1=.999999...?

[Atticus]

We've brought up finitism and non standard analysis. It seems that 0.999...=1 arises from the axioms chosen in the definition of real numbers. If you change the axioms you can do perfectly valid maths without the apparent paradox this thread is about.

Where else could they rise from? The real numbers are defined by the axioms of the real numbers. That's what the symbols in the equation mean.

With finitism it's at least misleading to say that 0.999... != 1, since it doesn't allow the expression on the left side. You could equally well say that 1/2 + 1/2 != 1, since there are no halves in the set of the natural numbers.

I don't remember you giving any reason to think that in NSA 0.999... !=1.

Also, you weren't arguing that there are consistent systems where 0.999... != 1. You were arguing that 0.999... != 1. (The first claim would be trivial anyhow, as shown before). You just changed your claim when it began to look like you can pull it off as a "victory".

Why would it be sufficient for you that there are such systems, but not for your opponents that there are systems where 0.999... =1?

Lastly, if you change the axioms, you also change the subject. You probably know that as a YEC.

(There are consistent languages where YEC = "Argumentative Brit")

 

[brennan]

And where are those people?

Changing the axioms changes the paradigm not the subject.

 

[dutchfire]

This is the third (fourth?) time I'm asking.

I would like to field a second question.

Does "0.999...8" also equal 1? Why or why not?

The construction of real numbers [usually] is via (Cauchy) sequences. The "number" 0.99999... is given by the sequence
a_n = 1 - 10^-n

What is the sequence you propose for 0.9999....8?

Why is this not valid?
a_n = (1 - 10^-n) + (1 - 10^-n - 1)

actually this even weirder to me. Are you all also implying there can be no such thing as 0.000...1? i.e. a unit of infinite smallness?

a_n = 1 -10^-(n-1) + 0.8/10^(n-1) ?

Or something?

Now, I'm not claiming that that's a valid sequence.

In that case, and I assume you mean something like
a_n = 1 -10^-(n) + 0.8*10^-(n)
Then yes, this Cauchy sequence represents the real number 1.

 

[Borachio]

Well, I put the (n-1) in there so that a_n has n terms after the 0, but I don't suppose it matters much one way or another. And I expect you're right anyway.

 

[onejayhawk]

You mean the people who think they know better than the whole world because they are too limited to understand infinity and pat each other on the back about how they see the flaw that Nobel prizes couldn't ?
Because THAT looks much more of a circle-jerk.

The people that think they know better than the whole world are the ones receiving the Nobel Prize. Innovators are an arrogant lot.

Then there are the jerks that are just wrong. We have some of both here.

J

 

[azzaman333]

The people that think they know better than the whole world are the ones receiving the Nobel Prize. Innovators are an arrogant lot.

Then there are the jerks that are just wrong. We have some of both here.

J

The ones who get the Nobel Prizes are able to prove why their definition/method/etc is better than the status quo though. There is evidence to back up their claims. They don't just change the definition and declare the status quo to be false because they say so.

 

[Hygro]

brennan, am I correct in identifying that your position this entire thread is that "0.999... = 1" is a totally valid mathematical claim, entirely reasonable, but that the reason it is so is because we choose it for the purposes of convenience and consistency—that it doesn't have to be so. Further, all you want is the insistent defenders of the concept to agree that, yes, the claim is based on a philosophical position of choice and not some immutable law of the universe that somehow tumbled its way into decimal notation?

 

[sophie]

Did this get posted here yet? I'm almost certain it has at some point but I'm not digging through 29.999...-some odd pages going back 1.9999... years to check.

 

[onejayhawk]

The ones who get the Nobel Prizes are able to prove why their definition/method/etc is better than the status quo though. There is evidence to back up their claims. They don't just change the definition and declare the status quo to be false because they say so.

Exactly.

J

 

[Akka]

The people that think they know better than the whole world are the ones receiving the Nobel Prize. Innovators are an arrogant lot.

There is much more arrogants fools than arrogant geniuses.

Then there are the jerks that are just wrong. We have some of both here.

J

I highly doubt we have any Nobel Prize materials here. Even the most intelligent and educated people in this thread haven't made any claim to worldwide reputation.

Jerks who are wrong, though, yes we have quite a bit, which is the reason this thread is going for so long despite the answer being unambiguous, known and proved.

 

[Mise]

brennan, am I correct in identifying that your position this entire thread is that "0.999... = 1" is a totally valid mathematical claim, entirely reasonable, but that the reason it is so is because we choose it for the purposes of convenience and consistency—that it doesn't have to be so. Further, all you want is the insistent defenders of the concept to agree that, yes, the claim is based on a philosophical position of choice and not some immutable law of the universe that somehow tumbled its way into decimal notation?

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.)

 

[Terxpahseyton]

As I said, the problem is with the definitions. In math 0.99... is defined as the goal line, not the process.
[...]
If you are using other definitions, then you have to tell us, before we can start a discussion that is not pointless.

The one where 0.999... simply stands for an infinite series of quantities, each quantity represented by a digit. That is the straight-forward real world basis of numbers, IMO.

@Aka
It appears you need to hear what I already told hoplitejoe

You get too hung up on those ancestors trying to reach 1. It is just an illustration of what 0.999... supposedly does. But as you note yourself, 0.999.. can't reach an end. Which is just another way of saying that it can never be one.

I find infinity perfectly easy to understand, since it is actually easy to illustrate what infinity looks like in the world. And endless progress / continuation of whatever kind of quality is infinite. Just as in my illustration

An infinite process, having no end, is effectively equal to its limit

EFFECTIVELY - oh god yes! Entirely agreed. But only effectively. That is my point.

(if it were in any way not strictly equal, it would mean the process stops at one finite point, which is not the case as it's infinite).

Well "strictly" is the wrong word, IMO. You had it with effectively, but otherwise agreed.

Terax: I came to think of this example that could be illuminating: If you don't accept that 0.999... = 0.9 + 0.09 + 0.009 + ... equals 1, you probably won't accept similar for any other series either. For example, while 1/2 + 1/4 + 1/8 +... equals 1 in maths, you'd claim that it in reality doesn't ever quite reach the 1.

True.

Now, you wanted a real life example, and this is the Zeno's arrow paradox: To reach one you have to reach first 1/2, then 3/4 and after that 7/8 and so on. Zeno's paradox is paradox because the real life events show us that arrows do hit their targets. Since that happens, the reasoning must be wrong.

If you think it a little further, you'll see that the question about 0.999... is the same thing with just different distances: To reach 1 you have to reach first 9/10, and then 99/100, after that 999/1000.

Zeno's paradox is, I believe, about the relation of the whole and its theoretically infinite parts. What we are discussing, on the other hand, is weither an infinite number can be identical with a whole number (and not just one of its parts). It has a similar mental construct to our debate in it, but a very different context.
In Zeno's paradox, we know that the overall speed of the arrow does not slow down (not in a relevant fashion, anyway). In the case of 0999... the "speed" of getting closer to 1 slows as much down as the number comes closer, which is the entire reason why it can't be 1.
So I am struggling to the see the relevancy.

I love how the "1 = 0.999999..." subject is a controversial one on the internet. Not only does it not involve deep-seated political, moral, or religious convictions like most things that spark controversy, it actually has an objective truth value. It's just a true statement.

It is that within the internal system of academic mathematics - no more.
Jesus, Jesus, Jesus....

It can never reach one if we restrict ourselves only to your non-mathematical world.

"My" non-mathematical world? :sigh:
All I base my argument on is what makes sense in the real world, what is logically in terms of the simple idea of quantities. I am not having a mathematical world of my own. You do And you continue arguing with it, whereas I think to have exhaustingly made clear why this artificial abstract world you keep referring to is in itself not relevant for discussing weather 0.999... = 1 actually makes sense.

With maths we are able to use techniques (most which have already explained in far more detail than I am capable of) to find what 0.999... is equal to in this theoretical limit.

Yeah as said a 1000 times, that is fine. I am just not interested in those techniques since they are under no obligation to reflect what makes sense but merely to reflect what, internally, makes sense in academic mathematics. Something all sides have by now acknowledged (with periodic exceptions springing up here and there).

I'm still kinda confused where your objection is, so perhaps you can clarify for me with the question I asked brennan,
Do you accept 0.999... (an infinitely repeating decimal) to be a meaningful concept,

Yes, in mathematics as in the real world. Of course it is. I see no reason whatsoever why I wouldn't, and it is depressing that you have to ask that, since it makes it all-too apparent that you are unclear on what is actually going on in this discussion.

Now can we get back to you explaining to me how my illustration supposedly wouldn't work? Because so far I haven't received this vital information, and if it does work, case closed (or so is my faint hope)

But can't you appreciate that I can't see how it wouldn't equal 1?

On a felt level maybe. But nope, not on a logical level. Especially after I just laid that level out.

My previous post expresses my position very well and succinctly, I feel.

In that case you do not so much seem to have a position but a gut further powered by intuitive "getting it"

 

[hoplitejoe]

"My" non-mathematical world? :sigh:
All I base my argument on is what makes sense in the real world, what is logically in terms of the simple idea of quantities. I am not having a mathematical world of my own. You do And you continue arguing with it, whereas I think to have exhaustingly made clear why this artificial abstract world you keep referring to is in itself not relevant for discussing weather 0.999... = 1 actually makes sense.

I'm going to need an actual example how it can exist in the real world then.
Note that your ancestors one really isn't working as any ancestor at any point in this future obviously couldn't have reached 0.999... as they will have written a finite number of 9's, regardless of how far in the future you ask.

 

[Terxpahseyton]

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. That 0.999... stands for a quantity which can not be known, since to correctly express that quantity requires infinite complexion. Infinite many 9' to exactly express what share of "one" 0.999... stands for. This is of course not possible, hence we can not write the actual full number but write 0.999... instead, which stands for a sequence which allows you to predict how the number "behaves" as you move along the digits. Since 0.999... has infinitely many digits, you will have to do this infinitely long. And doing something for eternity is a process! Hence 0.999... is a process, not a status. The status is unknowable! All we know is that it must be infinitely close to one, while also knowing that it can't be one, since then it would have a goal line.
So that in my example 0.999... is never "reached" but only various finite numbers is exactly because my example represents 0.999... and hence is process without a goal line.

And alternative example: To say that space is infinite, in practical terms, means to be able to forever move through space without reaching its end. You wouldn't, to disregard this argument, say that on such a travel they would never reach infinity and hence this wasn't infinity, would you? Because that made no sense and if anything would show a lack of understanding of what infinity means.

 

[hoplitejoe]

What else would you say this "process"ness applies to? I assume it would be to all repeating decimals (eg. 0.333, 0.111...) but how about irrational numbers such as pi?

How should we use your version of repeating decimals in a sum? I am sure someone has already presented you with something like this

how would you go about solving this?

There is still a lot i really don't understand about your position so hopefully this can clear it up a little.

 

[onejayhawk]

What else would you say this "process"ness applies to? I assume it would be to all repeating decimals (eg. 0.333, 0.111...) but how about irrational numbers such as pi?

How should we use your version of repeating decimals in a sum? I am sure someone has already presented you with something like this

how would you go about solving this?

There is still a lot i really don't understand about your position so hopefully this can clear it up a little.

Step 4 is wrong. The difference is not exactly 9. They do two at once to cover the slight of hand.

J

 

[TheMeInTeam]

All I base my argument on is what makes sense in the real world, what is logically in terms of the simple idea of quantities. I am not having a mathematical world of my own. You do And you continue arguing with it, whereas I think to have exhaustingly made clear why this artificial abstract world you keep referring to is in itself not relevant for discussing weather 0.999... = 1 actually makes sense.

By definition, you can't have an infinite string of numbers, or anything else for that matter, in the real world.

 

[onejayhawk]

By definition, you can't have an infinite string of numbers, or anything else for that matter, in the real world.

This is not entirely true. You cannot have imaginary numbers, but they are necessary to describe many real physical properties.

J

 

[brennan]

brennan, am I correct in identifying that your position this entire thread is that "0.999... = 1" is a totally valid mathematical claim, entirely reasonable, but that the reason it is so is because we choose it for the purposes of convenience and consistency—that it doesn't have to be so. Further, all you want is the insistent defenders of the concept to agree that, yes, the claim is based on a philosophical position of choice and not some immutable law of the universe that somehow tumbled its way into decimal notation?

Yes. (5char)

 

[Akka]

I find infinity perfectly easy to understand, since it is actually easy to illustrate what infinity looks like in the world.

The part before the comma is in contradiction with the part after.
No, infinity is not easy to understand, and that you think it's easy to illustrate in the real world strongly indicates you only think you understand it.

EFFECTIVELY - oh god yes! Entirely agreed. But only effectively. That is my point.

Well "strictly" is the wrong word, IMO. You had it with effectively, but otherwise agreed.

No, "strict" is the right word, and you're just putting too much meaning behind "effectively". You are adamant about the "it's a process, not a number", which is meaningless. ANY number can be expressed with "a process", as it was repeatedly shown in the thread.
After all, I can write "5" as a process : "5 x 1", or "(5/2) x 2", or "(5/3) x 3", and if I continue this toward infinity, I can end up writing something which would mean : "(5/infinity) x infinity" (of course it's not a mathematically rigorous example, as "infinity" is not a number, but it's to illustrate the point). At which point "5" is written as an infinite process, yet it's still strictly equal to 5.


The example given by Atticus, that you dismiss to casually, is actually a perfect way to show your reasoning is flawed. Regardless of how close the arrow is from the target, you can ALWAYS divide the distance into ten smallers increment, so you can argue that the arrow can never reach the target since there is ALWAYS 1/10 of the distance left after it goes through the first 9/10.
Yet the arrow strikes the target. It's a good real-world illustration.

Step 4 is wrong. The difference is not exactly 9. They do two at once to cover the slight of hand.

J

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.