0.9 repeater = 1

[BillyJoe]

BillyJoe, the thing is, when we say 0.999..., we mean that the 9's go on forever to infinity. It's implicit. So, it is valid to say that 0.999... is infinitely close to 1. That is to say, the difference between the two is infinitely small. I think that we can say that the difference between the two is infinitely small, than the two are in fact the same.

My understanding was that 0.999... is infinitely close to AND THEREFORE not exactly equal to 1. It seems you are saying that saying that 0.999... is infinitely close to AND THEREFORE exactly equal to to 1. (me)

Sorry, BiilyJoe (and everyone else who said the same thing), but if you gave that answer on a university exam, it would be marked wrong. (not looking good for me)

0.999... is exactly equal to 1. And the proof of this is airtight.

And apparently xooper agrees. (pay back for "Biily")

For those still not convinced that .999... is eactly equal to one, I have a question -

What is 1 - .999... ?

Here's another way to ask the same question -

If 1 > .999... then name a number between the two.

Well, if .000.....1 is really not a number, then I can not think of any number between 1 and .999...

There is no "problem" with decimal representation.

.333... is exactly equal to 1/3

Hmmmmm....

BillyJoe

I think Xouper nailed it with his proof, one that is actually provable.

As ceptimus said, it is just two different ways of expressing the same number. That's all.

1/3 = 3/9 = 0.3 repeated to infinity

3/3 = 9/9 = 1 = 0.9 repeated to infinity

It's all the same number just written differently.

So it's just true by definition???? Sheeesh!

People seem to have an easier time accepting that 0.3333... = 1/3 than that 0.9999... = 1. The explanation is the same for both, though.

The value of the infinite decimal 0.3333... is defined to be the limit of the infinite sequence 0.3, 0.33, 0.333, 0.3333, ... . The limit of that sequence is 1/3 because no matter how close you want to get to 1/3, you can get that close to 1/3 by going far enough in the sequence and you'll stay that close or closer no matter how much further you go in the sequence. The same cannot be said of any number other than 1/3. (A sequence can have at most one limit.)

An exactly analogous argument shows why 0.9999... = 1.

You can ask why a particular definition was chosen, or how a particular result follows from a particular definition; but it makes no sense to claim that something is false, if it's true by definition.

Just because "The same [that the limit of the infinite sequence 0.3, 0.33, 0.333, 0.3333, ... is 1/3] cannot be said of any number other than 1/3" doesn't mean or prove that 0.333... = 1/3 does it???

But if you are saying that it is true by definition, well then my argument slips away and I concede.
However, I think you are saying something more than this.

 

[xouper]

BillyJoe: .999... means that you keep tacking on another 9

To be pedantic (and in mathematics, this is allowed ), that's not what .999... means. The notation .999... means that all the 9s are already tacked on, simultaneously. Perhaps it helps some people to think of it as a process that can never be completed, but in doing so, you leave yourself open to being confounded by things like Zeno's Paradox. And I think that may be exactly what is happening in this thread.

There also seems to be some confusion about the phrase "infinitely close". Saying that .999... is "infinitely close" to 1 is a trick of semantics, since they are actually equal. That's like saying .5000... is infinitely close to 1/2.

In calculus, for example, the phrase "infinitely close" is a sleight of hand used to talk about a single number as if it were two different numbers. But in the end, they are the same number.

Just because "The same [that the limit of the infinite sequence 0.3, 0.33, 0.333, 0.3333, ... is 1/3] cannot be said of any number other than 1/3" doesn't mean or prove that 0.333... = 1/3 does it???

That's exactly what it means. It's a process that eliminates all other numbers except 1/3 (and I do mean ALL), and since 1/3 is the ONLY number left, it has to be that.

And apparently xooper agrees. (pay back for "Biily")

Touche.

 

[scribble]

My phrase was GETS THERE AT INFINITY = NEVER meaning that you never get there.

.999... means that you keep tacking on another 9......no, don't stop, keep tacking on another 9......yep, keep going......KEEP GOING.....

You, like many others, are thinking of numbers as some kind of a process. The number .999... is not a process, it doesn't change, you never add or remove digits from it. .999... *already is* an infinite number of 9s after the decimal place. You don't have to add any.

Does that clear things up at all? There is some fundamental misunderstanding of numbers going on here that is absolutely fascinating to me. It would be real interesting to understand what you are thinking -- but I don't, so if my explanation doesn't suffice to explain things, please let me know.

-Chris

 

[scribble]

Maybe you are too used to thinking of math in concrete terms, instead of as a descriptive language, which is what it is.

5+4=9 is a description, there is no 'action' there, the 5 and the 4 are not dynamic or changing, they do not become a 9, it's just a sentence like 'the brick is orange.'

Having five apples and getting four more means you now have nine apples - THAT is a dynamic concrete example of the language of math being used to express something that happens across time. In this case, perhaps you could think of the 5 and the 4 as becoming a 9, but in pure mathematics, that kind of thinking only leads to confusion.

Does that help any, or am I still missing what you're thinking?

 

[LibraryFox]

There also seems to be some confusion about the phrase "infinitely close". Saying that .999... is "infinitely close" to 1 is a trick of semantics, since they are actually equal. That's like saying .5000... is infinitely close to 1/2.
[/B}

Um, wouldn't .5 and 1/2 be finitely close, since a distance of zero is a finite number, and your addition of zeroes is a precision increase...
And I'd have to ask how you would define the number closest to, but less than one which does not equal one... I suspect your ansewr is that such a number cannot be defined, but I'd like to hear it...
-lf

 

[whitefork]

And I'd have to ask how you would define the number closest to, but less than one which does not equal one... I suspect your ansewr is that such a number cannot be defined, but I'd like to hear it...

If you're talking reals, there isn't any one real number N that is closest to one - there is always another number greater than N but less than 1.

"Infinitely close" should probably read "infinitessimally close" but the notion of infinitessimals got thrown out a while ago.

Suggested reading Understanding Infinity by Gardiner, which has nice explanations of the arithmetization of the calculus, etc.

 

[xouper]

LibraryFox: Um, wouldn't .5 and 1/2 be finitely close, since a distance of zero is a finite number,

Yep. The distance between .999... and 1 is also finite, for exactly the same reason.

and your addition of zeroes is a precision increase...

If I had added a finite number of zeros, yes. And this gets to the heart of the matter. If we assume that appending more nines to the number .999... increases its precision, we are already lost, since it already has an infinite number of nines and is already "infinitely" precise.

And I'd have to ask how you would define the number closest to, but less than one which does not equal one...

As Kullervo observed, there isn't one.

 

[rwguinn]

As a exercise in pure math, this has been a cute thread-and informative. I have read of what Xouper put forth, and much as I hate to agree with him, I think he's right.
As a practical matter, though- how "close to 1" do you need to be? Shooting at a 100 yard target, .9999 accurate means you're off by .036 inches. at the moon, .999999 accuracy means you miss by 1/4 mile, and so it goes.
As an engineer, I call out on drawings .999 as a dimension, costs X dollars to make. Call out .9999, and it is about 100X dollars. Call out .99999-and nobody will even attempt to make it.
So it all boils down to how accurate do you need?

RW

 

[rwald]

Not really. The main misconception is that 0.999... is an inexact number, or that it's "going somewhere," or something. It's not. 0.999... = sum(i=1,infinity,(9 * 10^(-i))), by the definition of repeating decimals. It's an exact number. It's exactly 1.

 

[rwald]

Update: If you don't believe all of our arguments saying that 0.999... = 1, than maybe you'll listen to Cecil Adam's latest Straight Dope column on the issue.

Oh, and could somebody who is already a member of the SDMB post a link to this thread in this thread over there? They might benefit from our discussion here.

 

[BillyJoe]

xouper,

To be pedantic (and in mathematics, this is allowed ), that's not what .999... means. The notation .999... means that all the 9s are already tacked on, simultaneously.

I think I do see .999... sitting there with all its 9 already tacked on. The only process I was thinking about is in trying to look at that number stretching out to infinity. You can't actually do it. It's too damn long.
So here I am casting my eyes along the long, long, looooooo...ooooooong line of 9s.
(Yeah, I know that ellipsis in the centre is not allowed )

Perhaps it helps some people to think of it as a process that can never be completed, but in doing so, you leave yourself open to being confounded by things like Zeno's Paradox. And I think that may be exactly what is happening in this thread.

Well, perhaps I have been seeing the number as a process just a little bit because I suddenly see the solution to that paradox.

There also seems to be some confusion about the phrase "infinitely close". Saying that .999... is "infinitely close" to 1 is a trick of semantics, since they are actually equal.

Okay, .999... is not "infinitely close to 1" it's "exactly equal to 1".
Good, that's definitely cleared up now......

In calculus, for example, the phrase "infinitely close" is a sleight of hand used to talk about a single number as if it were two different numbers. But in the end, they are the same number.

I did calculus in year 12 (or Form 6, Matriculation whatever it was called then) and scored an A. Yeah, hard to believe hey! And this example rings a bell.

That's exactly what it means. It's a process that eliminates all other numbers except 1/3 (and I do mean ALL), and since 1/3 is the ONLY number left, it has to be that.

Okay I've got it.

Thanks for clearing up my misunderstandings,
BillyJoe

PS: I do have a remaining suspicion that it's all a matter of definition.

 

[BillyJoe]

scribble,

You, like many others, are thinking of numbers as some kind of a process. The number .999... is not a process, it doesn't change, you never add or remove digits from it. .999... *already is* an infinite number of 9s after the decimal place. You don't have to add any.

Yes, grudgingly, I have to admit this.

Does that clear things up at all? There is some fundamental misunderstanding of numbers going on here that is absolutely fascinating to me. It would be real interesting to understand what you are thinking

On one level, of course, I know that numbers are not a process but it is easy to let this slip when thinking about numbers such as .999.... (that last . is a fullstop ) It's hard not to think of casting your eyes along this number adding 9's as you travel on and on on your way to infinity. Grazy

BillyJoe

 

[BillyJoe]

And this gets to the heart of the matter..... If we assume that appending more nines to the number .999... increases its precision, we are already lost, since it already has an infinite number of nines and is already "infinitely" precise.

Now you're rubbing it in, xouper.
Not that you don't need to.

 

[xouper]

BillyJoe: I do have a remaining suspicion that it's all a matter of definition.

In one sense, it is. In mathematics, every theorem is a logical consequence of the starting definitions and axioms (or postulates). If we started with different definitions and postulates, we would likely have a different set of theorems. In fact, this is how non-Euclidean geometries came about, by experimenting with mutually exclusive alternatives to Euclid's Fifth Postulate.

<blockquote>Jargon review - axioms (or postulates) are assumptions that can neither be proven nor disproven but are taken as self-evident. You could if you want, consider them as definitions.</blockquote>To put it another way, we could just define the notation .999... to be equal to one, but it is more satisfying to derive that fact from simpler definitions already in place (even if we have to take the long way and use a theorem from calculus).

Likewise we can also prove that the decimal notation .25 is equal to 1/4 but to most people that is already obvious (or trivial) and we don't need to see the proof. Most people probably don't need to see the proof that .333.... is equal to 1/3, but we could prove that too. It is less obvious, however, that the notation .999... is equal to one, so it's quite understandable to want to see that proof.

Now you're rubbing it in, xouper.

Oops. I didn't mean it to sound that way. I was trying to express it yet one more way, in case that helped.

I know that numbers are not a process but it is easy to let this slip when thinking about numbers such as .999....

Agreed. This may be because some numbers are easier to comprehend when they are described algorithmically.

For example, here's a number that is easier to describe as process, even though the number itself is not a process:

.101001000100001000001...

I assume you can guess the process (or algorithm) that describes this number. Each group of consecutive zero digits is one larger than the previous group.

Trick question - is the above number rational or irrational?

 

[xouper]

rwald: Oh, and could somebody who is already a member of the SDMB post a link to this thread in this thread over there? They might benefit from our discussion here.

I had not seen any of those discussions before now, but it seems the mathematicians in those threads have it pretty well nailed, and those threads go back to May 2001. One of their admins has a PhD in math. I doubt there's anything new our JREF thread could add to it. But it might be fun to let them know this is a recurring question on other forums.

I was also suprised at how many people in those threads posted mistaken notions about numbers (for example, the people who say 1/3 is not equal to .333...).

Just don't ask this question in sci.math (is .999... =1?). They are sick and tired of it there, since people are supposed to look in the faq first.

 

[whitefork]

For example, here's a number that is easier to describe as process, even though the number itself is not a process:

.101001000100001000001...

I assume you can guess the process (or algorithm) that describes this number. Each group of consecutive zero digits is one larger than the previous group.

Trick question - is the above number rational or irrational?

I believe this number is actually transcendental, hence irrational, but I can't prove it.

 

[pgwenthold]

Agreed. This may be because some numbers are easier to comprehend when they are described algorithmically.

For example, here's a number that is easier to describe as process, even though the number itself is not a process:

.101001000100001000001...

I assume you can guess the process (or algorithm) that describes this number. Each group of consecutive zero digits is one larger than the previous group.

Trick question - is the above number rational or irrational?

I would call it rational. The number is the sum of the series

10^-n, where n = 1 to inifinity. The sum of rational numbers is rational.

 

[whitefork]

Isn't the sum of 10^-n as n increases without bound, .1111111...
or 1/9?

The sum of rational numbers is rational.

Pi and e can both be represented as the sum of (an infinite series of) rationals.

 

[BillyJoe]

originally posted by xouper.101001000100001000001...
Trick question - is the above number rational or irrational?

Rational means recurring or terminating. This one doesn't terminate and doesn't really recur either. On the other hand, there is a pattern (unlike Pi which has no pattern, as far as we know, and is therefore probably irrational).

My guess is that it is rational.

 

[xouper]

pgwenthold: The number is the sum of the series 10^-n, where n = 1 to inifinity.

That number can indeed be written as a sum of an infinite series, but not the one you proposed, as Kullervo already observed.

The sum of rational numbers is rational.

Kullervo gave correct counter-examples. One of the many ways Pi can be expressed as the sum of an infinite series of rational numbers is as follows:

Pi = 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 ... )

Although this is a simple series, it converges too slowly to be of any practical use in computing Pi.

I will delay posting the answer to my trick question in case others wish to weigh in with an opinion.