.999 = 1?

Factual Questions
261

Limits don’t have to explain it as a satisfying concrete or metaphysical answer. We don’t have to get “comfortable” with it in that sense at all. The notion of limits is simply an idea that is consistent with itself; from that, we can then hang subsequent consistent ideas from it.

Instead of asking “limits” to “explain” or “understand” infinity. We only ask that limits be a usable tool for “working with” infinity. This is much more reasonable goal!

As a comparison, consider something as basic as a line mathematics.

Most of us just “accept” this mathematical idea of a “line” without giving it a lot of thought. But what exactly is a line? Can we construct it?

Is a piece of string drawn pull tight like a guitar string a “line”? No, that doesn’t meet the definition of a line. How about a thin laser beam? Nope. Even a line that is the width of 1 electron is not a true “line” in the the mathematical sense. A mathematical line has zero width.

If we say a line is is a set of infinite points with zero width, how can we even create statements such as “2 parallel lines never meet” … the non-mathematical mind can retort, “well sure, 2 of impossible and incomprehensible items of zero width never meet, duh!” Imagine a inquisitive child trying to drill down this line of reasoning to its very end. You must explain that it’s not possible to “think” of lines like that because it leads to writing English sentences that are meaningless.

If we can’t even construct lines, how do we comprehend it on some satisfying metaphysical sense? We don’t. A line is simply an idea we work with. (Same as limits.)

Another example… on a piece of paper, we can write the number 1 x 10[sup]81[/sup] which is a number larger than the number of atoms in the universe. This number is incomprehensible.

Even with that difficulty, how can we claim: (1 x 10[sup]81[/sup] + 1) > (1 x 10[sup]81[/sup])

It’s only the rules of addition that says that’s statement is true. We can’t arrange a pile of 1 x 10[sup]81[/sup] apples (or atoms) and visually see that one heap is obviously 1 bigger than the other.

If we can accept mathematical definitions of “lines” and rules of addition on incomprehensibly large numbers, we can also accept the concept of “limits” and that .999… = 1.

262

OK, let’s try one more thing.

For there to be a difference, you need to be able to subtract your decimal from 1 - correct?

Now, in order for you to do that, don’t you have to be able to find the tail end of the decimal?

But at infinity, there is no tail end because . . . wait for it . . . you’re AT infinity.

Are you feelin’ any love yet?

263

Between Giles and Allotrope they make a compelling case as to why intuitvely I should expect .999… to eqaul 1. But Limit theory just is less than satisfying. It says nothing about how or why… just that it does. It leaves me thinking perhaps .999… does = 1. Yet it seems such a fundimental issue, that the fact we have no proof troubles me. To me limits hide our ignorance in this matter.

As Allotrope says “It’s gone, buried in infinity” – okay maybe it is, but thats to me as vague as 1/infinity.

264

We can prove by though the mehtod of induction that since 1 + 1 = 2 that 1 + 2 =3 and so on. It is not just a given, only 1 + 1 = 2 is the given there. I am pretty sure set theory takes care of 1 + 1 = 2 to start things out.

So thats not a fair compairison to the assumptions of Limits.

265

How can you say it’s Zero then? That is if you can’t find the tail end to do the subtraction. You want to place the burden of proof on me to say that 1 - .999… does not equal zero. I could do the same in return. Saying you can’t find the tail end to do the subtraction therefor it must be just forgotten about doesn’t settle well with me.

266

No, what I’m saying is that since there is no tail, there is no difference to find.

I was using the subtraction example to make it feel more concrete.

But the point really is that to find that tiny difference you have to keep hopping from one 9 to the next ad infinitum. Therefore, if you can never actually FIND the difference, no difference exists.

267

Okay you guys have worn me down for now. I have to try and get an hour or so of sleep before work. :frowning:

268

So how do you “accept” the mathematical concept of a 1-dimensional line with zero width?

269

I dont think the numbers care if we can find them. They do their thing regardless :wink:

I think 1 + 1 = 2 before we ever convcieved of a number.

270

At this point I am not even sure what a line is. I mean every 2 points on a line has to have a point between them right? So pick any 2 points and there are so many levels of infinity between those points, I get dizzy. But none of them can touch each other right? Becuase if they did then that number - the number touching it would = 0, so they would be the same point… no? Not real?

You tell me what a line is.

271

Oh come now. Obviously I didn’t mean it in any kind anthropomorphized sense. Jeez.

Pls re-read what I said if that’s really what you believe.

272

Sorry, don’t have time to read all of the other comments, but if this is how you think non-terminating rational numbers are constructed, then the way you (not me, not anyone else) do long division is the following:

Let’s divide 1 by 9:

**0.111111111…1
**_______________
9/ 1.0000000000…0
*-0
**10
**-9
***10
***-9
****1
*******etc.
*************10
*************-9
**************1
STOP!! I (erik150) will arbitrarily stop here and discard the remainder so that I can have the result end with a 1.
This is also why my (erik150’s) intuition is skeptical that 0.333… is equal to 1/3, since the removal of the remainder necessarily makes 1/3 > 0.333… (or rather, 1/3 > 0.333…3)!

erik150, you really need to understand that there is NO END to the number of decimal places to the right of the decimal point. It’s like saying “infinity” is just some humongous finite number that you simply stopped bothering to count. I mean, what’s stopping me from just going one more step in the above division? Why the stopping point?

273

Ok first I haven’t read any posts since my last. I have been trying to sleep. But I had to post this while it’s in my mind.

Let’s assume Limit Theory is correct. .999 = 1
And for that matter 1/2 + 1/4 + 1/8 + … = 1

Now picture 2 horizontial parrallel line segements, A and B.
A direclty above B.
for fun lets say these lines are a mile long (it doesn’t really matter)

Each line segment is labeled A1 and B1 on the left most point
Each line segment is labeled A2 and B2 on the right most point

We can divide a line segment into an infinite number of points right?
So we do this for both lines. Making sure each point on the top line, lines up with the bottom line.

On the top line I will lablel each point as follows 1 . 0 0 0 . . .
On the bottom line I will label each point as follows 0. 9 9 9 . . .

How can I do this? Right how can I label 2 infinite lines?
Well to make things simple let’s assume I am marking line A with my left hand and line B with my right. So I only have to make one pass. And I will only decribe marking line A, but it assumed I am marking line B at each point as well.

I make my first mark at the half mile point, and put (1.). then at the 3/4 mile mark 0, then at the 7/8 mile mark 0, then at the 15/16 mile mark 0, etc…

Each mark takes me 1/2 the time to get to since I am traveling half the distance each time. So lets say it takes me 30 mins to get to the first mark then 15 mins to get to the second mark, 7 mins 30 secs for the 3rd and so on.

I think most can see where I am going with this, but I will go on…

According to limit theory I can travel to the end in 1 hour marking an infinite number of points.
1/2 HOUR + 1/4 HOUR + 1/8 HOUR + … = 1 HOUR

And that consists of an infinite number of points.

Now I have my lines both labeled. To review as such:
On the top line I will lablel each point as follows 1 . 0 0 0 . . .
On the bottom line I will label each point as follows 0. 9 9 9 . .

Can I really physycally do this? Of course not, but this is just a thought experiment so bear with me. I ceartainly can walk a mile in one hour and one could certainly imagine two lines with these points exisintg as they do on lines. Furthermore whether I actually labled them or not, does it really matter? The point is I can get to them and pass them all in 1 hour easily.

Now we reverse the process and start subtracting line B from A. To be honest do we really need to finish the whole calculation? Because once we see the 0 at the end of line A and the 9 at the end of line B, we know what the result will be:
.000…1

So it seems we can in a theoretical way find the end of these numbers and do the math.

But the math contradicts what limit theory tells us the answer should be which would be:
.000…0

I guess this is all a fanciful way of saying it in the land of math we can get to the end of the line as sure as we can walk across the room. Calculating an infinite number of decimal places is a snap.

I could be deleriously tired, but it makes sense to me.

274

Yeah, I don’t follow what point your tryign to make. You posted something about adding .999… to itself 10 times to simulate multiplication by 10. Which you can’t really do without limits. I think you need to go back to the post of yours I was repsonding to. And also the fact that in either case they are on shakey ground. But you want to just ignore all the decimal places beyond some unamed limit?

Maybe we are just not connecting with each other here. I don’t know why there is need to bring the construction of reals or divison into this? We were simply tlaking about adding already existing numbers. Namely .999… and 10 of them.

275

No, because by definition, that’s a term of art by the way, by definition, if you could label all of the points, then there were not an infinite number.

You know dude, if you just don’t WANT to understand, that’s fine, but at least be honest about it, m’kay?

276

As I said… do I really need to label them?

As we discussed before to get from point A to point B, you must go half way 1st right? And 1/2 of the remaining distance and so on… this is the classic Zeno’s paradox which is solved by saying
1/2 + 1/4 + 1/8 + … = 1
is it not? Is there not an infinite number of steps involved there? Do we clearly not move across the room form point A ot point B? Is there theoretically not an infinit number of points there? Do I really have to label them? All i need to do is reach the last one, labeling them is not really neccessary.

We have reached the end of an infinite number of points stecteched along this mile line line segment. We already know how they should be labeled, do we not? Do you susggest at some point in this process I would (if I was actually marking them), stop marking zero’s on top or stop marking nine’s on the bottom?

The point is we can find the end of an infinite number of points, and that’s all we really need to do here.

I’m not trying to not understand? Please tell me where my logic is wrong here?

277

Usually I would think it meant not more than countably many, which does not mean not an infinite number, so I don’t know what you are on about.

278

It’s not that the equation of .999… with 1 is “built in” to the system we use as an “assumption.” Rather, it’s a logical result of the definitions used at the foundation of that system. You can prove it–using the assumptions we all use when we use the decimal number system.

And it is in fact true, in the sense that in the system we are using when we use the decimal number system, the quantity represented by the expression “.999…” is the very same quantity as that represented by the expression “1”.

Maybe you already know both of those things, but you’ve said things lately which seem to indicate you think otherwise so I want to be sure this is clear.

In a system wherein .999… =/= 1, the expression “.999…” does not represent the same quantity that the expression “.999…” represents in the normal decimal number system. It represents some other quantity. (In fact, it may not represent a “quantity” at all in the colloquial sense. It depends on the model.)

I think.

279

Your logic is flawed because you still fail to understand what infinity means.

Let’s put a little ‘9’ at each halfway mark. There will still be an infinite number of them. As I said before, YOU NEVER GET TO POINT B EXCEPT AT INFINITY.

It’s not my fault if you don’t read my posts now is it?

280

I don’t know a lot of maths, but your explanation is concise, and I don’t find much to argue about here. When .999… is used, it’s used in place of 1. It indicates to me that a faulty algorithm that results in an endless loop, producing 9’s forever, but the value it’s trying to produce is 1.

As far as .999…1 goes, it’s pretty easy to understand that there are an infinite number of numbers between 1 and 2, yet 2 still follows 1. I don’t know what to do with .999…1, but it doesn’t seem difficult to understand.