Is there a limit to the decimal expression .999?

[ram1024]

since the other threads were locked i have to post a new one (much to the anger and rage of many people I'm going to assume having read through the other ones :D please don't kill me)

it's seeming to me that the whole argument stems around a misunderstanding as to the nature of numbers and our decimal notational system.

on one side people are saying .999 contains a limit and therefore it is equal to 1.

on the other side people are saying .999 has no limit and therefore will NEVER be equal to 1.

i am of the latter group of people. If a limit is "implied" by the expression .999 then it SHOULD be written out. not "assumed" in any way shape or form.

if you told someone lim(.999) = 1 no one in their right mind would argue with you.

but .999 as a limitless process drawn to infinity has NO end, and therefore will NEVER equal to 1.

i'll let some people offer their insight to the matter thus far before I continue

 

[Tango_dark]

Wow, took you a while to post this. Anyway, 0.9... is not a limit. It always has had and always will have an infinite number of nines. There are no more nines being added. 0.9... can be expressed as a sum of a series or as a limit, but this doesn't show that it is not equal to one. 0.9... is simply the terminology used to denote an infinite number of nines after the decimal place. Since it is impossible to actually write an infinite number of nines, we write ... (or correctly a bar which I don't know how to code).

 

[ram1024]

all non-terminating numbers "do not exist" they are created as the output of a process.

just as .333 is created by dividing one by three. as such they are non-quantifiable numbers (read: irrational) because of the inclusion of infinity in their nature. Just as the digits "never end" the process to create the digits never ends as well.

the sum of that infinite series can NEVER be 1. because no matter how many 9's you calculate out to you'll never get a 10 to complete it all the way back through and make the 1 whole.

the difference of 1 and .999 if beyond human comprehension such that calculus finds it convenient to just approximate the value to be 1, but never forget that it IS and approximation, or a limit, whichever you prefer

 

[Diane_]

.9_ is not a limit

It's a constant, a number. I know we're covering old ground, but try it this way: if two numbers are not the same, then it must be possible to find a number between them. There are no numbers between 1 and .9_, so they must be the same.

Some people try something like .0_1 as a number between .9_ and 1. This is absurd, as there can be no such number. The bar (the underscore here) indicates a digit or a group of digits repeating infinitely. If you put a "1" after it, you are saying that there is a digit 1 at the end of an infinite series of zeros. But if the series of 0's is indeed infinite, then there is no end to it, thus no place to put the 1. If you terminate the zeros at some point, then it ceases to be an infinite series.

.9_ can be expressed as a limit of a sum, which I will do when I'm better at the LATEX thing (translation: when I've played with it enough not to think I'll end up looking foolish). For now, think of it as the sum for i = 1 to infinity of $(9/10)^-^i$.

It seems to me that you're thinking of .9_ as a process of some sort - as something that comes closer and closer to 1 without ever reaching it. This is an error. The value can be expressed as the limit of a sum, but it is a constant, in the same way that the sums of certain infinite series can be constants.

 

[Integral]

since the other threads were locked i have to post a new one (much to the anger and rage of many people I'm going to assume having read through the other ones :D please don't kill me)

Are there other locked threads about this? There is currently an active thread in the Logic forum but I know of no locked threads. Other then the one on Anantech.

it's seeming to me that the whole argument stems around a misunderstanding as to the nature of numbers and our decimal notational system.

Perhaps the misunderstanding is yours?

on one side people are saying .999 contains a limit and therefore it is equal to 1.

on the other side people are saying .999 has no limit and therefore will NEVER be equal to 1.

Actually because .999... is limited it is = 1. To say that it has no limit the same as saying that it is not only greater then 1 but greater then any other Real number. Many people confuse the concept of an infinite number of digits with an infinite magnitude. These are completely different concepts and should be kept separate.

i am of the latter group of people. If a limit is "implied" by the expression .999 then it SHOULD be written out. not "assumed" in any way shape or form.

if you told someone lim(.999) = 1 no one in their right mind would argue with you.

The representation of a decimal fraction is given by
$$x = \Sigma _{n=1}^{\infty} D_n 10^{-n}$$
Where the $$D_n$$ represent an integer. So any decimal fraction implies an infinite series. You are talking about something a little bit difference. Let me define a sequence of numbers.
$$x_N = 9\Sigma_{n=1}^N 10^{-n}$$
What you are saying is acceptable.
$$lim_{N \rightarrow \infty} x_N = 1$$
If you think that is true why is it not true that
$$lim_{N \rightarrow \infty} x_N = .999...$$
is true. Just expand the sum that is what you get, how can it not be correct.

but .999 as a limitless process drawn to infinity has NO end, and therefore will NEVER equal to 1.

This is exactly WHY it is equal to 1

I'll let some people offer their insight to the matter thus far before I continue

Gee, Thanks.~^

 

[ram1024]

the limit of a sum is indeed constant

the NON-limited sum to infinity IS a process. a process such that at any given point "later" in the calculation i have obtained a value closer to 1.

There are no numbers between 1 and .9_, so they must be the same.

there are no KNOWN or Human Comprehensible numbers between them. Which doesn't mean they're the same, it just means you can't understand the difference.

 

[krab]

.9_ can be expressed as a limit of a sum, which I will do when I'm better at the LATEX thing (translation: when I've played with it enough not to think I'll end up looking foolish). For now, think of it as the sum for i = 1 to infinity of $(9/10)^-^i$.

I'll help

$$.9999...=\sum_{i=1}^\infty 9(10)^{-i}$$

 

[Diane_]

Thank you, Krab!

A gentleman indeed.

 

[ram1024]

Latex is cool

so is this latex thing creating the images from another site, or is there a latex image generator on this site.

it's very spiff

 

[ram1024]

but .999 as a limitless process drawn to infinity has NO end, and therefore will NEVER equal to 1.

This is exactly WHY it is equal to 1

1. if you stop the process at any decimal digit then it doesn't equal 1
2. if you take a limit on the process as a convergence then it DOES equal 1
3. if you DON'T stop the process and let it run forever then it doesn't equal 1

you're talking about 1 I'm talking about 3 :D

 

[Integral]

click on the equations to see the code used to generate the equation. See the link at the top of the General Physics forum to get more specific instructions.

 

[ram1024]

god that is too cool :D

much thanks Integral

 

[Hurkyl]

The technical definition of a decimal number is that it is a function from the integers into a set of digits (e.g. {0, 1, 2, ..., 9}) (and a few other details).

This function is, intuitively, a list for the digits. If a function a is a decimal number, then we say a(n) (or more commonly, $a_n$) is the n-th digit. To say it another way, if n is nonnegative, then a(n) is the (n+1)-th digit to the left of the decimal place, and if n is negative, a(n) is the n-th digit to the right of the decimal place.

Boy, that was a mouthful! An example is in order:

The decimal number 12.34 is really the function f where:

f(n) = 0 whenever n > 1
f(1) = 1
f(0) = 2
f(-1) = 3
f(-2) = 4
f(n) = 0 whenever n < -2


This may seem a strange and mysterious way to define things, but if you think about it, it makes sense; all we're doing when we write a decimal number is listing digits and their positions, so it makes sense that the technical representation would be a function that says what digit is in what position.


Anyways, now that we have seen (part of) the actual definition of a natural number, we can clear up some things. When we write $0.\bar{9}$, it is the function f where:

f(n) = 0 for n >= 0
f(n) = 9 for n < 0

Whether you want to think of a function as a generator, a process, a lookup table, a set of ordered pairs, or anything else, the point is that, in all technicality, a decimal number is the function, not the digits it spits out.



My favorite way to actually carry out the construction of the decimal numbers involves doing all of the work on these functions and never saying what real number they're supposed to represent. This approach simply defines $0.\bar{9} = 1$ (and all similar situations); in other words, it's true simply because that's the way we've defined equality of decimal numbers.



It's late so I'm going to stop here. I need a sleepy smiley.

 

[Integral]

1. if you stop the process at any decimal digit then it doesn't equal 1
2. if you take a limit on the process as a convergence then it DOES equal 1
3. if you DON'T stop the process and let it run forever then it doesn't equal 1

you're talking about 1 I'm talking about 3 :D

A number is not something that moves. An infinite number of 9s represents a fixed point on the number line, It is NOT a changing thing, that is what you are implying. There must be an infinite number of 9s to reach 1, any finite number does not make it. You seem to be thinking that infinity is simple a large finite number which keeps moving. That is not the way to think of it. It is not a large finite number, it does not move.

 

[ram1024]

An infinite number of 9s represents a fixed point on the number line, It is NOT a changing thing

an infinite number of decimal sequential digits of 9 even if it WERE to simply EXIST, it is a unique number and quite simply NOT equal to 1. it is certainly NOT rational whereas 1 is

 

[Tango_dark]

Ram a repeating infinite decimal is rational. It can be expressed as a fraction, and so is rational. The definition of a rational number includes repeating infinite decimals. On this point you are simply wrong.

 

[ram1024]

infinitely repeating decimals can only be approximated as fractions because of the inclusion of infinity in their nature.

i'm not budging on this one

http://www.math.fau.edu/Richman/html/999.htm

Perhaps the situation is that some real numbers can only be approximated, like the square root of 2, whereas others, like 1, can be written exactly, but can also be approximated. So 0.999... is a series that approximates the exact number 1. Of course this dichotomy depends on what we allow for approximations. For some purposes we might allow any rational number, but for our present discussion the terminating decimals---the decimal fractions---are the natural candidates. These can only approximate 1/3, for example, so we don't have an exact expression for 1/3.

pure theory as is everything else with this imaginary number, but at least someone else "gets" what I'm alluding to.

 

[ram1024]

Consider my good friend Zeno (all hail)

he once defined a problem to where motion could not exist because in order to traverse a distance you must first travel half that distance then half the remaining distance then half of THAT remaining distance ad infinitum

this is unequivocally true. sure it's provably false that motion doesn't exist, but within the confines of the problem it can be concluded that given INFINITE number of half distances you could never reach your destination.

the LIMIT of that expression is of course the total distance, the sum is undefined or indeterminable but CONVERGES to the total distance.

so apply what we can from this example to what the situation of $$.99\bar{9}$$ and we can conclude that it is a very good approximation of 1 but not quite 1. the difference being indeterminate, yet real.

that's my thoughts on the matter at least :(

[edit]stupid spellings

 

[Integral]

On most web sites on the INTERNET, you can have long arguments about this. However this is not MOST websites. We have active mentors who are chosen because we KNOW WHAT WE ARE TALKING about. There will be no protracted argument about this fact.

You need to change modes. Get out of the "I think I know what I am talking about" mode and get into the learning mode. You may actually learn something you did not know about the the Real Number line if you read and attempt to understand what you are being told. You have been given good information by several others in this thread. I posted a link in the similar thread in Logic on this topic go read that pdf and think about it.

You need to modify your concept of infinity. Zeno did not know about convergent infinite sequences, with that knowledge there is no paradox.

 

[matt grime]

Will it never end? Ram, please look up the definitions of the real numbers. They have nothing to do with what you can write down as the result of some process like this. I dont' know why but this is the single biggest point of misapprehension amongst the, erm, keen but ill-educated, perhaps. (And I mean mathematically, I don't mean intellectually.) Constructibility, Turing machines and fear of the infinite are uniting themes in the crank world and do not affect in the slightest the truth of the statement 0.99... =1 because that truth is inherent in the meaning of all those symbols and the under lying assumption that we are working in base 10 decimal expansions of R.

 

[Integral]

Matt,
It goes beyond base 10, since .111... (binary)=1 and .222... (base 3)=1 etc. this is more another facet of the wonders of the largest digit in what ever base you choose.

Infinite representations are just a fact of life in the real number system. Think about this. .1(base 10) = .0001100110011... (base 2) now is .1 a fixed definite point on the number or is it not!

Ram,
You really should not put so much faith in a single quote, why not make an effort to learn more about the construction of the Real number line before being so argumentative.

 

[Zurtex]

Consider my good friend Zeno (all hail)

he once defined a problem to where motion could not exist because in order to traverse a distance you must first travel half that distance then half the remaining distance then half of THAT remaining distance ad infinitum

this is unequivocally true. sure it's provably false that motion doesn't exist, but within the confines of the problem it can be concluded that given INFINITE number of half distances you could never reach your destination.

the LIMIT of that expression is of course the total distance, the sum is undefined or indeterminable but CONVERGES to the total distance.

so apply what we can from this example to what the situation of $$.99\bar{9}$$ and we can conclude that it is a very good approximation of 1 but not quite 1. the difference being indeterminate, yet real.

that's my thoughts on the matter at least :(

[edit]stupid spellings

You assume the universe is continuous?

As you have said it would take an infinite number of half distances to get there and that is similar to $$.999...$$ reaching 1, but there are an infinite number of 9s in $$.\bar{9}$$

 

[HallsofIvy]

ram1024: would you please state clearly what DEFINITION of "real number" you are using? I know several definitions that could be used- they all lead immediately to the conclusion that 0.9999... is exactly equal to 1.

 

[Hurkyl]

an infinite number of decimal sequential digits of 9 even if it WERE to simply EXIST, it is a unique number and quite simply NOT equal to 1.

As strings of digits they would be unequal. Why do you think they would be unequal as numbers?

i'm not budging on this one

Then you have already defeated yourself.

pure theory as is everything else with this imaginary number, but at least someone else "gets" what I'm alluding to.

So, how much further than this sentence did you read? Did you catch, for instance,

So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning.

?

but within the confines of the problem it can be concluded that given INFINITE number of half distances you could never reach your destination.

No; it can be concluded that when considering successive half-distances that reaching the desintation will not be included in the consideration.

so apply what we can from this example to what the situation of $.9\bar{9}$ and we can conclude that it is a very good approximation of 1 but not quite 1. the difference being indeterminate, yet real.

Ok, then, what is the number halfway between $.9\bar{9}$ and 1? It exists because we can compute it via ($.9\bar{9}$ + 1)/2.

 

[matt grime]

Matt,
It goes beyond base 10, since .111... (binary)=1 and .222... (base 3)=1 etc. this is more another facet of the wonders of the largest digit in what ever base you choose.

Infinite representations are just a fact of life in the real number system. Think about this. .1(base 10) = .0001100110011... (base 2) now is .1 a fixed definite point on the number or is it not!

I will bite back my natural inclination and point out that 0.999... is not equal to one in base 11. I am perfectly well aware of all that.

 

[Diane_]

an infinite number of decimal sequential digits of 9 even if it WERE to simply EXIST, it is a unique number and quite simply NOT equal to 1. it is certainly NOT rational whereas 1 is

Ram, you keep saying things like this, but you never offer anything resembling proof. I presume that's because you consider it self-evident, but it isn't. You need to ask yourself why you believe that - I think the answer is that the two numbers don't look the same, so the back of your mind says that they aren't the same. That's not how mathematics works. There are certain things that must be taken as postulates in order for a mathematical system to get started - once you've postulated those things, though, the rest of the system must flow strictly from logic. If you choose to postulate that .9_ and 1 are, as you put it, "unique" numbers, that is certainly valid. You must be aware, though, that doing so will put you into a system of mathematics which is in disagreement with what the rest of the world has been doing for the past several centuries.

So, as specifically as you can, without reference to numbers which are not "known" (like the number between .9_ and 1) and without reference to intuition - why are .9_ and 1 two different numbers?

 

[ram1024]

but within the confines of the problem it can be concluded that given INFINITE number of half distances you could never reach your destination.

No; it can be concluded that when considering successive half-distances that reaching the desintation will not be included in the consideration.

same thing different words. it does not change the fact that the destination will never be reached :P

As you have said it would take an infinite number of half distances to get there and that is similar to reaching 1, but there are an infinite number of 9s in

no i said an infinite number of half-distances will NOT reach a destination :P

ram1024: would you please state clearly what DEFINITION of "real number" you are using? I know several definitions that could be used- they all lead immediately to the conclusion that 0.9999... is exactly equal to 1.

A number inclusive in the set that is comprised of numbers less than infinity and greater than negative-infinity

You need to modify your concept of infinity. Zeno did not know about convergent infinite sequences, with that knowledge there is no paradox.

there is no paradox because Zeno is absolutely correct. you cannot reach the destination EVER within the confines of his equation.

You really should not put so much faith in a single quote, why not make an effort to learn more about the construction of the Real number line before being so argumentative.

you should really open your mind to the possibility that you and many other very very smart people could be wrong as well. I'm not argumentative, I'm just having a rational calm discussion with other people who (seem to) have interest in the same subject

 

[J'Dona]

Whoa, where have I been?

Just to explain things to the native forum users: this is a continuation of a thread from the Warcraft 3 Off-Topic forum which was closed because 0.999 = 1 threads are banned there. Sorry to bother you with all this. :/

What happened is that there was a particularly vicious round of 0.999 = 1 related posts, and then ram1024 made a thread asking everyone to come to him with their questions, and in which he 'explained' to us that 0.999 does not equal 1. The thread, or rather, flames went on for a page or two before the thread was closed, so apparently ram1024 has decided to bring them here.

I guess he wasn't counting on anyone/everyone here actually posting to disagree with him, since he's convinced tha he's right. Actually, I'd just like to the say thanks ram1024... I've been posting on www.scienceforums.net[/url] with my biology/chemistry relates posts, but this is good too. :) (use this url if you want ot help me out by giving me referrals :) [PLAIN]http://www.scienceforums.net/forums/index.php?referrerid=764 )

Okay, I'll try to join in here by dissecting ram1024's post (making references to another post that I made on the other forum which, I hope, he read before they were deleted, otherwise I'll just have to type them out again):

same thing different words. it does not change the fact that the destination will never be reached :P

This being in reference to the two quotes above it. I already proved before that infinity does not act similarly to other real numbers, in that you cannot divide and multiply by it, (as would be suggested by your earlier claim that infinity/infinity = 1), because then you would end up with statements like 1 = 0 x infinity, which lead to, as I showed before, contradictions like 2 = 3. So while it is true that the series which halves with each term will not reach 1, because there is only a finite number of terms, can you confidently say the same of infinity?

no i said an infinite number of half-distances will NOT reach a destination :P

Same reason as above, and the same as your first point; by taking one point out of context, that being someone else's conclusion, you can't state your own unverified conclusion to 'prove' that they are wrong.

A number inclusive in the set that is comprised of numbers less than infinity and greater than negative-infinity

Here's another definition... "One of the infinitely divisible range of values between positive and negative infinity" taken from here: http://dict.die.net/real number/
That's not the same as less then positive infinity and more than negative infinity. Here's one from www.dictionary.com: "any rational or irrational number". It doesn't specify limits. The one before does, and those are infinity, but they're not really necessary because there is NOTHING beyond infinity or negative infinity.

However, not all rational and irrational number behave the same. 1/0 is undefined, for example. Wait... I forgot. You said on the other forum that zero is not a number, right? Then these definitions are incorrect, as is yours... you defined a real number as one being less than infinity and more than negative infinity, but you forgot that you don't think zero is a number and it lies between those two!

there is no paradox because Zeno is absolutely correct. you cannot reach the destination EVER within the confines of his equation.

The confines of HIS equation, which was based without the knowledge of these series. As I said before, you are assuming that when you multiply by any number, including infinity, the series never reaches its destination, which as I said before is based on the assumption that infinty acts like a real number (even though your own definition denies that it is one).

you should really open your mind to the possibility that you and many other very very smart people could be wrong as well. I'm not argumentative, I'm just having a rational calm discussion with other people who (seem to) have interest in the same subject

Yes, we are having a calm discussion, but you are not being rational... you have not desproved a single point we have made and yet still claim to be correct without proof. The quotes you have selected are countered by repeats of your own unproven assumptions, and when someone points out that your single (outdated and fallacious) source may be wrong you tell them that it is the rest fo the modern mathematical community which may be wrong. Remember, Zeno clearly didn't know what infinity was, because they probably hadn't invented it then.


Sorry to all the native forum users about this thread. ram1024 made his thread personally asking us to come to him with questions, and then he tries ot give us these 'arguments' which, even though the subject is actually very trivial, is still quite offensive. I don't usually argue this much. :(

 

[NateTG]

infinitely repeating decimals can only be approximated as fractions because of the inclusion of infinity in their nature.

According to you $$\frac{1}{1} \neq 1.\bar{0}$$.

Do you think that 0.99999... is a real number?

If so, what is 0.99999... + 0.99999... equal to?

If the answer is 2, then you've got :
(0.99999...+0.99999...)=2
so
2(0.9999...)=2(1)
0.9999...=1

If the answer is 1.9999... then you've got
0.9999...+0.9999...=1.9999...
(0.9999...+0.9999...)-0.999...=1.9999...-0.9999...
0.9999...+0.9999...-0.9999..=1
(1+1-1)(0.9999...)=1
(1)(0.99999...)=1
0.9999=1

 

[MiGUi]

1/3 = 0.33...
3*(1/3) = 3/3 = 1
3*(1/3) = 3*0.33... = 0.99...

Then... 1 = 0.99..

 

[matt grime]

If we make a reasonable assumption about zeno's paradox, that both speeds are constant, then it just states that they will not meet before some fixed time (dependent on the ratio between the speeds). given that the observation that zeno's paradox does not stop you overtaking on the motorway, perhaps you ought to look at it and figure out where it's wrong for yourself, ram.

 

[ram1024]

If we make a reasonable assumption about zeno's paradox, that both speeds are constant, then it just states that they will not meet before some fixed time (dependent on the ratio between the speeds). given that the observation that zeno's paradox does not stop you overtaking on the motorway, perhaps you ought to look at it and figure out where it's wrong for yourself, ram.

Zeno's Observation is ABSOLUTELY correct. the conclusion he came to is incorrect because of the times he was living in this was a relatively unexplored field.

Tell me where I'm wrong in saying you will NEVER reach 1 in this series outlined in this problem

Object A traveling From Point B to Point C which is a measured distance of 10 meters, and moving at a speed of 10 meters per second.

Object A must first travel 1/2 the distance (5 meters) then 1/2 the remaining distance (2.5 meters) then 1/2 that remaining distance (1.25 meters) etc ad infinitum.

the series has a LIMIT of 10 meters, but the SUM of the series will NEVER be 10 meters. there will ALWAYS be some portion "half distance" left over to travel, infinitely small and beyond human comprehension.

SOMETHING cut into 2 parts will ALWAYS yield SOMETHING. 1/Infinity ≠ 0

 

[Integral]

A number inclusive in the set that is comprised of numbers less than infinity and greater than negative-infinity

This is pretty funny. Clearly all that you know about the real number line can be written on a decimal point.
The world of mathematicans defines Infinity as an addition to the Real Number line. In doing so the field nature of the Real numbers is lost. In reality Infinity is not a number, but a definition, part of the definition is how Infinity behaves under each of the operations. It is by DEFINITION that

$$\frac 1 {\infty} = 0$$

no i said an infinite number of half-distances will NOT reach a destination

But Since that is a convergernt series, Math says that it DOES reach the end after an infinite number of steps.

 

[Integral]

http://home.comcast.net/~rossgr1/Math/one.PDF

I have a sneaking suspicion that you simply do not have enough mathematical sophistication to understand formal proofs, so I doubt that the above link will be meaningful. If you can get yourself out of the lecturing mode into the learning mode you may actually be able to get a glimmer of how mathematic ans look at this problem. In this thread you are arguing with at least one Mathematics PhD and several graduate level math students. Given acess to this level of knowledge it is a shame that you cannot make some effort to learn something.

 

[ram1024]

This is pretty funny. Clearly all that you know about the real number line can be written on a decimal point.
The world of mathematicans defines Infinity as an addition to the Real Number line. In doing so the field nature of the Real numbers is lost. In reality Infinity is not a number, but a definition, part of the definition is how Infinity behaves under each of the operations. It is by DEFINITION that

$$\frac 1 {\infty} = 0$$

well that definition is CLEARLY wrong then. I don't know why you guys are using "an infinity" that has clear limits as such, but it's ridiculous.

it's a good thing you guys live in seclusion and never come out to see the real world, because in the real world, something is never nothing, not matter how you cut it.

find me the guy that invented your infinity cause i wish to verbally abuse him.