your attempt to constantly think of numbers in a physical sense baffles me. They are abstract concepts, that can be applied to reality, but are not necessarily representative. They are born of axioms, not empirically studying the world. The fact that the numbers 1,2,3,... (the "natural numbers") are so easily applied to the world does not make them any more special than a real number like pi^-1 or something. You'll note that there are many different number systems with different governing axioms.

The natural numbers (1,2,3,...) don't have additive inverses. The integers (...,-2,-1,0,1,2,...) do. The rationals have multiplicative inverses. The reals have "completeness." The integers mod 5 have only 5 elements, 0,1,2,3,4. The complex numbers have square roots of negative numbers. The quaternions aren't commutative, and neither are the set of 3x3 matrices.

There's tons of number systems, all abstractly defined. Some of these (the natural numbers) are easily applied to reality. Others are more difficult, but have some niche utility. Others are entirely useless. All of them, however, are defined by axioms with theorems that follow deductively from the axioms and are necessarily true. The number 1 is actually a vague statement, because you aren't stating whether it is the real number 1, the complex number 1, the integer mod 5 1, the rational number 1, the Identity matrix
[1 0 0
0 1 0
0 0 1]

or some number in another system. It is generally easy to figure out based on the context, however.

 

[dutchfire]

Sorry. Knee jerk response there. Thanks Dutchfire.

What faults does this system have?

What a subjective question.

 

[brennan]

I meant when you said "it willnot have many of the properties you would want from numbers". What would be missing?

 

[Akka]

...and your post neatly illustrates my point.

That you imagine a problem where there isn't ?
You wonder if 0,999... is necessarily the same thing as a limit. Well, what is the definition of a limit and how doesn't it apply to 0,999... ?

 

[Smote]

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 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.

You are on to something here. I think there is a way to do it all with "infinitesimals." I saw a professor prove theorems with them once. It adds a lot of a different sort of complexity though that I am very unfamiliar with. This wiki page might cover it. https://en.wikipedia.org/wiki/Non-standard_analysis

I am unsure whether .999... = 1 in this system.

 

[warpus]

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.

You would actually have to define such a system for anyone to be able to begin to refute it.

 

[brennan]

Thanks! Looks like I have some reading to do.

 

[dutchfire]

@brennan:
Well, so far, as a set of Cauchy sequences, only addition and scalar multiplication with a rational number are defined. Defining a larger than operator is nontrivial. Consider these sequences:
a = 0.9 , 0.9 , 0.99, 0.99, 0.999, 0.999 ...
b = 0.8, 0.95, 0.98, 0.995,0.998,0.9995...

Is a>b, is b<a or is a=b?

Also, without the equivalence relation, these sequences
x_n = 1 - 10^-n
y_n = 1 - 10^{-2n}
are not the same. In the context of this thread, we would want to write the first one as (x_n)=0.999... and the second one as (y_n)=0.999... , rather confusing I'd say since (x_n) != (y_n).

 

[brennan]

You would actually have to define such a system for anyone to be able to begin to refute it.

Gimme a couple days i'll see what I can come up with...

 

[uppi]

I will quibble with the definitions. A segment is open if it has no defined endpoint. You also resort to the same sort of squeezing that is tied up in the definition of limits. This is equivalent to the nonsensical "An open segment has no end point." Of course it does, you simply cannot find it. If it were true it would be a line (or a ray), not a segment.

If 0.99... was an element of (0,1) it would be the endpoint and it would be defined. In consequence, there would be no open intervals and even no need for open intervals, because we could just use [1-0.99...,0.99...] instead.

 

[Atticus]

Shouldn't probably have opened this thread, but since I did, and happened to read something akin to a personal attack, and also saw an opportunity to be constructive, I will break also my promise of shutting up.

I love the 'this is first year uni math/calculus', though. Makes one think that in your and Atticus' view a uni math graduate is at least Gauss or something to us mere mortals.

Oh, you mean like someone who thinks publishing a collection of short stories makes him a mighty author?

No, that's not the case. If it were we'd bring in all kinds of fancy words that a first year student wouldn't understand. The point is to tell you that if you went through a little trouble, not much, you'd understand the whole thing. This doesn't require you to dedicate your whole life to maths, just a few weeks of part time studying, reading a book or listening to the lectures and doing the exercises.

You simply can't learn maths from the top down. You have to know the basics, you have to do things by yourself and not just read them. That way you can learn the implicit lessons too. For example, that maths isn't the only discipline where this is true.

The point isn't that laymen shouldn't present their thoughts on maths. Their thoughts are often very good. Though, it looks to me like the best layman thoughts comes from the people who don't think they have it all figured out.

Onejayhawk, I really believe that you on the other hand want to understand maths, but it seems like you haven't learnt it the rigorous way (the definition of the open interval you quoted above is an example of that).

The halo you described above, that's the infinitesimals. People used them around 17th century, but they were highly controversial and were banned when the rigorous concept of the limit was introduced in mid 19th century, although they're still popular today in the maths taught to the appliers.

However, in the 1960s it was shown that infinitesimals can be rigorously constructed. The maths involving them is called nonstandard analysis. This is a thing I am very hesitant to tell, since IMO people should learn the standard maths properly first before getting into nonstandard analysis (=NSA). It's very easy to form all kinds of misunderstandings of it and for many people the license to use infinitesimals is a license to do bad maths. For example the infinitesimals you encounter in the maths for appliers courses are rarely the consistent infinitesimals.

Now, one thing to notice about these infinitesimals is that they are not real numbers. They are an addition to the real numbers, just like complex numbers are. They are something stuffed in between the reals. So, in real analysis there is no way any sum, limit or series can be an infinitesimal (or to differ by an infinitesimal from 1, as it would be in the case of this thread).

Another good thing to know is that they don't change anything in the real analysis: everything you can prove of the real numbers with them you can prove without, and vice versa. So, it's a matter of taste whether you use them or the standard analysis. The NSA people are very few, and to much of them it's only a curiosity. Some on the other hand remind more religious fanatics.

The infinitesimals carries a baggage though: If you accept the construction of them, you must accept also the axiom of choice, and if you accept that, you have to accept the Banach-Tarski paradox. Most mathematicians don't have anything against that axiom, but I thought you might. At least the ultrafinitists in this thread would.

Notice also that this doesn't change pretty much anything said in this thread: your intuition isn't necessarily contradictory, but the way you tried to force it in the normal maths is. Your proofs were still faulty, 0.999... still is equals to one, it still isn't in ]0,1[. A limit still can be exactly 1 although the elements of the sequence are strictly less. There still is just one limit of a real number sequence (with the usual topology). A series is still a number and not a sequence, and the sum of it is still by definition the limit of the partial sums.

Then, Brennan: It is trivially true that there can be a consistent system where 0.999... doesn't equal to 1. For example, you can define
R= {0,999..., 1} and the =-symbol like this:
0.999... != 1
1 =1
0.999.... = 0.999...

Or you can define the natural number so that 1+1=3. Here's one example:
N={1,3} with the addition
1+1=3
1+3 =1
3+3 =3
3+1 = 1.
This is even a commutative addition. You can define a multiplication there too.

It's trivial to ask if words could mean something else than they mean. It's as trivial as the above mentioned fact that God exists, if you define God to mean the Eiffel tower.

Now, when someone speaks about maths, it is assumed that he uses the words like they are in maths. It is assumed that he states clearly if he invents his own words or things like that. Just like in theological discussions it is assumed that people actually speak about the existence of a perfect omnipotent and benevolent being rather than the Eiffel tower. Although, Eiffel tower could be perfect, omnipotent and benevolent, I'm not so sure about that. It surely hasn't done anything bad to anyone. It just stood there watching while the Nazis marched to the Paris. If it's omnipotent and benevolent, why didn't it stop the Nazies? Maybe the whole tower is a hoax?

Anyhow, the point is: if you ask whether 0.999... = 1, it's your fault if you meant something else than you wrote.

 

[Terxpahseyton]

0.999... is a mathematical concept, it cannot exist in reality.

I just gave you a real-world-example of this, one that doesn't end with a specific digit. This example was a family which never dies out (which is extremely unlikely, and most likely impossible in practical terms, but not in principle) and generation after generation continues the number of 0.999..., forever. Infinity right there. And we know that also in this infinity they will not reach 1. All that infinity leads to is that 0.999... can never be known in its exact quantity in the real world, you are right on that one. Math ultimately just presumes this was not so. That is cool, you people can stop noting that Math doesn't have to acknowledge it, as a self-contained system which does not need to bow to the real world. It is just something I said from the get-go - something I may have been unclear about, my bad - and it just would be nice to either have peope acknowledge this or argue against this position (rather than falling back on math to fight this since nobody seems to be arguing that the academic math behind it was wrong as such, just that it fails to reflect the reality of it,a tiny infinitely small bit)

 

[Smote]

Or you can define the natural number so that 1+1=3. Here's one example:
N={1,3} with the addition
1+1=3
1+3 =1
3+3 =3
3+1 = 1.
This is even a commutative addition. You can define a multiplication there too.

Hey, that's Z mod 2 (3 = 0).

1+1=0
1+0 =1
0+0 =0
0+1 = 1.

 

[Atticus]

Hey, that's Z mod 2 (3 = 0).

Easiest thing to do, just had to make it more blatant for this purpose.

Also, the thing reminded me of people sometimes saying "in maths 1+1 isn't necessarily true" based on modulo arithmetics. It's annoying because it has the undertone that people in the maths (/sciences/university) are completely cut off from the real world and perhaps their findings and views contradict the common sense. (Although many people tell it as a mere anecdote too).

Of course, it's a false statement, since the 1 and 1 in it don't mean what they usually mean. Sure you don't always write the indices to mark that you're talking about the 1 of Z₂, but in those situations it's understood what you're talking about. There are homonyms elsewhere in language, and it's cheating to use them wrong. Maths isn't different in this respect.

 

[Hygro]

I see what you're saying here: It's not immediately clear that a*lim(f(x)) = lim(a*f(x)).
This is like saying it's not immediately clear that 5*.999... = 4.999...

However, if our function is sum(9*10^-n), n = 1,2,3,..., then it is clear from the distributive property that 5*(a+b) = 5*a+5*b. This means 5*(a+(b+c)) = 5*a+5*(b+c) = 5*a+(5*b+5*c). Thus the distributive property can extend to any amount of numbers being added together like this. So, 5*sum(9*10^-n) does actually equal sum(5*9*10^-n) = sum(45*10^-n).

This looks like 4.5+.45+.045+.0045+... = 4.9999999...

The question that's still not quite answered, although I keep seeing the contours of the answer in bits and pieces and dutchfire did a concise job poking nearby, is that could it not simply be an error in the notation we use that allows the distributive property to prove that 0.9...=1, even if a difference is functionally meaningless?

I almost wonder if this conversation is arguing the precise location of the two opposite face sides of a two dimensional object. By definition they are in the same place, yet counting up to it could lead to differently notated results if we're discussing where it is and where it ends.