.9999... = 1

[Mentat]

I'm sure most of you already know this. The real point of the thread is finding different ways of approaching it. You see, I have a friend who refuses to accept what seems to me to be so obvious: .9 repeating (infinite 9s after the decimal) is exactly equal to the whole number 1.

Here are the proofs that I've used so far:

1) 1/3 is precisely equal to .333... (to which he agreed). 3*.333... equals .999... and 3*1/3 equals 1. Therefore, since I multiplied equal numbers by the same number, they should still be equal...ergo 1 = .999...

2) If 1 is greater than .999..., then by how much, exactly, is it greater? After all, 1 is greater than .9 by .1. It is greater than .99 by .01. It is greater than .99999999999999999999 by .00000000000000000001. And so on, and so on. However, if the number of nines is infinite, then the number of zeros preceding the 1 will be infinite, and 0.000... is obviously equal to 0, which means that 1 is greater than .999... by exactly 0.

3) After this point, my friend revealed that he doesn't really believe in actual infinites. He thinks you will eventually reach the 1, at the end of the infinite series of 0s. Ridiculous as that may sound, I respected that as his opinion and posed this next point: If there are no infinites then the term equal to 1/3, which is usually considered .333..., would actually (eventually) terminate in a 4. I asked him if that was right. After some thought, he said "yes". So, I asked, if we multiplied this number by 2, would you logically expect to get the same number 6s as you had 3s, and then terminate in an 8 (since 4+4 is 8)? He said, "yes", not realizing that he had just implied that the infinitely repeating number .666..., which is equal to 2/3 must now terminate in an 8, instead of a 7 (as his reasoning would naturaly have assumed).


Are there any other points I can use? I wasn't going to ask for help on this, until he talked to his math teacher, and the teacher agreed with him!! Now he's got this quaint little ipse dixe argument, "well, the math teacher said I'm right".

Any new points or comments on my previous ones are appreciated.

[Mentat]

btw, I don't know if this was the right place to have posted this thread, but I couldn't think of a better place for it.

[confutatis]

The real point of the thread is finding different ways of approaching it. You see, I have a friend who refuses to accept what seems to me to be so obvious: .9 repeating (infinite 9s after the decimal) is exactly equal to the whole number 1.

You have provided your friends some arguments as to why .9999.... must be the same as 1, otherwise math would not be perfect. Your friend is saying math is not perfect. Countering his argument with an unsubstantiated claim that math is perfect, end of discussion, doesn't seem to do much for him.

But you can still get him to think about the issue from a different perspective. If he doesn't accept that numbers can be infinitely small, then he must agree that there exists a minimum quantity which cannot be divided. For instance, if he thinks the smallest number possible is N, then you can ask him, how much is N/3? Is it N or is it 0? There is no known answer to that question, but if the problem is real then an answer must necessarily exist. And that means the answer can be determined with probabilities. The probability of N/3 being N is 1/3, and the probability of it being 0 is 2/3.

So the correct answer to his problem is that the result of 3 * 0.3333...3 must also be determined by probabilities, and 1, not 0.9999...9, is by far the most likely answer.

[NateTG]

There is a fundemental question that is generally skipped in the discussions of 0.\bar{9} = 1, which is the assumption that 0.\bar{9} is a real (or rational) number.

A second shortcoming is that 'bar' notation is generally not well explored. If your friend is comfortable with bar notation, the following might be interesting for him:

1
=\frac{1}{9} + \frac{8}{9}
=0.\bar{1}+0.\bar{8}
=0.\bar{9}

Although this is, of course, identical to the 3 \times \frac{1}{3} example that you gave.

The traditional elementary school method is:
9 \times 0.\bar{9} = 10 \times 0.\bar{9} - (1)0.\bar{9} = 9.\bar{9} - 0.\bar{9} = 9
so
9 \times 0.\bar{9} = 9
so
0.\bar{9}=1

You can also talk about infinite sums, but getting satisfactory answers with sums would (eventually) involve invoking the completeness of the real numbers - which is not necessarily an accessible subject.
0.\bar{9} = \sum_{i=1}^\infty \frac{9}{10^i}

You can ask your friend what:
\frac{1}{1 - 0.\bar{9}}
is equal to. Since 0.\bar{9} \neq 1 the denominator is non-zero right?

[Integral]

Give this link a look (http://home.comcast.net/~rossgr1/Math/one.PDF)

[Dissident Dan]

x = 0.\bar{9}
9x = 10x - x
9x = 9.\bar{9} - 0.\bar{9}
9x = 9
x = 1

Of course, I tend to think that the idea of a value of anything repeating is illogial--after all, it's just a defect of the number system (decimal, aka base-10). The proof that I just listed relies on the fact that:
infinity - 1 = infinity

0.\bar{9} cannot be viewed as anything other than a defect of the number system. It is not a value, as it involves infinity.

[Integral]

0.\bar{9} cannot be viewed as anything other than a defect of the number system. It is not a value, as it involves infinity.

I don't see it that way. Just the opposite if we were UNABLE to assign a value to any infinitely repeating number that would be a defect in the number system.
This is NOT a facet of DECIMAL numbers, it is a facet of the density of the Real number line. No matter how you choose to represent the points of the real number line there will always be numbers which require an infinite number of digits to be specified.

The Base does not matter. for example

.1_{10} = .00011001100... _2

So in binary .1 has an infinitely repeating representation which MUST be truncated in all computations. This fact was first made apparent to me years ago when I did the following loop in Applesoft Basic
10 x= 0
20 For i = 1 to 10
30 x= x+ .1
40 Print x
50 next i

The final number would not be 1 but .999999, in only 10 steps the round off error would become apparent.

(Round off error has nothing to do with the fact that 1=.999....)

[Tango_dark]

If we allow .0...1 (sorry not sure how to code for the bar notation) to have a non-zero value, then it must be the smallest positive number. If we allow for numbers to have a minimum value then the number line ceases to be continuous as any 2 numbers no longer have an infinite range of values between them. Now we have x, x+.0...1, x+.0...2, etc as our number line. There can be no number between these values because then there would be a smaller number then .0...1 Since this causes a contradiction, we can determine that .0...1 cannot have a non-zero positive value.

[Hurkyl]

If we allow .0...1 (sorry not sure how to code for the bar notation) to have a non-zero value, then it must be the smallest positive number.

If we allow this, then why is .0...01 not a smaller number? After all, if we're going to allow ourselves "decimals" with infinity + 1 digits, then why not decimals with infinity + 2 digits?

Before we ask "what is .0...1?" we should ask "does the notation .0...1 even make sense?"


Are there any other points I can use?

Try asking him how many digits 1/3 has.

[Tango_dark]

Since there are already an infinite number of zeroes before the one, adding one more will not make a difference (property of infinity). Thats part of the problem. Giving the 1 at the end of infinity a value means a) allowing the infinity to end and b) since the value of the one is equal to 1/10^(infinity+1) we are allowing 1/infinity to have a non-zero defined positive value. Neither of these is consistent with the way which math works. The number .0...1 cannot possibly exist in any form which would allow it to be added to .99... to equal 1 (and hence prove that the two are not equal).

[hypnagogue]

Mentat, you have your friend stuck on a pair of horns. If 1/3 = 0.333...4, then 3 * 1/3 = 1.000...2, contradicting 3 * 1/3 = 1. So if your friend continues to hold that 1/3 = 0.333...4, then he must believe that

a) 1 = 1.000...2, which is just as bad (in his eyes) as the original position he was seeking to refute, or
b) 1/3 * 3 = 1 is false, or
c) that mathematics is not internally consistent, which essentially means that statements like 1 = 2 can be shown to be true.

If this forces him to reject that 1/3 = 0.333...4, then you can return to the conventional arguments. If not, then your friend should have no faith in mathematics whatsoever.

Another plan of attack would be to convert the numbers to a base where we don't get infinitely repeating decimals. For instance, 3_{10} = 10_3, and 1/3_{10} = 0.1_3. So if he accepts that 1/3 = 0.333..., and therefore that 3 * 1/3 = 0.999..., then he must accept that 0.999... = 1, since in base 3, 10_3 * 0.1_3 = 1_3, and 1_3 = 1_{10}.

[Mentat]

Thanks for all the replies, guys :smile:. All very useful stuff (even those who seem to agree with my friend, as I can use my own rebuttals to their reasoning as added arguments against his).

Hypna, you are absolutely right. He should indeed have no faith in math whatsoever. The problem is, his math teacher agreed with him...which means I'm going to have to talk to the math teacher first, thus removing his ability to argue from authority.

[Dissident Dan]

[QUOTE=Integral]I don't see it that way. Just the opposite if we were UNABLE to assign a value to any infinitely repeating number that would be a defect in the number system.
This is NOT a facet of DECIMAL numbers, it is a facet of the density of the Real number line. No matter how you choose to represent the points of the real number line there will always be numbers which require an infinite number of digits to be specified.

The Base does not matter. for example

.1_{10} = .00011001100... _2
QUOTE]

Well, yes, all number systems have their own flaws. I didn't say that flaws were unique to decimal.

The invention of repeating numbers is a result of the fact that some values cannot be exactly represented using certain bases. A repeating number is just an attempt to express a number using partitions that do not allow for exact representation.

[leto]

This has bothered me for years. My own logic tells me they aren't equal, while math tells me they are. I haven't decided yet if my logic is flawed or the number system is. I never got anywhere with it when I ask a math head because they are stuck on the definitions they are taught and seem to be oblivious to anything else. I personally think it is as close to 1 as you can get without ever being 1. In any pratical use of math it can be called 1, so it really doesn't matter anyway.

[Hurkyl]

I personally think it is as close to 1 as you can get without ever being 1.

Suppose that's true. Then how does (0.\bar{9} + 1)/2 compare?

[Jeebus]

Suppose that's true. Then how does (0.\bar{9} + 1)/2 compare?

D00d, you got served! :biggrin: Just kidding, of course. I just had to say that. :rolleyes:

[Tango_dark]

Also, if we assume that they are not equal and that it is "as close to one as possible without being one". Then we define must define a smallest number which represents their difference. If we assume a smallest possible number, bad things happen. Like the number line ceases to be continuous, distance ceases to exist, and other bad stuff.

[Integral]

Consider this inequality.

1 - .1^n < x < 1+ .1^n

It seems clear that there is only a single number for which this is true for ALL values of n>0, x= 1 . This is a simple statement that any number added to x results in something greater then 1, or any number subtracted from x results in something less then one. I doubt that you will find many people who will argue with the truth of the statement, x=1.

Now, in the link I posted above, I show that using simple arithmetic, involving only valid rational numbers, one can construct this inequality.

1- .1^n < .999.... < 1+.1^n

notice that is the exact inequality as above, thus we have x=.999... If the original statement is correct that the inequality can only be satisfied by 1 you must be lead to the conclusion that
1=.999...

[honestrosewater]

What if you make a distinction between 1.000... and .999..., i.e. so that they are different numbers, and then figure out "how infinite" this set of numbers is. If this new set is "more infinite" than the set of real numbers, case closed. That is, if your friend doesn't object to 1-1 correspondence, countablilty, and the various ways of comparing infinite sets ;)

I'm not sure how to make such a distinction. Perhaps the sets of all arrangements, where an "arrangement" is an infinite string of digits and one decimal point, ex.
1010= ...0001010.000...
98765.4321= ...00098765.4321000...
1= ...0001.000...
.999...= ...000.999...

The set of all arrangements should include all real numbers, plus two strings for each terminating decimal, but may not be more infinte than the reals. Just a thought. The answer is probably already known. Though now I find it interesting in itself :)

The whole idea is that 1=.999... implies that there must be more representations (numerals) than numbers. This may be something your friend has not realized. And it may be a way to get him/her to take another look at the axioms and theorems. Of course, "more" becomes vague when dealing with the infinite, but...
Happy thoughts
Rachel
EDIT- Nevermind, I see my mistake.
But I would like to know more about the sets of all numerals.

[Imparcticle]

Does your friend deny the fact that .9999... when rounded is 1? Is that the problem he/she seems to have difficulty with?

[Math Is Hard]

Does your friend deny the fact that .9999... when rounded is 1? Is that the problem he/she seems to have difficulty with?

when rounded, yes, but, I don't think rounding enters into this. This is .9999... with the decimal places stretching out to infinity. No rounding allowed- we're going all the way to the bitter end*.
My personal visualization of this is that 1 is reached only at the infinite decimal place and not anywhere before that. (But let me add the caveat that I am a million times lower on the rungs of mathematics than the majority of posters to these types of threads - so I may be completely full of baloney here.) :smile:

[Polly]

And I am definitely worse than Math Is Hard and just shooting some hot air here. But I am thinking, assuming 1 to be infinity, will taking a particle from it make it less than infinity? Probably not.

[pig]

What many people don't understand is that decimal representation is just that, a method for representation of quantities. That's why they think that any possible combination of numbers, dots, etc, should represent some quantity, and they invent meaningless stuff like "0.000...0001". What it means is important, not what it looks like when written down. We don't "allow" 0.000...0001 to have a non-zero value, or a zero value - it is simply not a number.

Polly - taking a finite amount out of something infinite doesn't reduce its size. This is the most important difference between infinite and finite sets. This is why the 0.999... * 10 = 9.999... demonstration makes sense - you moved all the nines one place to the left, but there is still the same number of them after the decimal point.

[quddusaliquddus]

Can Fuzzy Logic help here? - or am I confusing the topic?

[NateTG]

Can Fuzzy Logic help here? - or am I confusing the topic?

The technical term 'fuzzy logic' refers to systems of logic, for example electronics, that have more than 2 states, or many inputs. It is not at all apropos.

If you mean fuzzy logic in the sense of what mathematicians refer to as handwaving -- that is, saying that something makes sense, and leaving it there -- then it's also not appropriate for a discussion of this nature.

[quddusaliquddus]

Lol - not hand writing. I once read a book on it (quiet a while ago), and from what I think I understood - fuzzy logic allows one to have in-between states like 80% true. It probably doesn't help though ... :D

[Imparcticle]

This is why the 0.999... * 10 = 9.999... demonstration makes sense - you moved all the nines one place to the left, but there is still the same number of them after the decimal point.

No, you move one nine to the left.


We don't "allow" 0.000...0001 to have a non-zero value, or a zero value - it is simply not a number.

What is a number? .000...0001 is a possible measurement for some subatomic entity such as an atom or something.

[Hurkyl]

No, you move one nine to the left.

Which 9?


.000...0001 is a possible measurement for some subatomic entity such as an atom or something.

How do you figure?



What is a number?

What kind of number? I presume you mean "decimal number".

A decimal number is (leaving out some details), a function from the integers into the set {0, 1, 2, ..., 9}. The intuition is that the function takes as input a position and it returns the digit in that position.

0.00....001 ("infinite" zeroes) whatever it means, cannot be a decimal number. There has to be some integer which says where the '1' occurs, but that would mean there are only a finite number of zeroes between the '1' and the decimal point.

[Integral]

No, you move one nine to the left.


What is a number? .000...0001 is a possible measurement for some subatomic entity such as an atom or something.
A real number is a very well defined "thing". Numeric representations of real numbers consist of an infinite sum, you MUST be able to assign a integer to each digit. What is the integer assigned to the above 1? To say an infinite number of zeros then a 1, is an self contradictory statement.

Physical objects do not enter into the definition of the real numbers. Even if it did Planck's length, the theoretic smallest physical length, is not infinitely small.

Edit: a few words for clarity.

[Sariaht]

It's true if all pieces are shareable. I guess they are.

[HallsofIvy]

This has bothered me for years. My own logic tells me they aren't equal, while math tells me they are. I haven't decided yet if my logic is flawed or the number system is.

I know that feeling. Just the other day I was driving down a one-way street and all the other cars were going the other way! For the life of me, I just could decide whether I was going the wrong way or everyone else was!

I never got anywhere with it when I ask a math head because they are stuck on the definitions they are taught and seem to be oblivious to anything else.

Exactly! That blasted cop was so stuck on HIS definition of "one-way" he just wouldn't listen to me! (I was, after all, only going one way!)

[Math Is Hard]

Exactly! That blasted cop was so stuck on HIS definition of "one-way" he just wouldn't listen to me! (I was, after all, only going one way!)

I tried to apply similar logic. "Officer, the sign said 'No, U turn'." So I did.

[Math Is Hard]

Dear Marilyn,

If it is a fact that 1/3 = .333 repeating, why is it true that 1/3 + 1/3 + 1/3 = 1?

-Daniel

Dear Daniel,

"... no matter how far you extend the .333 repeating, the part you didn't write down (the repeating part) totals 1/3 of the next decimal place. And no matter how far you extend a .666 repeating, the part you didn't write down totals 2/3 of the next decimal place. And no matter how far you extend a .999 repeating, the part you didn't write down totals 3/3 of the next decimal place. This means that you can stop at any point and add it to what you have already written, and the result will be 1.000."

[leto]

I know that feeling. Just the other day I was driving down a one-way street and all the other cars were going the other way! For the life of me, I just could decide whether I was going the wrong way or everyone else was!
Your sarcastic one-way street analogy is interesting enough, but it doesn't have any value aside from your apparent need to mock me. We both know what appeal to popularity is. Traffic rules require no justification beyond keeping order. My problem with .999 repeating = 1, is simple - There is NEVER a point where you can stop the repetition and say that number = 1. I understand that the repetitions never stop, however, the amount of places from the decimal point also never stop. You are infinitely taking smaller and smaller steps closer to one. If infinity at some point arbitrarily fills the gap then why can't .999 repeating = 2, or any other number? Certainly the repetition must continue to go on even after 1 is approached. I can not give a value between .999 reapeating and 1, I understand this - This is because I cannot stop the repetition; just as you cannot stop the repetition and have 1.

I don't doubt I may be wrong. What I will not do is mindlessly accept what someone tells me unless I understand the logic behind it.

[Hurkyl]

0.999~ is a (decimal) number. It doesn't change, it doesn't approach anything. It doesn't have any steps, nor repetitions. It simply has a 9 in all of the positions to the right of the decimal point.

[honestrosewater]

Leto,
I had conceptual problems with this too. Maybe I can help.
Conceptually, can you begin or end at infinity? Does that make sense?

If infinity at some point arbitrarily fills the gap then why can't .999 repeating = 2, or any other number? Certainly the repetition must continue to go on even after 1 is approached.

What do you mean when you say "infinity at some point"?

Happy thoughts
Rachel

[leto]

Conceptually, can you begin or end at infinity? Does that make sense?
No, you cannot. However, lets suppose something existed forever and never changed. I am able to observe that entity at a finite time in my existence. If I can do this, wouldn't that tie it to my concept of time regardless of how absurd it might be from the opposite point of view? Wouldn't I be able to observe it and be accurate in describing what it has and always will look?

What do you mean when you say "infinity at some point"?

I believe we have to break it to observe it.

[Integral]

I believe we have to break it to observe it.

Please make an effort to free your mind of physical objects when thinking of Mathematical infinity. Mathematical infinities do not exist in the the physical world, any Physical analogy will fail. You do not need to break anything to consider an infinite number of 9s after the decimal. Please go back to near the beginning of this thread where I posted a link to a Mathematical proof of this. Be sure to look at and read ALL 4 pages, not just the first.

[NateTG]

I don't doubt I may be wrong. What I will not do is mindlessly accept what someone tells me unless I understand the logic behind it.

Well, for rational numbers, bar notation can be defined to represent a particular fraction. It takes a bit of work, but it's quite easy to see that this approach leads to 0.\bar{9}=1. Note that the decimal representations of all rational numbers eventually repeat, so they are all candidates for bar notation. (Although \bar{0} is usually omitted when writing numbers.)

Now let's take a brief look at decimal notation for real numbers:

Real numbers are often written as unfinished decimals. A real number 0\leq r \leq 1 might be written as:
0.d_1d_2d_3d_4d_5...
where each of the d_i is a digit -- for now let's just use (0-9).

This decimal notation can be considered a shorthand for:
\sum_{i=0}^{\infty} \frac{d_i}{10^i}

Which leads to
0.9999... = \sum_{i=0}^{\infty} \frac{9}{10^i} = 1.0000...

Once again 0.\bar{9}=1

Another option is to look at:
0.\bar{9} \times 0.\bar{9}=0.\bar{9}
but
0.\bar{9} \times x = 0.\bar{9}
and if you divide out by 0.\bar{9} you get
x=1
so now we have x=1 and x=0.\bar{9}. So either it's unsafe to divide by an unknown, or 0.\bar{9}=1

Alternatively, let's take a look at the most likely reason that you wonder about 1=0.9999...: the notion that each decimal sequence uniquely represents a real number in the usual representation. So, let's circumvent that notion with a slightly different approach to the real numbers that explicitly allows many different representatives for each real number:

(This may be a bit heavy)

Let's denote [x_n]=x_1,x_2,x_3,x_4... to be a sequence of rational numbers. Then we say that [x_n] is cauchy if given \epsilon > 0 there exists N so that n_1,n_2 > N \rightarrow |x_{n_1}-x_{n_2}| .

Now, define \doteq that [x_n] \doteq [y_n] if the sequence x_1,y_1,x_2,y_2... is cauchy. It's easy to show that \doteq is an equivalence relation.

Then I can define the real numbers to be equivalance classes of these sequences under \doteq. Addition and multiplication is simply componentwise addition and multiplication. (I'm too lazy to prove that they work properly on equivalence classes right here.)

Now it's easy to see that a decimal representation of a real number 0.d_1d_2d_3... readily translates to a representative sequence [r_n]=\frac{d_1}{10}, \frac{10 d_1 + d_2}{100}, \frac{100 d_1 + 10 d_2 + d_3}{1000}... that is cauchy.

Then the representations 1.0... and 0.\bar{9} lead to the sequenes 1,1,1,1... and 0.9,0.99,0.999,0.9999,0.99999... repsectivevely. But the sequence 1,.9,1,.99... is cauchy, so those two sequences represent the same real number. So 0.\bar{9} = 1

[honestrosewater]

I agree with Integral, though Integral certainly doesn't need my agreement in order to be correct ;)

However, I know that reading a proof umpteen times will not necessarily make it any clearer. Understanding the proof should be your goal, but if you don’t understand it yet...

There is NEVER a point where you can stop the repetition and say that number = 1.

**The language really needs to be precise, and it isn't here, so we have to be especially careful and keep in mind that the conversation is informal and aimed at conceptual understanding.**

That said, the above quote is what I’ll try to get you to understand, because it doesn't seem like you actually understand it.

I understand that the repetitions never stop, however, the amount of places from the decimal point also never stop.

You don’t need the “however”, in fact, you need to get rid of the “however”. On the right side of the decimal point, every decimal place has a 9 in it. The “amount of places from the decimal point” is the same as “the repetitions”. See?

You are infinitely taking smaller and smaller steps closer to one. If infinity at some point arbitrarily fills the gap then why can't .999 repeating = 2, or any other number?

The gap is filled when infinity ends! :) And you already know that infinity doesn’t end. So the gap is never filled.

Another way of looking at it is that, if you are taking smaller and smaller steps, you never take a *last* step.

Note that, if the first step is .9, each successive step is 1/10 the size of the previous one, that is, multiply the size of the last step by 1/10 to get the size of the next step. 9/10, 9/100, 9/1000, 9/100000, ...
The size of each step is very important. Just getting smaller will not do, they have to get smaller by 1/10.

You cannot get to 2 because you never pass 1. You never pass 1 because the size of your steps is getting smaller by a very important amount, 1/10.

(Notice that you have already said "There is NEVER a point where you can stop the repetition and say that number = 1." You admit that .9 is less than 1. In order to get from .9 to 2, you must pass 1, since 1 is between .9 and 2. And so you must have assumed that the successive values 0.9, 0.99, 0.999, ..., "jump over" 1, or pass 1 without landing exactly on 1, or without being exactly equal to 1. Had you assumed that?)

You also never pass 1 because you never reach 1. This is the same as saying the gap is never filled or you never take a last step.

You can start at .9 then step to .99 then step to .999 and so on. Each step is closer to 1. And after each step, you only have to take a step “this size” to reach 1. But you can never take a step “this size” because you have to take a step 1/10 the size of your previous step, AND 1/10 the size of your previous step is always smaller than “this size”.
When you are at .9 you need to take a step of size .1 to reach 1.
But you can only take a step of size .09
See?

There is NEVER a point where you can stop the repetition and say that number = 1.

I hope that cleared up any misunderstanding you had so far.
The next thing to understand is why 0.9999... = 1.000...
Perhaps taking another look at Integral’s proof or reading through other's posts will make this clear.

Happy thoughts
Rachel

[leto]

honestrosewater, your reasoning for why .999.. =1 is my exact reasoning for why it isn't. I did fully understand all of the concepts you mentioned. I may not have been clear, but that was exactly what I was trying to express for why .999 would not equal 1. As you put it, "since the gap is never filled." How can the two be equal when the gap is never filled? I am actually kind of annoyed you walked me through my own reasoning, but I guess I didn't express it well enough.

I know that .999.. = 1 can be proven mathematically in many ways. I think my inability to grasp it is, as integral mentioned, I am thinking abstractly in physical terms. Although, the only value math has to me is in its ability to describe the physical world. I still don't understand why the two shouldn't correspond. Reading the link now.

Edit: I finally got done downloading acrobat so I could see the link. I've seen those proofs before and never qestioned whether it could be proven mathematically. I think I've mentioned I thought it was either the number system itself or my irrationality.

[honestrosewater]

Leto,
Are you faulting me for misunderstanding what you meant to express? (Please don't read any sarcasm into that question.) If so, I think that is unfair. I over-explained myself in order to avoid misunderstanding; it wasn't my intention to annoy you. I am sorry it made you feel that way.

To clarify- my comments were not aimed at convincing you that 0.999...=1.000...
"I hope that cleared up any misunderstanding you had so far.
The next thing to understand is why 0.9999... = 1.000..."

Do you still want to understand it? I will try to think of a way to explain why they are equal. Let me know if you're still interested :)

Happy thoughts
Rachel

[Hurkyl]

How do the decimals not correspond to the physical world?

Suppose I walk .9 meters, then .09 meters, then .009 meters, et cetera.
During this sequence of tasks, I've crossed every point that lies between my starting point and 1 meter from my starting point. (and nothing beyond)

From one point of view, I've travelled a distance of .9 + .09 + .009 + ... = .999... meters, as computed by adding up all of the distances I travelled during my tasks. From another point of view, I've travelled a distance of 1 meter, the size of the interval I've crossed. So how could suggesting .999... is unequal to 1 correspond to reality?

[quartodeciman]

If someone actually tries to step off .9 meter, then step off .09 meter, then step off .009 meter and so on, then people tend to think the someone will be unable to proceed this way after a while. On the other hand, stepping off 1 meter is no problem. Therefore, they tend to think of them as two different operations. Fractionations of 1 stride are conceived as distinct from the summation of smaller strides. That is where both the Achilles and the Dichotomy get off to a bad start.

[leto]

Leto,
Are you faulting me for misunderstanding what you meant to express? (Please don't read any sarcasm into that question.) If so, I think that is unfair. I over-explained myself in order to avoid misunderstanding; it wasn't my intention to annoy you. I am sorry it made you feel that way.
You did misunderstand what I meant to express.

You don’t need the “however”, in fact, you need to get rid of the “however”. On the right side of the decimal point, every decimal place has a 9 in it. The “amount of places from the decimal point” is the same as “the repetitions”. See?
The however was meant to separate the two patterns. It was my attempt at expressing there was a gap because, not only was it getting closer, it was getting closer in smaller and smaller steps.

The gap is filled when infinity ends! :) And you already know that infinity doesn’t end. So the gap is never filled.
I never had a problem with this concept. I was trying to express that there was a gap, and if infinity somehow filled the gap then why couldn't it continue from there? (Since there is no end.) I was trying to express why I don't think the two are equal, and didn't understand you could somehow admit there was a gap and still believe the two to be equal. This seems irrational to me. Feel free to continue your explaination.

Edit: I still see no flaw in my reasoning, but I just thought about it from a different perspective and I can understand how the two could be equal. I may still just be crazy, but it's begining to seem like a paradox to me. The gap between the two is never filled, but there is no measurable distance between the two. I am satisfied now, but I am still happy to hear why I am wrong with my initial reasoning.

[honestrosewater]

You did misunderstand what I meant to express.

I agree, my point was the misunderstanding was not intentional :)

The gap between the two is never filled, but there is no measurable distance between the two. I am satisfied now, but I am still happy to hear why I am wrong with my initial reasoning.

I don’t know that your initial reasoning was wrong. I noticed some things that seemed wrong to me, but, as we’ve agreed, I misunderstood and you have explained yourself.

Once you accept 1) there is no last step, 2) each step is smaller than the last, and 3) each step always gets closer to 1, then the conclusion is that they are equal (because there is no smallest step).
The only thing left for me to explain was why that conclusion follows, but you seem to have gotten it already.

Have you taken another look at the proofs Integral provided?

Happy thoughts
Rachel

[leto]

I don’t know that your initial reasoning was wrong. I noticed some things that seemed wrong to me, but, as we’ve agreed, I misunderstood and you have explained yourself.

What about my gap with no measurable distance?

[honestrosewater]

What about my gap with no measurable distance?

It seems to say the same thing as my "steps" explanation, and I granted you this already.
There are of course problems with both because they are not precise. They are just conceptual devices which need to be refined and put back into the mathematical context of the original problem. The language we're using is not precise enough for the problem.

The biggest problem with saying "there is no measurable distance between the two" is that it is also true if the distance is infinitely large, as well as infinitely small. That's why I added the "smallest step" bit- just to be clear. I also prefer the "step" explanation because it involves the idea of order (<, >, =). But it still has problems.

BTW A good example of a language goof is when I said: "2) each step is smaller than the last." Because "last" is ambiguous here, I should have said "each succesive step is smaller than the previous step" or something.

Happy thoughts
Rachel

[NateTG]

What about my gap with no measurable distance?

It's unclear whether you mean measurable in the technical sense, or you mean that the 'length of the gap' is zero.

You are indeed correct in your assessment that 0.\bar{9} = 1 is in some sense an artifact of the real numbers. There are more exotic number systems where it's possible to have distinct numbers that are adjacent zero, but these number systems are more difficult to deal with than the real numbers.

To give you an idea of the kinds of problems this leads to, consider, for a moment, the following:

Let's assume, for a moment that 0.\bar{9} is not equal to 1. Then \frac{0.\bar{9}}{2} = 0.4\bar{9} \neq 0.5.
Now, if you convert this into base 3, you get 0.\bar{9}_{10}=0.\bar{2}_3 and 0.5_{10}=0.\bar{1}_3 and 0.4\bar{9}_{10}=\frac{0.\bar{9}_{10}}{2_{10}}=\fra c{0.\bar{2}_3}{2_3}=0.\bar{1}_3 so we have 0.4\bar{9}_{10} \neq 0.5_{10} \rightarrow 0.\bar{1}_3 \neq 0.\bar{1}_3

[Grizzlycomet]

I spoke to my math teacher about this issue today, and he was quite firm in his belief that 0.\bar{9} does not equal 1. He didn't provide any proof though.

[BoulderHead]

I spoke to my math teacher about this issue today, and he was quite firm in his belief that 0.\bar{9} does not equal 1. He didn't provide any proof though.
The public should understand education. And it would do no harm if teachers and professors understood it, too.
-Hutchins

[Grizzlycomet]

The public should understand education. And it would do no harm if teachers and professors understood it, too.
-Hutchins
Indeed. I am planning to make a small document with various proofs that 0.999... does equal 1. Any links anyone could provide would be most helpful.

[quartodeciman]

I spoke to my math teacher about this issue today, and he was quite firm in his belief that 0.\bar{9} does not equal 1. He didn't provide any proof though.

It would be a good idea to first ask teacher what endless repeating decimal expressions actually mean, inasmuch as addition is a finite (originally binary) operation between numbers.

[Grizzlycomet]

It would be a good idea to first ask teacher what endless repeating decimal expressions actually mean, inasmuch as addition is a finite (originally binary) operation between numbers.
Further discussion with my teacher would indeed be a good idea. However, my school year ends this week so there's not exactly a lot of time for this.

[TENYEARS]

I was posting on the math thread before it was closed. .99999 can only be 1 if indeed it was generated by the division of three equal parts of a whole. It would preclude you knowing that the .999999 was generated by this act and then could be equated to one. Other wise no one has a right to make .999999 = 1 because it is not so and non relative at this point. So it could have been generated by three parts of a whole or it was a selected number. If it is just a selected number it is not equal to 1.

[ahrkron]

Tenyears, as I told you in the math thread, the fact that 0.999...=1.00... is not a matter of convention, "acceptance", or authority. It is a logic inevitability from the definitions of real numbers. The issue is not controversial at all among professional mathematicians, and is based on solidly established branches of math (in particular, Real Analysis).

[europium]

quick random q
where do you get all your mathematical symbols from on the keyboard... thanks

K_

[Grizzlycomet]

quick random q
where do you get all your mathematical symbols from on the keyboard... thanks

K_
The matematical symbols is created using a code in the forum knows as Latex. It's usage is described in This Thread (http://www.physicsforums.com/showthread.php?t=8997)

[Njorl]

I am firmly in the .999...=1.0 camp, where 1.0 equals the real number one. Is it allowable to say that the real number one is equal to the integer number one? Is this like mixing apples and oranges (one apple does not equal one orange)?

I feel that the integer one does equal the real one, but I don't know of a rigorous logical foundation for it. Is there one? Is it just defined to be so? Am I wrong?

Njorl

[honestrosewater]

I am firmly in the .999...=1.0 camp, where 1.0 equals the real number one. Is it allowable to say that the real number one is equal to the integer number one? Is this like mixing apples and oranges (one apple does not equal one orange)?

I feel that the integer one does equal the real one, but I don't know of a rigorous logical foundation for it. Is there one? Is it just defined to be so? Am I wrong?

Njorl

I have only seen two contructions of R (Dedekind cuts & Cauchy sequences) and both construct R from Q. And Q is defined by Z and Z by N. So I would think that is rigorous logical foundation, but what do I know? :uhh:

Happy thoughts
Rachel

[TENYEARS]

I don't care what proof you come up with.

1 - I take a geometric object a cicle or a square and divide it into three parts and then add the decimal values the number is .999999 but is 1 relative to the object as a whole. This is correct for it is the totality of the object.

2 - I take the number .9999.... out of the blue with no reference to a geometric representation of an object this is not 1. If I take a geometric reference to the universe, divide it into three equadistant rays starting from a central point and extending into infinity running a wall of ray for the length of extension so you have three defined parts, make that a decimal value of .333.... added together, then I will equate .99999..... to one with repsect to the universe.

If there is no geometric reference, and it is just a number, it is not equal to 1.

[NateTG]

Really, this seems like a funky philosophy of math question but, there is an obvious subset of \Re that is isomorphic to the integers with the appropriate operations. As long as you think of it more as an instatiation of the integer 1 rather than as the only integer 1, you should be fine.


I have only seen two contructions of R (Dedekind cuts & Cauchy sequences) and both construct R from Q. And Q is defined by Z and Z by N. So I would think that is rigorous logical foundation, but what do I know?


You can also construct the real numbers from things like the set of all countable sequences of zeroes and ones that do not end in reapeating 1's, or something, but a construction like that has a PITA factor while Cauchy sequences and Dedekind cuts can readily be shown to have the desired properties. I bring this up because it allows for 0.\bar{1}_2 <binary notation>, not to be a real number.

I expect that, the (mistaken) notion that 0.\bar{9} and 1 are distinct is a result of the mistaken assumption that decimal representations are unique.

[honestrosewater]

I don't care what proof you come up with.

I think you are giving ahkron too much credit.
If ahkron has seen further it is by standing on the shoulders of Giants.

Now, if you cut a giant into 3 parts... only kidding, in good fun :biggrin:

Okay, now I can't help myself ;)

"If I have not seen as far as others, it is because giants were standing on my shoulders." -- HalAbelson

"In the sciences, we are now uniquely privileged to sit side by side with the giants on whose shoulders we stand." -- GeraldHolton?

"If I have not seen as far as others, it is because I was standing in the footprints of giants"

"If I have seen farther than others, it is because I was standing on a really big heap of midgets." -- EricDrexler (Nice for those of us who believe the inspiration of giants isn't the only engine of progress.)

"If I have seen further than others, it is because I was surrounded by dwarves." -- attributed to MurrayGellMann?, possibly maliciously.

"I cannot see very far, because my eyes are full of midgets."

:rofl:

[honestrosewater]

You can also construct the real numbers from things like the set of all countable sequences of zeroes and ones that do not end in reapeating 1's, or something, but a construction like that has a PITA factor while Cauchy sequences and Dedekind cuts can readily be shown to have the desired properties. I bring this up because it allows for 0.\bar{1}_2 <binary notation>, not to be a real number.

Great, now I'm confused too :yuck: Is there a quick way to explain how that construction proceeds? Oh, countable is a clue methinks. No, the set is countable? Or the sequences are countable? Yeah, :confused: What is this PITA factor you speak of? (I get the PITA, but what is it?)

[Hurkyl]

Basically, you define all the arithmetic operations by the method of elementary school arithmetic, but the trick is that you have to perform addition from left to right. The "PITA" factor is in cases where 'ambiguous'
whether you should have a carry or a borrow when doing an operation.

E.G. when adding 0.1100... and 0.011000..., you can "look ahead" to see that the second place to the right of the decimal point generates a carry, and it's propagated through the next place, so you can set the one's digit to be a 1, and then so on.

However, when adding 0.101010... and 0.010101..., there's nothing to say that there should or should not be a carry. Thus, you make a definition; you either say that in this situation you will always consider there to be a carry, never consider there to be a carry, or define both options as being equal.

[honestrosewater]

However, when adding 0.101010... and 0.010101..., there's nothing to say that there should or should not be a carry. Thus, you make a definition; you either say that in this situation you will always consider there to be a carry, never consider there to be a carry, or define both options as being equal.

1) carry-> 1.000...
2) no carry-> 0.111...
3) equal-> 1.000...=0.111...

Yes? No? So why can .111... not be real? I am still missing something; hopefully I will see clearer after I some shuteye.

Happy thoughts
Rachel

EDIT- Oh duh- if you decide to carry, you cannot ever get the "noncarry" number, and vice versa.

[Swigs]

My understanding of infinte numbers is that you can not treat them like any other number. Like what 'tenyear" was talking about.

Infinite numbers are just a mathimatical idea and dont realy exist in the real universe as we know it. I thought that is why "limits" were created anyway.

[mikesvenson]

infinity, as I've been thinking about it, is whithin every real number as a result of mathamatical processes. e.g, divide to infinity, add to infinity, subtract to infinity, .....etc. Even whithin 0, there is infinity.

Infinity is the true nature of all things from greatly massive to micro. "Limits", are just easier ways of looking at things and are nothing more than a generalization of reality.

[lvlastermind]

infinity, as I've been thinking about it, is whithin every real number as a result of mathamatical processes. e.g, divide to infinity, add to infinity, subtract to infinity, .....etc. Even whithin 0, there is infinity.


every real number has an infinite number of digits but no real number has an infinite magnitude. e.g. ...000000001.0000000...

[mikesvenson]

it has an infinite divisible magnatude

[steersman]

1/3 is a non-mathematical representation of a potential infinity: 0.333...etc. There is no such thing as an actual infinity. I refer to the Hilbert's hotel paradox. Potential infinities exist in mathematics because it is a theoretical tool.

Let me put it this way:
does 0.9 = 1? No
does 0.99 = 1? No
does 0.999999999999999999 = 1? No

In each case we are getting closer to 1, but will we ever get there? No.

[Erazman]

0.9~ is not 1.
If you can think of space as infinately large,
then just imagine the concept of infinately small.
Distance in space can always be cut in half. And in half again.
Our mind just has a hard time comprehending it...

[Integral]

There is no such thing as an actual infinity.

Mathematically speaking there is and it is carefully defined. Maybe you should take a few minutes to read some of the posts in this thread.
0.9~ is not 1.
Certainly is! I guess your concept of the universe does not apply to the real number line. Perhaps you should modify your concept of the universe.

[Hurkyl]

Let me put it this way:
does 0.9 = 1? No
does 0.99 = 1? No
does 0.999999999999999999 = 1? No

Does 0.9 = 0.9~? No.
Does 0.99 = 0.9~? No.
Does 0.999999999999999999 = 0.9~? No.

In each case, we are getting closer to 0.9~ but will we ever get there? No.

Why do you think your observation has any bearing on whether 0.9~ = 1?


Have either of you, steersman and Erazman, read through this thread?

[steersman]

Have either of you, steersman and Erazman, read through this thread?

Do I have to?

Mathematically speaking there is and it is carefully defined.

So what? Mathematics uses potential infinities not actual ones. This experiment...getting to 1 is a problem because in maths potential infinities exist whereas in reality they don't. This experiment is reality based, despite its mathematical content. It involves the quantization of measurement. You need to take it out of context to see it has no meaning. You can divide a ruler into an infinite of potential parts - does this mean that the ruler itself is infinite? No.

[e(ho0n3]

Is .4999... + .4999... = 1? I mean, how does one define an addition or any arithmetic process on these kinds of numbers without first changing them into rational form. Are we stuck using the rational form?

[jcsd]

Yes 0.4999.... + 0.4999....... = 1.

I'm not sure what you mean by rational form as any repeating decimal number is rational

[selfAdjoint]

Is .4999... + .4999... = 1? I mean, how does one define an addition or any arithmetic process on these kinds of numbers without first changing them into rational form. Are we stuck using the rational form?

You can define them rigorously via modern methods, or by Eudoxos' theory of proportions, which occupies Book X of Euclid's Geometry, together with a completeness axiom.

[Hurkyl]

Do I have to?

No; but you may find it enlightening. And it might save others from being irritable that they have to make the same responses over and over.

[balrog]

It's sort of like how no matter how many times you cut a number in half it will never equil zero... but it will get pretty damn close... same thing: no matter how many 9s you add to the end of the decimal you will always get closer to one, but you could never ever reach it! Either I'm an idiot, or you will never ever reach it.

[AKG]

It's sort of like how no matter how many times you cut a number in half it will never equil zero... but it will get pretty damn close... same thing: no matter how many 9s you add to the end of the decimal you will always get closer to one, but you could never ever reach it! Either I'm an idiot, or you will never ever reach it.First of all, you aren't approaching 1, one digit at a time. You have an infinite number of digits. You can make sort of an inductive proof that for all natural numbers n, if 0.999... has n digits, it's still not going to be 1. So with that think you'll never reach 1. But the point is that although for no n will 0.999... (with n digits) equal 1, 0.999... doesn't have a natural number of digits, it has infinite digits. Second of all, 0.99... = 1 for the very strange reason that 0.99... is defined as the infinite sum of the terms t_k, where t_k = 9/10^k. Then, this infinite sum is defined as the limit as X approaches infinity of k=1 to k=X of the sum of t_k. It can be proven that this infinite sum cannot be any real number other than 1, and we simply define the sum to be the real number approached as X approaches infinity. It is useful to define this sum as a real number, and we choose the limit, i.e. the only number it can be, to be that real number. I say this is strange because 0.999... is something you don't use past grade 4, and you don't learn infinite sums and limits until high school.

So, in summary, we choose to define an infinite sum to be a real number (assuming it represents a sequence of converging partial sums), and the number we choose is called the limit, and it's a decent choice because we can prove that it can be no other number.

[balrog]

sure there are an infinite number of placeholders after the decimal, I accept that, but even infinity isn't enough to make it 1. As the number of nines after the decimal approach infinity the value will come pretty damned close to 1... but there is no end. It's still not one, it's just undeterminably close to one.

[Erazman]

Maybe the real debate here is: What's infinity?

The dictionary describes it is "An indefinitely large number or amount"
the term "indefinate" means having NO distinct limits. Undefined. It's simply a term made up by man to describe an indefinate value.

0.99~ is not definate. If we all of a sudden give a DEFINED VALUE to infinite by calling 0.999~ 1 instead of 0.999~, then we have just destroyed the whole concept of infinity, because infinity HAS NO DEFINED VALUE.

Not only that but...

When i look at 0.999~ i see a 0 before the decimals. the 0 means it cannot possibly be 1. If the number is LESS THAN 1, then a 0 will indicate it, in this case it does...

[Hurkyl]

same thing: no matter how many 9s you add to the end of the decimal you will always get closer to one, but you could never ever reach it!

Also, no matter how many 9s you add to the end of the decimal, you will always get closer to 0.9~ but you could never reach it!


As the number of nines after the decimal approach infinity the value will come pretty damned close to 1... but there is no end. It's still not one, it's just undeterminably close to one.

Ah, but the number of 9's in 0.9~ is infinite; it is not some nebulous thing that "approaches infinity".

And it is equal to one; the distance between 0.9~ and 1 is zero.


Maybe the real debate here is: What's infinity?

The dictionary describes it is "An indefinitely large number or amount"
the term "indefinate" means having NO distinct limits. Undefined. It's simply a term made up by man to describe an indefinate value.

And mathematicians are somewhat more precise. In particular... it is mathematically specified that each of the places after the decimal place corresponds to a unique positive integer.


When i look at 0.999~ i see a 0 before the decimals. the 0 means it cannot possibly be 1. If the number is LESS THAN 1, then a 0 will indicate it, in this case it does...

And if that's how < worked, you'd be right.

[AKG]

sure there are an infinite number of placeholders after the decimal, I accept that, but even infinity isn't enough to make it 1. As the number of nines after the decimal approach infinity the value will come pretty damned close to 1... but there is no end. It's still not one, it's just undeterminably close to one.Well, then you didn't read what I wrote. For one, the addition of an infinite number of terms is defined as the limit of the sequence of partial sums, this is proven to be 1 in this case. Second of all, you have no reason to believe that it is not 1 if it has an infinite number of digits after the decimal. Because that means adding an infinite number of terms, and that's something different from adding a finite number of terms. In short, you have no reason to say "infinity isn't enough to make it 1." There's a difference between going on interminably/infinitely and actual infinity. For example, dividing a number by 2 infinitely would not get you zero, because you can keep dividing "forever" and not reach zero. But even if you divided forever, you wouldn't have divided an infinite number of times. I suppose the wording's confusing. What does it mean to cut a number in half infinity times? You can continue cutting it in half for infinity or forever, and never reach zero, but you can never have actually cut it an infinity number of times. Similarly, you can add 9's on to the end of 0.9999 forever and never reach 1, but that's different from having an actual infinite number of 9's. If some number x is divded by 2 n times, then it becomes x/2^n. Now, what the heck is x/2^\infty? We don't have a way to deal with that, we can only calculate limits, any other answer is meaningless (it's not that it's not zero, it's nothing, at not a real number because reals don't deal with infinities, maybe a surreal number though).

[Erazman]

Also, no matter how many 9s you add to the end of the decimal, you will always get closer to 0.9~ but you could never reach it!




Ah, but the number of 9's in 0.9~ is infinite; it is not some nebulous thing that "approaches infinity".

And it is equal to one; the distance between 0.9~ and 1 is zero.




And mathematicians are somewhat more precise. In particular... it is mathematically specified that each of the places after the decimal place corresponds to a unique positive integer.




And if that's how < worked, you'd be right.


sorry, LESS THAN 1 AND GREATER THAN ZERO. i messed up. and this "unique positive integer" your talking about is exactly what shows up in the number itself on paper. 0.9999~ shows the 9999~ corresponding to a ZERO. If it was truely 1, then that number before the decimal place would NOT be zero.

[Erazman]

Well, then you didn't read what I wrote. For one, the addition of an infinite number of terms is defined as the limit of the sequence of partial sums, this is proven to be 1 in this case. Second of all, you have no reason to believe that it is not 1 if it has an infinite number of digits after the decimal. Because that means adding an infinite number of terms, and that's something different from adding a finite number of terms. In short, you have no reason to say "infinity isn't enough to make it 1." There's a difference between going on interminably/infinitely and actual infinity. For example, dividing a number by 2 infinitely would not get you zero, because you can keep dividing "forever" and not reach zero. But even if you divided forever, you wouldn't have divided an infinite number of times. I suppose the wording's confusing. What does it mean to cut a number in half infinity times? You can continue cutting it in half for infinity or forever, and never reach zero, but you can never have actually cut it an infinity number of times. Similarly, you can add 9's on to the end of 0.9999 forever and never reach 1, but that's different from having an actual infinite number of 9's. If some number x is divded by 2 n times, then it becomes x/2^n. Now, what the heck is x/2^\infty? We don't have a way to deal with that, we can only calculate limits, any other answer is meaningless (it's not that it's not zero, it's nothing, at not a real number because reals don't deal with infinities, maybe a surreal number though).

so if 1 / 2~ = not a real number, a surreal number
then 0.99~ = not a real number, a surreal number
adding .x9 to .999~ and calling it "1" is the same thing as 1 / 2~ and calling it "0"
right? therefore 0.99~ doesn't equal 1. It doesnt equal anything other than what it is. 0.99~.

[AKG]

so if 1 / 2~ = not a real number, a surreal number
then 0.99~ = not a real number, a surreal numberNo. 0.999... is defined to be a real number, the limit of the converging sequence of partial sums. 1/2^\infty is not defined to be a real number. I suppose it depends on how you interpret the notation. \sum _{n=1} ^\infty t_n looks like nonsense except when you realize it's a compact notation for \lim _{x \rightarrow \infty} \sum _{n=1} ^x t_n. Similarly, if you wanted, you could say that 1/2^\infty is just compact notation for \lim _{x \rightarrow \infty} 1/2^x. Both the sum and the 1/2^\infty have no meaning as real numbers unless you clarify that it's shorthand notation for some limit, as the real numbers don't deal with infinities, i.e. \infty \notin \mathbb{R}. It might be more sensible to have an alternate definition for the infinite sum and 1/2^\infty as surreal numbers because they deal with infinites and infinitessimals, but for now, we have a logically consistent, useful definition for infinite sums as a limit of a converging sequence of partial sums. Perhaps, don't think of it as proving what it means to add an infinite number of terms together, it is simply defining it to be a certain way.

[Integral]

When the real line is extented to include infinity you must include definitions for the arithemetic operations. The ususal definition is

\frac 1 {\infty} = 0
and
x^ {\infty} = \infty \ \forall \ x>0 \ x \in \ R
so when these definitions are applied you have:

\frac 1 {2^ {\infty}} = \frac 1 {\infty} = 0

Just for the record, when extending the real line to include infinity the definition is something like.

\infty > x \ \forall \ x \ \in R

With a similar definition for negitve infinity.

[AKG]

When the real line is extented to include infinity you must include definitions for the arithemetic operations.
Affinely Extended Real Numbers (http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html)
Projectivly Extended Real Number (http://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html)
Pretty interesting, but notice that with these systems, by adding infinities, you lose some of the properties that the real numbers have.

[Integral]

You are absolutly correct, the extended Reals are not a field because of the defined properties of infinity.

[Hurkyl]

nd this "unique positive integer" your talking about is exactly what shows up in the number itself on paper. 0.9999~ shows the 9999~ corresponding to a ZERO. If it was truely 1, then that number before the decimal place would NOT be zero.

You entirely misunderstood.

The correspondense is that there is a 1st digit, a 2nd digit, a 3rd digit, and so on; there is an n-th digit for each positive integer n.

Furthermore, every digit can be labelled in this way; for every digit to the right of the decimal place, there is a positive integer n for which you can say that it's the n-th digit.

0.9~ is a sequence such that for each integer n, the n-th digit to the right of the decimal point is a 9.


And, as a sequence of digits, 0.9~ is, indeed, inequal to 1. Furthermore, 0.9~ < 1. according to the lexical ordering of sequences of digits. But 0.9~, as a decimal number, is equal to 1..

[e(ho0n3]

I don't know if this has been said already, but the reason I think of why a lot of people don't 'buy' that 0.999... = 1 is simply because these two numbers are syntactically different. In other words, if the numbers don't look the same then they are not the same. I think people should just realize that numbers aren't as straightforward as they think they are. So far nobody has given a convincing logical argument that 0.999... and 1 are different numbers so why is this thread getting so large.

[Hurkyl]

Yet, oddly, nobody seems to have any trouble accepting that \frac{4}{3} = \frac{8}{6} = 1\frac{1}{3}, and most people can understand that 1 = 13 \mod 12. I still haven't understood why so many have trouble with 0.\bar{9} = 1.

[Erazman]

alright ive changed sides...
makes more sense now

[steersman]

Yet, oddly, nobody seems to have any trouble accepting that , and most people can understand that . I still haven't understood why so many have trouble with

1 and 1/3 are different forms of representation. The example you gave is correct, 1/3 is predicated on there is a 1/3 that can be found and definately measured. It is a potential representation. This is a predicate in mathematics not reality. The case is different with 0.9~=1. The number 1 is a reality representation.

See, the thing is 0.9 maybe the same as 0.9~. It just depends on how far you are willing to measure. So in a sense 0.92 is more than 0.9 in the way you are justifying your case, despite this being just a matter of measurement.

It is not possible to travel at the speed of light, why is this? because you must expend an infinite amount of energy - which is not possible, because actual infinities do not exist. You would need an actual infinity in 0.9~=1.

This is a philosophical problem not a mathematical problem. It is to do with the philosophical meaning of infinity not any mathematical definition of it.

It's like saying: well if this were possible then this would equal 1. But its not possible.

[Hurkyl]

Measurement? Reality? Where did you get any of this stuff?


0.9~ does not depend on any choices whatsoever. It has a 9 in the n-th place for ALL positive integers n, and there exist no other places to the right of the decimal places.

It is not formed by starting with 0. and adding 9's one by one.

[steersman]

It is not formed by starting with 0. and adding 9's one by one.

Who said it was?

[Hurkyl]

You sort of implied it with "See, the thing is 0.9 maybe the same as 0.9~. It just depends on how far you are willing to measure."

[steersman]

You sort of implied it with "See, the thing is 0.9 maybe the same as 0.9~. It just depends on how far you are willing to measure."

No no. The point I was making was that every number that isn't whole is already (potentially) infinite in terms of the infinite regress involved when trying to quantize something.

[Integral]

1 and 1/3 are different forms of representation. The example you gave is correct, 1/3 is predicated on there is a 1/3 that can be found and definately measured. It is a potential representation. This is a predicate in mathematics not reality.
The sentence I have bolded makes no sense to me, could you provide a translation in to relevant language?

The case is different with 0.9~=1. The number 1 is a reality representation.

Just what is a reality representation? Is there a mathematical definition for that?

See, the thing is 0.9 maybe the same as 0.9~. It just depends on how far you are willing to measure. [\quote]
I don't care how far you measure, as long as we are talking bout Real numbers 0.9 in NEVER = .9~
[quote]
So in a sense 0.92 is more than 0.9 in the way you are justifying your case, despite this being just a matter of measurement.

Could you please demonstrate a "sense" were 0.92 is NOT greater then 0.9?

It is not possible to travel at the speed of light, why is this? because you must expend an infinite amount of energy - which is not possible, because actual infinities do not exist.

Wrong thread, discuss Relativity and the speed of light in the correct forum. (BTW you are not exactly on the mark with that statement either!) This is logic applied to Math, no Physics needed or wanted.

You would need an actual infinity in 0.9~=1.

Just so happens that Math HAS a definition of infinity and it means that 0.9~ =1. Perhaps if you knew even the basics of Real Analysis you would have known this.

This is a philosophical problem not a mathematical problem. It is to do with the philosophical meaning of infinity not any mathematical definition of it.

You could not be more wrong. This is a property of the Real Number system, This system has been very carefully constructed on well known and understood axioms followed by careful and through proofs of each and every theorem.

It's like saying: well if this were possible then this would equal 1. But its not possible.

News to me, and every mathematician in the world, perhaps you know something we don't or.... Just maybe...

You don't know a LOT that Mathematicians DO KNOW.

[steersman]

You could not be more wrong. This is a property of the Real Number system, This system has been very carefully constructed on well known and understood axioms followed by careful and through proofs of each and every theorem.

Maths has limited utility in answering this problem because it's answer does not correspond with reality. That's why I call it a philosophical problem. I do concede though that in maths 0.9~=1. But if that's all you care about then you'll never learn anything.

[Integral]

The problem has meaning ONLY in Math. Out side of the definitions of math the the string of symbols .999... has no meaning what so ever.

[steersman]

Noone said this was a math problem. It could easily be a philosophical problem. Indeed, many philosophers have pondered this very question in different words and symbols

[Hurkyl]

Well, the fact that 0.9~ = 1 (and everything about the decimals!) is chosen so that they're a model of the real numbers. (*sigh* "real" was a poor choice of name, but anyways...)

The real numbers are defined to be a complete ordered field.

To put it loosely, "ordered field" simply means that +, -, *, /, and < all work "properly". The definition is "complete" is more difficult, and is not needed for what follows.


The first thing to notice is that there cannot be any numbers between 0.9~ and 1; if I change any of the digits of 0.9~ into something other than a 9, I'll get a smaller number.

The next thing to notice is that, if we assume 0.\bar{9} \neq 1:


0.\bar{9} < \frac{0.\bar{9} + 1}{2} < 1


which contradicts the fact that there are no numbers between 0.9~ and 1.


Thus, in order for the decimal numbers to fulfill the purpose for which they are created... that is, to be a model of the real numbers... it cannot be the case that 0.\bar{9} \neq 1. In other words, 0.9~ = 1 must be true.



There are, indeed, a lot of interesting philosophical questions that relate to mathematics, but they aren't about whether 0.9~ = 1 in the decimal numbers.

[wtfc]

I'm sure most of you already know this. The real point of the thread is finding different ways of approaching it. You see, I have a friend who refuses to accept what seems to me to be so obvious: .9 repeating (infinite 9s after the decimal) is exactly equal to the whole number 1.


Here is how I look at it.
You start of with .9999.... and continue until you either run out of time or get tired. Now you are at a crossroads. At this stage you decide if it is still .9999.... or 1. Conclusion is what one reaches when one is out of time. Just because you are tired of travelling through the infinite, .9999 does not equal 1 irrespective of all the mathematical equations.

Rounding off is only to solve equations. To further extend this, you either use whole numbers or fractions. You should not criss-cross domains like this.

[Hurkyl]

You start of with .9999.... and continue until you either run out of time or get tired.

What is there to "continue"? .9999.... is a decimal with an infinite number of nines; it is not a suggestion that you start with .9, then go to .99, then to .999, and continue in this fashion.

[balkan]

i can't wait till scientists figure out the smallest quantization of all matter and then use it to construct a new mathematical system without infinities :biggrin:
that'll rid the world of all this nonsense... or at least, i'll be a very happy man...

[Prometheus]

What is there to "continue"? .9999.... is a decimal with an infinite number of nines; it is not a suggestion that you start with .9, then go to .99, then to .999, and continue in this fashion.

I sympathize with people who have trouble understanding infinity in mathematics.

Mathematical infinites exist by definition. It is not that there is any actual correlate to these infinities in the real world. For example, the series of numbers, 1, 2, 3, ..., is not infinite in the real world, where real people count them. Numbers are only infinite in the world of mathemeatics, where they can be defined as infinite, and thereby given attributes of this infinity.

When a person attempts to add actual instances of the number 9 to the end of this number, the world of mathematics is being substituted by the real world, and this is a mistake.

[Hurkyl]

i can't wait till scientists figure out the smallest quantization of all matter and then use it to construct a new mathematical system without infinities

(a) The real numbers are a mathematical "system" without "infinities".
(b) There are lots of other "system"s, each with their own pros and cons.
(c) You say it like the use of "infinities" is bad.
(d) What does science and quantization of matter have to do with anything?

[hello3719]

i can't wait till scientists figure out the smallest quantization of all matter and then use it to construct a new mathematical system without infinities :biggrin:
that'll rid the world of all this nonsense... or at least, i'll be a very happy man...

I'll take that as a joke.

[Prometheus]

I'll take that as a joke.

Are you laughing at his phraseology, or at the deeper meaning that he is trying to convey?

Or, do you understand the difference?

[loseyourname]

He's laughing because there is nothing nonsensical about infinity in mathematics. In fact, infinity, and particularly infinite sequences and series, can be quite useful.

[HallsofIvy]

He's laughing because there is nothing nonsensical about infinity in mathematics. In fact, infinity, and particularly infinite sequences and series, can be quite useful.


and has nothing to do with physical discoveries.

[balkan]

well, to elaborate:
i'm perfectly fine with the concept of infinities within contemporary mathematics, it works fine for now and is quite usefull...
but i suspect that a full understanding of physics can only be made through a parallel evolvment of a new mathematical system... and i suspect this system not to involve infinities due to the simple fact that nothing in this universe is infinite, as far as we know... most likely not infinitely small either... only time will tell though...

the old mathematical system will still be appropriate for most tasks, just like classical physics compared to quantum mechanics, and it will have coherence, but it will not be usefull to describe the mechanisms of the universe to a sufficiently precise degree...

[hello3719]

well, to elaborate:
i'm perfectly fine with the concept of infinities within contemporary mathematics, it works fine for now and is quite usefull...
but i suspect that a full understanding of physics can only be made through a parallel evolvment of a new mathematical system... and i suspect this system not to involve infinities due to the simple fact that nothing in this universe is infinite, as far as we know... most likely not infinitely small either... only time will tell though...

the old mathematical system will still be appropriate for most tasks, just like classical physics compared to quantum mechanics, and it will have coherence, but it will not be usefull to describe the mechanisms of the universe to a sufficiently precise degree...

hahaha, This would mean that our "old"(includes finite and infinite concepts) mathematical system would be more complete than the "new"(includes only finite concepts) one.

[Didd]

use your calculator to calculate 1/9.
It is 0.1111.......... and then 9/9=9*1/9=9*0.1111... but this means adding 0.1111...
itself 9 times. You will get 0.9999.......
so, 9/9=9*1/9=0.9999.........
But 9/9=1. So 1=0.999.........

[Hurkyl]

And I'll reiterate; the theory of the real numbers has no infinities, and the only infinitessimal is 0.

[balkan]

hahaha, This would mean that our "old"(includes finite and infinite concepts) mathematical system would be more complete than the "new"(includes only finite concepts) one.

it would be more "complete", yes, but not more precise. i'm not just talking about a new array of functions, i'm talking about an entirely new system...
just like quantum mechanics are more precise dispite the fact that it deals with quantization and probability, where classical physics claims to predict precise locations and have fluent change of energy levels...
ironically, the "imperfect"/"unprecise" theory is more correct...

hurkyl:
i know, but that's not what i'm talking about... i'm talking about a new way of calculating...

[loseyourname]

Kind of like the way cellular automata have been used recently to model biological systems?

[balkan]

Kind of like the way cellular automata have been used recently to model biological systems?

are you asking me?
in that case, i'd say... erhm... something like it... but cellular automata uses regular math of course... on the other hand, cellular automata is time dependant and deals with uncertainties aswell, am i right? so i guess it could be paralleled...
it's hard to imagine another kind of math, since we're so used to think in the old mathematical patterns, but i think at some point it migth become usefull or even neccessary to evolve it, just like classical physics...

[Hurkyl]

Speaking of quantum mechanics seems to be antithetical; mathematically, QM didn't replace anything, but it did push existing analysis to its limits (haha) and spurred developments in the subject. As far as I know, LQG and ST do the same.

[balkan]

Speaking of quantum mechanics seems to be antithetical; mathematically, QM didn't replace anything, but it did push existing analysis to its limits (haha) and spurred developments in the subject. As far as I know, LQG and ST do the same.

did i say it replaced anything? no... fine then... did i say the new math would replace the old? certainly not...
but it didn't just push existing analysis to its limits. classical physics were 100% unable to explain the phenomenoms that quantum mechanics did... so it did much more than that...

seriously, what is your problem with the idea of a new way of calculating? like i said, it should be coherent, so it wouldn't replace what you seem to hold so dear...

[Hurkyl]

classical physics were 100% unable to explain the phenomenoms that quantum mechanics did

Yes, but I was talking about the mathematics, remember?


seriously, what is your problem

I get the impression that you are calling for a change, not for any particular theoretical or practical reason, but because for whatever reason you think "infinities" are "bad".

This sort of affliction is far too common, and is a great contributor to the popular misunderstanding of mathematics and science, so I like to spend a good deal of effort trying to combat it. If I'm seeing it where it does not exist, I apologize.

[hello3719]

are you asking me?
in that case, i'd say... erhm... something like it... but cellular automata uses regular math of course... on the other hand, cellular automata is time dependant and deals with uncertainties aswell, am i right? so i guess it could be paralleled...
it's hard to imagine another kind of math, since we're so used to think in the old mathematical patterns, but i think at some point it migth become usefull or even neccessary to evolve it, just like classical physics...

Mathematics is not at all like physics. In mathematics we establish the axioms, in physics, we do not even know if there is any or what are the axioms of the system.That's why we can develop ANY theory that is compatible with experiments, observations and the deductions. So even we change the mathematical system we would have to add axioms which won't make it a new kind of math.

[balkan]

Yes, but I was talking about the mathematics, remember?

I get the impression that you are calling for a change, not for any particular theoretical or practical reason, but because for whatever reason you think "infinities" are "bad". If I'm seeing it where it does not exist, I apologize.
you are... infinities works splendidly... i would hate to do quantum mechanics without it... it is for a practical change however... to fully develop string theory e.g., many physicists believe a new mathematical system is neccessary...

Mathematics is not at all like physics. In mathematics we establish the axioms, in physics, we do not even know if there is any or what are the axioms of the system.That's why we can develop ANY theory that is compatible with experiments, observations and the deductions. So even we change the mathematical system we would have to add axioms which won't make it a new kind of math.
this makes no sense... since we don't know if there are axioms at all, we can develop any model and any theory using our mathematical system, which has inevitably got axioms involved?

anyway...
we wouldn't have to "add" axioms... the new system should simply just be backwards compatible with the new system (so that a proper adjustment of the new system would give the same results as working in the old one). But it should be more capable of explaining the physical phenomenons that seems to care **** about being in the dimension visible to humans...
and that's my entire point: there are phenomenons which a are very difficult to model and create theory for using our dimensional dependent math system...

look, i'm not expecting you to understand what i mean. it's obviously not that easy to swallow, but just like new concepts like the integral, infinity and zero have been involved, so can other new concepts... perhaps these concepts will be developed so that our present math system cannot copy it very well, while the new system could be used like the old one...

[arivero]

0.999999999... is not equal to the whole number 1 but to the real number 1.000000000000...

Check the definition of real number; a pair of converging sequences is actually involved.

[selfAdjoint]

Actually you don't have to sum the series. The completeness property of the real numbers guarantees there is a real number, which you can prove to be unique, as the least upper bound of the set of numbers { xi : xi = 1.999...999 for i decimal places} . Proving that the number is neither greater nor less than 2 is then easy.

[hello3719]

you are... infinities works splendidly... i would hate to do quantum mechanics without it... it is for a practical change however... to fully develop string theory e.g., many physicists believe a new mathematical system is neccessary...


this makes no sense... since we don't know if there are axioms at all, we can develop any model and any theory using our mathematical system, which has inevitably got axioms involved?

anyway...
we wouldn't have to "add" axioms... the new system should simply just be backwards compatible with the new system (so that a proper adjustment of the new system would give the same results as working in the old one). But it should be more capable of explaining the physical phenomenons that seems to care **** about being in the dimension visible to humans...
and that's my entire point: there are phenomenons which a are very difficult to model and create theory for using our dimensional dependent math system...

look, i'm not expecting you to understand what i mean. it's obviously not that easy to swallow, but just like new concepts like the integral, infinity and zero have been involved, so can other new concepts... perhaps these concepts will be developed so that our present math system cannot copy it very well, while the new system could be used like the old one...

Do you know what is an axiom ? I guess you can't swallow well what i said.

you are... infinities works splendidly... i would hate to do quantum mechanics without it... it is for a practical change however... to fully develop string theory e.g., many physicists believe a new mathematical system is neccessary...

Define "new mathematical system ", we are not on the same track.

[balkan]

Do you know what is an axiom ? I guess you can't swallow well what i said.

Define "new mathematical system ", we are not on the same track.

yes i do know what an axiom is... but your sentence made no sense...
"new mathematical system" like imaginary number theory was... just "bigger"... it can explain phenomenoms that the "old" theory can't, but is still coherent with the old... i wouldn't quite call it a new paradigm or anything, since it would still just be "math"...
they're trying it with string theory, but i'm having my doubts about it... but you gotta start somewhere, right?

[MathematicalNewbie]

I was doing some research about .999~ = 1 tonight and I came across this page. I agree completely that .999~ is indeed equal to 1. However, I have question. Based on what I read, I concluded that .999~ can be expressed as an infinite geometric series such as 9/10 + 9/100 + 9/1000... etc. From what I've learned in math class, this should converge to 1, but not equal it. However, the proofs still show that .99~ is 1. I was wondering that since this series "approaches" 1, but still equals it, then can the sum of any infinite geometric series actually equal a sum instead of converging towards one? For example, if we are given a list of numbers in a series and I find that the sum of these numbers converges to say.. 30, then would there be a way I could prove that the sum is EQUAL to 30?

I may be confused about the whole concept of series in general, so please correct me if I'm wrong about anything here.

[chroot]

If a sequence converges to a number, then, when summed over an infinite number of terms, it would equal that number.

This is the meat of the entire debate that usually surrounds the 0.999... = 1 issue. 0.999... only equals one when you tack on an infinite number of nines, which makes some people uneasy. There is nothing mathematically wrong with summing an infinite number of terms, however, no matter how conceptually difficult it is for some people to grasp. The notation 0.999... specifically means an infinite number of nines, and is equal to one.

- Warren

[Hurkyl]

I find that the sum of these numbers converges to say.. 30, then would there be a way I could prove that the sum is EQUAL to 30?

Yes, it's true by definition. The term "sum of an infinite series" means "the value to which the partial sums converge".

So, if the partial sums converge to 30, then the sum is equal to 30.

[MathematicalNewbie]

Yes, it's true by definition. The term "sum of an infinite series" means "the value to which the partial sums converge".

So, if the partial sums converge to 30, then the sum is equal to 30.

Well, I guess that answers my question, but I still wonder why my math teacher specifically marks points off for writing "the sum of the series is 30" as opposed to "the sum of the series approaches 30 or converges to 30." :confused:

[unicorn]

I am new around here, so I don't know if anybody presented this already, but here goes:
0.999999...=x
10x=9.99999...=9-0.99999...=9-x
9x=9
x=1
And yes, sum of infinite geometrical series is one particular value when b(n+1)/b(n)<1. It's like in Zono's paradox with Achileus and a turtlle...

[musky_ox]

IMO its just rediculous to try to say that 0.9 repreating = 1. We KNOW that it is infinately close to, but will NEVER actually EQUAL 1. Maybe using mathematics we can say that, but it is not logically true, so there must be flaws in mathematics. You can give all the arguements you want to try and prove it, but its kind of a circular arguement. You have to assume that math is flawless, and i dont see why would would assume this when it contradicts itself.

Also, wierd things happen with infinity. For example, what is infinity/intinity... what is infinity/infinity squared. How do you enter notation in this forum? :mad:

[chroot]

IMO its just rediculous to try to say that 0.9 repreating = 1. We KNOW that it is infinately close to, but will NEVER actually EQUAL 1.
It will be equal to one when there are an infinite number of nines, which is precisely what the notation 0.9... means.
Maybe using mathematics we can say that, but it is not logically true, so there must be flaws in mathematics.
Mathematics does not have flaws. It begins with a set of axioms, and derives a set of conclusions. You can start with different axioms if you like, and you'll get different conclusions. The axioms normally used in mathematics can be used to prove 0.9... = 1. There is no question about it, nor is there any room for debate.
You can give all the arguements you want to try and prove it, but its kind of a circular arguement. You have to assume that math is flawless, and i dont see why would would assume this when it contradicts itself.
It does not contradict itself. Once again, all you must do is select your axioms, and the results follow from them.
Also, wierd things happen with infinity. For example, what is infinity/intinity...
Indeterminate.
what is infinity/infinity squared.
Indeterminate.
How do you enter notation in this forum? :mad:
Like this:

\frac{\infty}{\infty}

- Warren

[musky_ox]

It will be equal to one when there are an infinite number of nines, which is precisely what the notation 0.9... means.

1-0.999...=1/infinity

It is infinitly close to 0, but will NEVER reach it. It is like some immortal person writing down zeros forever and ever after the decimal point, with the idea that he is going to write a 1 as his last digit. We know that he will never reach his last digit, but his number will never equal zero, because it isnt complete... even though he gets infinitely close to it.

[chroot]

1-0.999...=1/infinity
It is infinitly close to 0, but will NEVER reach it. It is like some immortal person writing down zeros forever and ever after the decimal point, with the idea that he is going to write a 1 as his last digit. We know that he will never reach his last digit, but his number will never equal zero, because it isnt complete... even though he gets infinitely close to it.
This is a shaky argument, and I suspect you know it. No real person ever needs to be capable of writing down an infinite number of zeros to make an expression true. The operative word is if. If a person could write down an infinite number of nines, the resulting number would be equal to one. That's what the notation "0.99..." means, regardless of whether or not a human hand can write it out digit for digit.

- Warren

[HallsofIvy]

Not an argument at all! 1- 0.999... "is infinitely close to 0 but will never reach it" only means that the poster does not understand what 0.999... (or 1- 0.999...) means. Numbers are fixed, specific things- they are not "reaching" something and don't depend on any person, real or hypothetical, writing down digits. Digits are something we use for convenience- every number has an existance of its own completely independent of "digits". 0.999... IS 1.0 because the notation "0.999..." means "the limit of the sequence 0.9, 0.99, 0.999, etc." Notice that the definition is clear: an infinite decimal is the limit of such a sequence, not the sequence (a string of digits) itself.

[Tom McCurdy]

It seems like the same principles used here are the ones used to solve zeno's paradox... I posted some of zeno's paradoxes in another part in the math forum... correct me if I am wrong.

[chroot]

Tom,

Yep, that's correct. A proper understanding of the sum of infinite series solves this problem and Zeno's paradoxes the same way. An infinite series can have a finite sum.

- Warren

[Rogerio]

Well, 0.999... is just a representation.
It means infinite 9's after the decimal point. It's not a growing number - all the 9's are already there! It's not required a high level of abstraction to understand it (or is it?!), and children commonly accept it.

So, anyone who can understand that
1/3 = 0.333... ,
should understand that
3 * 1/3 = 0.999... , too.

And then, they should understand that
1 = 3 * 1/3 = 0.999...

But, incredibly, some people have serious difficulties in understanding these equalities. Personally, I believe if an adult refuses to accept the first equality, then even all the arguments in the world are not enough to go beyond.

[Integral]

I think that in general the people who have trouble with these concepts cannot see infinity as anything but a large finite number. Until they can be convinced that infinity is NOT simply a large finite number there is no argument that they will be able to comprehend.

[musky_ox]

Okay let me answer this dumb thread for you guys. 0.9 repeating does NOT =1. The problem comes in the base 10 math system. 1/3 is not exactly 0.3 repeating. 1/3 cannot be accurately represented by the base 10 system, base 12 would be much better than base 12 since we could represent 1/3, and 1/4.

I can understand that infinity means that it has forever 9s. It has no limit to the number of 9s after the decimal point. However, what some thick headed idiots dont realize is that 1 is a limit.

[Integral]

You have just called everyone with a reasonable Math degree a thick headed idiot. I take that personally and have issued you a warning for insulting behavior.

Please take a math course beyond high school calculus.

Edit: Meanwhile, study the proof starting on page 2 of the link I posted in Post #5 of this thread. It does not rely on taking any limit, but does rely on the definition equality on the real number line.

[musky_ox]

Okay 1 = 5, 3 = 7. Whatever you want to think. However, if 0.9 repeating = 1 then they wouldnt have named them differently in the first place. They are 2 numbers that could be represented as 2 lines, one line being just a 'point' longer than the other.

[Hurkyl]

However, if 0.9 repeating = 1 then they wouldnt have named them differently in the first place.

Ah, I see. So \frac{6}{4}, \frac{3}{2}, 1\frac{1}{2}, and 1.5 are all different numbers! I can't believe I thought they were all the same. :uhh:


They are 2 numbers that could be represented as 2 lines, one line being just a 'point' longer than the other.

How can a number be represented as a line? Oh, I see, you meant line segment. Well, as I'm sure you know, between any two distinct points there is another point. If both have their left endpoint at the origin, what lies between the right endpoint of the line segment representing 0.999... and the line segment representing 1?

[musky_ox]

Consider this inequality.

1 - .1^n < x < 1+ .1^n

It seems clear that there is only a single number for which this is true for ALL values of n>0, x= 1 . This is a simple statement that any number added to x results in something greater then 1, or any number subtracted from x results in something less then one. I doubt that you will find many people who will argue with the truth of the statement, x=1.

Now, in the link I posted above, I show that using simple arithmetic, involving only valid rational numbers, one can construct this inequality.

1- .1^n < .999.... < 1+.1^n

notice that is the exact inequality as above, thus we have x=.999... If the original statement is correct that the inequality can only be satisfied by 1 you must be lead to the conclusion that
1=.999...

I dont get what you are saying here. It seems to me that while 1- .1^n = .999.... maybe be true, this doesnt make sense... 1- .1^n < .999.... < 1+.1^n. You are saying that 1cm - (and infinitely small amount of space aka a point) is greater than 0.999...cm, which is the same thing as far as i can tell.

[musky_ox]

Ah, I see. So \frac{6}{4}, \frac{3}{2}, 1\frac{1}{2}, and 1.5 are all different numbers! I can't believe I thought they were all the same. :uhh:




How can a number be represented as a line? Oh, I see, you meant line segment. Well, as I'm sure you know, between any two distinct points there is another point. If both have their left endpoint at the origin, what lies between the right endpoint of the line segment representing 0.999... and the line segment representing 1?

Alright, maybe i have a bad example there, if space is quantisized. Anyways, since mathematics is abstract, i dont see why there cant be an infinitely small amount. Wouldnt this be 1/infinity? If you had an infinite amount of 1/infinities you would get 1, but if you have (infinity - 1) amount of 1/infinities you would get 0.9... so there is a difference theoretically.

[Integral]

I dont get what you are saying here. It seems to me that while 1- .1^n = .999.... maybe be true, this doesnt make sense... 1- .1^n < .999.... < 1+.1^n. You are saying that 1cm - (and infinitely small amount of space aka a point) is greater than 0.999...cm, which is the same thing as far as i can tell.
Simply pick an n, any n in the integers and do the arithmetic. The whole point is that for ANY n you choose (you must pick an n) the relationship holds.

Please show me an integer for which the bolded statement holds? Remember that infinity is not a valid integer, or is it a valid real number. I am not talking about anything physical, we are discussing math and not physics here.

[musky_ox]

Simply pick an n, any n in the integers and do the arithmetic. The whole point is that for ANY n you choose (you must pick an n) the relationship holds.

Please show me an integer for which the bolded statement holds? Remember that infinity is not a valid integer, or is it a valid real number. I am not talking about anything physical, we are discussing math and not physics here.

Well you say that n cannot be infinity, but we are dealing with infinity and infinitly small values here so i dont see how you can have this stipulation on the equation...

[Hurkyl]

Anyways, since mathematics is abstract, i dont see why there cant be an infinitely small amount.

Just because it's abstract doesn't mean you can do whatever you please. Mathematical objects and ideas usually have very precise definitions.The definition of the real numbers (and thus the decimal numbers) permits no infinities or infinitessimal numbers.

If you so desired, you could start talking about some different abstract idea in which you do have infinities and infinitessimal numbers. You may or may not be able to show that 0.999... is different than 1 in one of these different abstract systems. But the point is that you're speaking about something other than the real numbers.

Incidentally, in one of the more useful alternate systems, the hyperreal numbers, there are two analogues of 0.999...: you could have a hyperdecimal number that has some specified transfinite number of 9's (followed by zeroes). Such numbers would then be infinitessimally close to 1, but not equal.

However, the direct analogue of 0.999... is the one with a 9 in every position, and it is equal to 1, just as in the reals.

[musky_ox]

So basically you are saying that mathematics (or whatever system of it we are talking about) is incapable of handling infinities, and is thus not capable of representing our universe? Then what is the use of everyone studying it, if there are other systems that can better account for everything in life? :confused: In higher level mathematics/physics do they switch to different systems so we can handle the infinite?

[musky_ox]

In base 5, 1/4 = (0.111...), and 4·(0.111...) = (0.444...). In every base, dividing 1 by the largest digit of the system gives 0.111... In hex, 1/(F) = (0.111...), and (F)·(0.111...) = (0.FFF...).

0.999... = 1
(0.444...)5 = 1
(0.FFF...)16 = 1

Its just a flaw that exists in any base system, because you can never acount perfectly for all the fractions. All you are proving is that the mathematical system rounds 1/3 to 0.33....

[Integral]

I am not sure how you arrive at that conclusion? Why cannot we handle the infintiies correctly? Perhaps it is you who cannot deal with the infinities correctly?

[musky_ox]

Well i can see that 0.99... is infinitely close to 1. It is just the way that our base system rounds up 0.33... to be 1/3. Obviously whatever mathematics we are talking about cannot handle infinity if it has to precisely say that infinity doesnt have a place in it. We know that infinity has a place in abstract ideas and in the universe.

[Hurkyl]

So basically you are saying that mathematics (or whatever system of it we are talking about) is incapable of handling infinities

I'm saying this: zero is the only infinitely small real number, and no real number is infinitely large.


Well i can see that 0.99... is infinitely close to 1.

Well, you see wrong. Or, more precisely, your intuition is not in agreement with the definitions.


Its just a flaw that exists in any base system, because you can never acount perfectly for all the fractions.

No, this is a flaw in your version of the decimals. The very purpose of the decimals is to exactly represent any real number (and thus all fractions). The properties of the decimals were carefully chosen so they fulfill this purpose.

You state that 0.333... is not an exact representation of 1/3, and similarly for other decimals, but you will always get exactly the right answer if you replace your fractions with decimals, compute, then convert back to fractions. (this includes using 0.999... = 1, and similar equalities)



We know that infinity has a place in abstract ideas and in the universe.

And that place is not as an element of the real numbers.

[Rogerio]

It is just the way that our base system rounds up 0.33... to be 1/3. Obviously whatever mathematics we are talking about cannot handle infinity if it has to precisely say that infinity doesnt have a place in it.

Commonly, anyone who knows divide by hand, knows that 1/3 is 0.33... EXACTLY!
The three points at the end (...) means infinite 3's , and there is no rounding here.
It is not the same result we get with a calculator.

[Locrian]

So basically you are saying that mathematics (or whatever system of it we are talking about) is incapable of handling infinities, and is thus not capable of representing our universe? Then what is the use of everyone studying it, if there are other systems that can better account for everything in life? :confused: In higher level mathematics/physics do they switch to different systems so we can handle the infinite?

There are tools to deal with infinities. For instance, limits. You probably learned about them in high school.

The original poster, however, clearly did not.

[musky_ox]

Yea sure i learned about limits. However, in limits you are saying that the limit to something is for example 0 as you approach x=infinity, not that it actually equals 0. Anyways, limits has nothing to do with it. Here is what im saying.

"In base 5, 1/4 = (0.111...), and 4·(0.111...) = (0.444...). In every base, dividing 1 by the largest digit of the system gives 0.111... In hex, 1/(F) = (0.111...), and (F)·(0.111...) = (0.FFF...).

0.999... = 1
(0.444...)5 = 1
(0.FFF...)16 = 1

Its just a flaw that exists in any base system, because you can never acount perfectly for all the fractions." You can only pick on this example as long as you use the decimal system.

[chroot]

You cannot find a number between 0.999... and 1, and therefore, by the definition of the real numbers, 0.999... and 1 are the same number. It's just that easy.

- Warren

[Hurkyl]

you can never acount perfectly for all the fractions.

I challenge this. Would you care to demonstrate an actual error that arises from using decimals instead of fractions, even if you accept 0.999... = 1?

[musky_ox]

I can think of a number between them...

0.99...9 + 0.00...1 = 1
0.99...9 + nothing = 0.99...9

[musky_ox]

I challenge this. Would you care to demonstrate an actual error that arises from using decimals instead of fractions, even if you accept 0.999... = 1?

The error is in the tread name:

"0.99... = 1"

In base 12, 1/3 is a terminating number... so you dont get the small rounding error when you use it for calculations.

[Hurkyl]

I can think of a number between them...

0.99...9 + 0.00...1 = 1
0.99...9 + nothing = 0.99...9

Warren didn't ask for a number between 0.{terminating sequence of 9's} and 1. He asked for a number between 0.{9 in every allowable position} and 1.

(And, incidentally, you didn't produce a number between 0.99...9 and 1)



In base 12, 1/3 is a terminating number... so you dont get the small rounding error when you use it for calculations.

This doesn't address my challenge. Maybe if I restate it, it will be more clear.


I am asking you to produce an arithmetic calculation involving only fractions.
It must have the property that, if I convert the fractions to decimals, do all the arithmetic according to decimal arithmetic, then convert back to a fraction (including the use of 0.999... = 1), I get the wrong answer.

[musky_ox]

Warren didn't ask for a number between 0.{terminating sequence of 9's} and 1. He asked for a number between 0.{9 in every allowable position} and 1.

(And, incidentally, you didn't produce a number between 0.99...9 and 1)

Okay try this then. Using your logic:

0.99...9 = 0.99...8
0.99...8 = 0.99...7

Now, i can think of a number between 0.99...9 and 0.99...7, which is 0.99...8.

In base 12, 1/3 is a terminating number... so you dont get the small rounding error when you use it for calculations.

This doesn't address my challenge.

I am challenging you will this. I have just shown you the answer to your "challenge." 0.99... = 1 is the error that it causes.

[Hurkyl]

It's easy to find a number between two different terminating decimals of the same length: simply append a 5 to the lesser of the two.

0.999... is not a terminating decimal. It has a 9 in every allowed position. There is no place left to put a 5.

[chroot]

Okay try this then. Using your logic:

0.99...9 = 0.99...8
0.99...8 = 0.99...7
But these numbers do not (and cannot) exist. There is no such thing as a number with an infinite number of nines, followed by an eight. There is no position to put an eight in "after" an infinite number of nines, because an infinite number of nines never ends.

- Warren

[musky_ox]

They dont exist? Okay. So an infinitly large number exists, but an infinitly small number doesnt? Theoretically, what is 1/infinity? Its limit is 0, but we know that it will never reach 0, doesnt this make it 0.00...1?

Its like my example with the guy counting. He is counting out 0 forever after his decimal point, always with the concious idea that he is going to say 1 as his last digit, even though he will never make it.

Also, please address my posts about using base 12 to eliminate the 1/3 glitch.

[BoulderHead]

I can think of a number between them...

0.99...9 + 0.00...1 = 1
0.99...9 + nothing = 0.99...9
Let me guess...somewhere, wayyyyy out there at the end of infinity there's room for just one more digit ?

:rofl: :rofl:

[Hurkyl]

Okay. So an infinitly large number exists, but an infinitly small number doesnt?

No, no infinitely large (real) number exists, and no infinitely small nonzero (real) number exists.

[musky_ox]

How do you get from 1.00... to 1.10...? Do you not have to first pass through 1+infinitesimal to get there?

[musky_ox]

No, no infinitely large (real) number exists, and no infinitely small nonzero (real) number exists.

So how many 9s are after the decimal point in 0.99...? There is no real number for them, so the number cant exist?

1 2 3 4 5 6 ... infinity
2 3 4 5 6 7 ... infinity + 1 = ?

Why cant we add an 8 after the 9s? Would there still be an infinite number of decimals after the decimal point?

[Hurkyl]

So how many 9s are after the decimal point in 0.99...?

I don't know*. All I know is that there is a 9 in every legal position.


Why cant we add an 8 after the 9s?

There is a 9 in every legal position in the decimal 0.999...; in particular, there is no "after". Any decimal number (with a 0 to the left of the decimal point) with an 8 in it is smaller than 0.999..., because 0.999... has a 9 where that number has an 8.



*: actually, I do know, but this question is irrelevant to the discussion at hand. A different number system, called cardinal numbers, is used for counting the "size" of things. This number system is an extension of the natural numbers, and has rather poor arithmetic. For example, x + x = x for nearly every cardinal number.

[chroot]

So how many 9s are after the decimal point in 0.99...? There is no real number for them, so the number cant exist?
Infinity exists, it's just not a real number -- specifically, I mean [itex]\infty \nin \mathcal{R}[/tex].

1 2 3 4 5 6 ... infinity
2 3 4 5 6 7 ... infinity + 1 = ?
The notation "infinity + 1" is nonsense.
Why cant we add an 8 after the 9s?
Because there are an infinite number of nines. No matter where you stuck the eight, you'd by definition no longer have an infinite number of nines before it.

- Warren

[musky_ox]

1) How do you get from 1.00... to 1.10...? Do you not have to first pass through 1+infinitesimal to get there?

2) In base 12, 1/3 is a terminating number... so you dont get the small rounding error when you use it for calculations. If 0.99... truely equals 1, then it wouldnt be confined to just our base 10 (decimal) system.

[chroot]

1) How do you get from 1.00... to 1.10...? Do you not have to first pass through 1+infinitesimal to get there?
"1 + infinitesimal" doesn't exist, so the question is meaningless.
2) In base 12, 1/3 is a terminating number... so you dont get the small rounding error when you use it for calculations. If 0.99... truely equals 1, then it wouldnt be confined to just our base 10 (decimal) system.
Numbers are numbers, no matter what base you put them in. Numbers exist independently of representation, as has been shown with numbers like 6/4 and 1.5. There is no "rounding error" involved anywhere.

The fact that 1/3 is a terminating number is base 12 actually helps our case, not yours. If 0.333... did not truly equal 1/3, then it would truly not equal 1/3 in any base, not just decimal.

- Warren

[chroot]

Besides, musky_ox, you have still not answered my question:

Can you find a number between 0.999... and 1?

- Warren

[musky_ox]

1 - 0.99... = infinitesimal

As you say that a nine occupies every space in 0.99..., in infinitesimal, a 0 occupies every space except the last one. I dont see how infinity is any more real than infinitesimal.

[chroot]

1 - 0.99... = infinitesimal
No, it equals zero, because 1 and 0.999... are the same number.

- Warren

[Hurkyl]

in infinitesimal, a 0 occupies every space except the last one.

Why do you think there is a "last space"?


I dont see how infinity is any more real than infinitesimal.

I repeat, the real number system has neither infinite numbers nor infinitessimal numbers.


Incidentally, I get the feeling the terminology is misleading you; the "real" in "the real numbers" is unrelated to the English word "real".

[musky_ox]

asymptote

\As"ymp*tote\ (?; 215), n. [Gr. ? not falling together; 'a priv. + ? to fall together; ? with + ? to fall. Cf. Symptom.] (Math.) A line which approaches nearer to some curve than assignable distance, but, though infinitely extended, would never meet it

So is there no such thing as an asymptote?

Incidentally, I get the feeling the terminology is misleading you; the "real" in "the real numbers" is unrelated to the English word "real".

So 0.99... is not a real number in the first place? Math must deal with more than just real numbers then.

[chroot]

So is there no such thing as an asymptote?
What? Why are you bringing another concept into this muddled discussion? Please answer the questions that have already been asked of you, rather than trying to complicate it further.

- Warren

[chroot]

So 0.99... is not a real number in the first place? Math must deal with more than just real numbers then.
0.999... is a real number in the sense that it is a member of the field \mathbb{R}[/tex]. Infinity is not a real number in the sense that it is a not a member of the field [itex]\mathbb{R}. That is what mathematicians mean by the term "real." That is what we in this thread mean by the term "real."

- Warren

[musky_ox]

I see no questions to answer.... And no, im not bringing a totally unrelated topic in here. Read the definition of an asymtote. By your reasoning, there is no such thing as one. Id like to know so next time someone starts talking about an asymptote i can tell them that it doesnt actually exist because there is no such thing as infinitely approaching a number without being defined at it.

[chroot]

I see no questions to answer.... And no, im not bringing a totally unrelated topic in here. Read the definition of an asymtote. By your reasoning, there is no such thing as one. Id like to know so next time someone starts talking about an asymptote i can tell them that it doesnt actually exist because there is no such thing as infinitely approaching a number without being defined at it.
From what I can tell, this has nothing to do with the discussion to this point. I fear you are misreading what people are saying.

The only question I have to ask you is this one:

Can you find a number between 0.999... and 1?

- Warren

[musky_ox]

Sorry i looked for a graph of this but couldnt find one.

Say you graph a function that has a horizontal asymptote of 1. By definition, as the x value goes to infinity, the y value is infinitely approaching 1, but will never be 1. I see no reason that just because i cannot identify a number between 0.99... and 1 that you can say it equals 1.

[chroot]

By definition, as the x value goes to infinity, the y value is infinitely approaching 1, but will never be 1.
Except at infinity, where it (might) be 1.

The notation 0.999... does not mean "a lot of nines," it means "an infinite number of nines," which is completely different.

- Warren

[chroot]

BTW, I must mention that Webster's dictionary is an abhorrent place to find the definitions of mathematical or scientific words. Rarely are the definitions provided in a common English dictionary adequately precise for technical communication.

- Warren

[musky_ox]

I never said that it was "a lot of nines." I said as x-> infinity that y is infinitely close to 1. (infinitesimal away from 1)

Think of this analogy.
Lets assume that space is quantisized and just say that a quark is the smallest distance of space. Does length of 1 quark = 2 quarks just because there is no length between it?

[chroot]

I never said that it was "a lot of nines." I said as x-> infinity that y is infinitely close to 1. (infinitesimal away from 1)
And its behavior as it goes to infinity might have nothing at all in common with its behavior at infinity.
Think of this analogy.
Lets assume that space is quantisized and just say that a quark is the smallest distance of space. Does length of 1 quark = 2 quarks just because there is no length between it?
We're not talking about physics, for the last time. We're talking about pure math. The real number line is not quantized, and there is no "smallest number."

I'm going to ask you again, for the third time:

Can you find a number between 0.999... and 1?

- Warren

[musky_ox]

What is 2(.99...)? Isnt is 1.99...8? So theoretically if you could have this number, would it equal 2? I can think of a number in between it and 2, 1.99....

[chroot]

No, because once again numbers like 1.99...8 do not, and cannot exist! We've already covered this ground. 0.999... does not have a lot of nines, it has an infinite number of nines.

2(0.999...) = 2

I'm really beginning to believe you are just a troll. When things have been explained to you clearly, yet you continue to just repeat yourself and ignore what has been explained, you are trolling.

- Warren

[chroot]

I'm going to ask this question again for the fourth time. If you do not answer it directly, I will take action against you for trolling.

Can you find a number between 0.999... and 1?

- Warren

[musky_ox]

I guess i cant find any number between 0.99... and 1 then. :biggrin: I dont think it has any implication, but it does appear to be a glitch in the decimal system to me. The only reason i kept arguing is that in base 12 it seems to me that doing 1/3*3 gives you the same answer as 0.4*3.

BTW - What do you mean by "Troll?" I got a warning for saying someone had an IQ below 25 (an idiot) and i hope you werent indirectly calling me stupid! :cry:

[chroot]

If you cannot find a number between 0.999... and 1, then 0.999... and 1 are the same number. There are literally dozens of proofs in this very thread, which I suggest you read in its entirety.

Your concept that numbers behave differently in different bases is simply wrong. There is no room to argue this. What is true in one base must be true in all other bases. The definition of the real numbers has nothing to do with what base you choose to represet them in. The definitions deal with the properties of the numbers themselves, independent of representation.

- Warren

[musky_ox]

Alright, i will admit defeat then, however 0.99... only equals 1 mathematically. Theoretically, it is still an infinitely small distance away from being 1. I have a question for you: How can a problem involving infinities be represented with equations?

[chroot]

Alright, i will admit defeat then, however 0.99... only equals 1 mathematically.
What else have we been talking about besides math? This is absurd. :rofl:

- Warren

[musky_ox]

Physics. :biggrin:

- musky ox

[BoulderHead]

I always appreciate it when [other] people can admit defeat. :rofl:

Three cheers for musky_ox not being a diehard !!!

Cheers - Cheers - Cheers

[Hurkyl]

I have a question for you: How can a problem involving infinities be represented with equations?

Generally by figuring out what you really mean to ask. (which usually turns out not to involve anything infinite)

For example, the statement "the limit of f(x) as x approaches infinity is L" is actually defined in a way that doesn't involve infinity at all. The thing it is "really" saying is:

You can make f(x) as close to L as you want simply by picking any sufficiently large x.

The precise definition is: for any positive number e, there is a number M such that:
if x > M, then |f(x) - L| < e.



And as I've already mentioned, when one says "0.999... has an infinite number of nines", what they "really" mean is

0.999... has a 0 in every position left of the decimal point, and a 9 in every position to the right of the decimal point.

[musky_ox]

Thanks Capt'n Obvious. :wink:

I was talking about an asymptote. It is never defined, doesnt matter if x=infinity, it is defined as NEVER being defined.

[arildno]

I was talking about an asymptote. It is never defined, doesnt matter if x=infinity, it is defined as NEVER being defined.
Worthless crap; learn some math.

[phoenixthoth]

Sorry if this has already been posted in some other form but I have to ask this question to anyone thinking that 0.9... does not equal 1:

Ok so .9... != (does not equal) 1. Then express the difference between the two different numbers in decimal form. Since they're different, their difference must be non-zero. Or, if you can't express the difference between them in decimal form, say what decimal that difference must be greater than. For example, if it was .9 and 1, you can say the difference is greater than .0001.

[Adam]

0.9 * 100 = 90

1.0 * 100 = 100

Big difference. Clearly not the same number. However, before you bother bombarding me with the proofs, I'll remind you that I am absolutely terrible at maths, don't know any of it, and this very thread has proofs aplenty of all manner of strange beast.

[Adam]

Does 0.98 = 0.99 then?

[Integral]

Adam,
You are missing a key symbol.

.999... Notice the 3 dots, known as an ellipsis, after the 9s. This is short hand for repeat this pattern (999) infinitely. So

.99 != .98

but it is true that:

.99 = .98999...

That infinite string of 9s makes a difference.

[Adam]

Oops, sorry. That was very silly of me. With the ellipses in use, I forgot that you meant the dot over the last digit instead. We don't use that here.

What is the Latex code for placing the dot over the digit?

Also, can someone show me some more about the reciprocal of that, and the value of that reciprocal?

[Adam]

For that matter, how do I write the reciprocal for an infinite series (0.9 with the dot over the 9)?

[Integral]

lets see if this works.

.99\dot{9}

I am not sure what you mean by the reciprocal? The Reciprocal of what?

[Adam]

1 is to 0.\dot{9}
As 0 is to... ?

Is there some symbol or way to write that it's an infinitesimally small value? Apart from zero, I mean.

[phoenixthoth]

I believe that 0.\bar{9}=1 is what we're after.

As for the reciprocal, the reciprocal of an infinite series is generally difficult. But for our case, let's see...

0.\bar{9}=0.9+0.09+0.009+0.0009+...
0.\bar{9}=9\cdot \frac{1}{10}+9\cdot \left( \frac{1}{10}\right) ^{2}+9\cdot \left( \frac{1}{10}\right) ^{3}+9\cdot \left( \frac{1}{10}\right) ^{4}+...
0.\bar{9}=\sum_{k=1}^{\infty }9\cdot \left( \frac{1}{10}\right) ^{k}

For finite [i]n[i], we have that
ax+ax^{2}+...+ax^{n}=a\frac{x\left( 1-x^{n}\right) }{1-x}
Then to do the reiprocal, we can flip both sides:
\frac{1}{ax+ax^{2}+...+ax^{n}}=\frac{1-x}{ax\left( 1-x^{n}\right) }.

Now take limits of both sides:
\lim_{n\rightarrow \infty }\frac{1}{ax+ax^{2}+...+ax^{n}}=\lim_{n\rightarrow \infty }\frac{1-x}{ax\left( 1-x^{n}\right) }
\frac{1}{\sum_{k=1}^{\infty }ax^{k}}=\lim_{n\rightarrow \infty }\frac{1-x}{ax\left( 1-x^{n}\right) }.

Now let a=9 and x=1/10.

\frac{1}{0.\bar{9}}=\lim_{n\rightarrow \infty }\frac{1-1/10}{9\left( 1/10\right) \left( 1-\left( 1/10\right) ^{n}\right) }. This reduces to
\frac{1}{0.\bar{9}}=\frac{1-1/10}{9\left( 1/10\right) \left( 1-\lim_{n\rightarrow \infty }\left( 1/10\right) ^{n}\right) } and further to \frac{1}{0.\bar{9}}=\frac{9/10}{9\left( 1/10\right) \left( 1\right) } and therefore \frac{1}{0.\bar{9}}=1.

There is a way to represent infinitesimal quantities. In fact, in nonstandard analysis there are no limits. We would say that \sum_{k=1}^{n }9\cdot \left( \frac{1}{10}\right) ^{k} is infinitesimally close to 1 for all unlimited n.

In standard analysis, real numbers are constructed (or can be) from the rational numbers. Your question is similar to "how does one express or write a real number if all that is known are rational numbers?" And we can usually do only a pathetic attempt at representing real numbers because their decimal expansions rarely repeat (in some sense). The only thing I can think of is if you get into "equivalence classes" (which are just divisions of a set into parts) then you can say that a nonzero infinitesimal is given by, for example,
[{1,1/2,1/3,1/4,...}]. This is nonzero and smaller than every real number x which is represented by [{x,x,x,x,x,x,x,x,...}]. Note the brackets around the sequence. That means really it's the set of all things equivalent to that sequence. But what I mean by equivalent is hard to describe (in fact I barely remember). If you're really interested look up nonstandard analysis--goldblatt is i believe a good book--and get ready for some things call ultrafilters.

[Adam]

WTF is an ultrafilter? I'll go look it up...

[Adam]

You know, there really should be a simple symbol or something to show it, rather than writing out a set or such.

[Adam]

Oh, for those who use the dot above the last digit, as I do, teh code is: ( tex ) 0 . \ dot { 9 } ( / tex )
But use the square braces and remove the spaces.

[phoenixthoth]

pi stands for [{3,3.1,3.14,3.141,...}]. If you like, you can let any symbol you choose represent the infinitesimal indicated above. But considering what the "long" symbol stands for, it really is compact notation. However, if it's not compact enough, you can let x equal it and then you have just one letter.

You might want to check out
http://en.wikipedia.org/wiki/Nonstandard_analysis
and
http://en.wikipedia.org/wiki/Hyperreal_number
and
http://en.wikipedia.org/wiki/Infinitesimal

[Adam]

Thanks very much for the work you put into the explanations.

[krusty the clown]

Nobody in my family but me believes that 1/3 is exactly equal to 0.333...
When I asked what it does equal they told me that you can't write 1/3 as a decimal, even after I showed them a paragraph in a mathematics "encyclopedia" that says all common fractions can be represented by either a finite or an infinite repeating decimal.

I looked for the definition of "common fraction" but I couldn't find one; except possibly that a common fraction is a fraction whose numerator has a smaller magnitude than the magnitude of the denominator.

1)Is 1/3 a common fraction?
2)What is the definition of a common fraction?
3)Are there any definitions that show 1/3=.333...?


-Thanks, Erik

[phoenixthoth]

You're welcome.
Nobody in my family but me believes that 1/3 is exactly equal to 0.333...
When I asked what it does equal they told me that you can't write 1/3 as a decimal, even after I showed them a paragraph in a mathematics "encyclopedia" that says all common fractions can be represented by either a finite or an infinite repeating decimal.

I looked for the definition of "common fraction" but I couldn't find one; except possibly that a common fraction is a fraction whose numerator has a smaller magnitude than the magnitude of the denominator.

1)Is 1/3 a common fraction?
2)What is the definition of a common fraction?
I've never heard of a "common" fraction. Maybe they mean common as in a fraction that typically appears on a wrench or in the stock market or a recepie. Truth is any fraction (of integers) has a repeating decimal, common or not.
3)Are there any definitions that show 1/3=.333...?


-Thanks, Erik
Let me give you two ways, one that a third grader would believe (which is not to say that's a bad or good thing) and one that requires some second semester calculus.

1.
Divide 3 into 1 using pencil and paper. You always get the same remainder and there's no reason why the remainder should suddenly change becuase you're working with the same numbers.

2.
0.\bar{3}=0.3+0.03+0.003+0.003+.... Then turn that into an infinite series and sum it. You get 1/3 thus showing that 0.\bar{3}=1/3.

[krusty the clown]

Thanks phoenixthoth, the problem with "common fraction" might be that I got it out of a really old book, but nevertheless all fractions with integers can be written exactly with decimals, correct.

I have already tried explanation one, and they continue to insist that 1/3 can't be written as a decimal. :grumpy:

As I am currently taking calc II and haven't gotten to infinite series I don't fully understand the second way. I see the general idea behind it, but hopefully I can spend some time on this tonight and figure out how to do it myself; and also how to explain it to my parents.

-Thanks, Erik

[phoenixthoth]

Check out specifically geometric series in your book. A geometric sequence has the form a, ax, ax^2, ax^3, ... and a geometric series is the sum of a geometric sequence. All repeating decimals are really geometric series (don't know what the plural of series is). In your book, it will discuss how limits are involved in series and the main point is that an infinite series is the limit of a sequence of partial sums. In other words, if S_1, S_2, S_3,... is a sequence where the subscript denotes how many terms have been added, then the limit of this sequence is the sum of the infinite series.

In the case of 1/3=0.3(bar), you have the sequence
0.3, 0.33, 0.333, 0.3333, ... .

This is the sequence of partial sums of the infinite series
\sum_{k=1}^{\infty }3\cdot \left( \frac{1}{10}\right) ^{k}.

In your book, they'll prove that
\sum_{k=1}^{\infty }a\cdot r^{k}=\frac{ar}{1-r} or something equivalent to that.

In this case, a=3 and r=1/10:
\frac{3\left( 1/10\right) }{1-\left( 1/10\right) }=\frac{3/10}{9/10}=1/3.

Geometric series are cool for a few reasons, not the least of which is that all repeating decimals are really ones. Another thing is that their sum is actually findable; often infinite series have sums that can only be approximated but not found exactly in a simple way.

[Alkatran]

Do it by the difference.

First, ask him if he agrees that:
1/infinity = 0
x^-y = 1/x^y
x*0 = 0
x^infinity = infinity

Then do this:
1 - 0.9 = 1 - 9*10^-1 = .1 = 10^-1
1 - 0.09 = 1 - 9*10^-2 = .01 = 10^-2
skip ahead...
0.999~ = 1 - 9 * 10^-infinity = 1 - 9 * 1/10^infinity = 1 - 1/infinity = 1 - 0 = 1

The difference is 0. The have the same value but a different representation.

[Adam]

You know I asked about representing the reciprocal of the 0.\dot{9}, earlier? Would that be the limit i\rightarrow0 ?

[Alkatran]

You know I asked about representing the reciprocal of the 0.\dot{9}, earlier? Would that be the limit i\rightarrow0 ?

Latex isn't working, please print out your equation

[Adam]

You know I asked about representing the reciprocal of the 0.9..., earlier? Would that be the limit i->0 ?

[phoenixthoth]

Division is an operation that is defined for real numbers and infinity is not a real number. Therefore, 1/infinity is not equal to 0.

Similarly, other operations are not defined for non real numbers such as exponentiation. You might as well say 1/rabbit=0 or :smile: + :surprised =1/ :confused: .

That's the standard comeback but now let's turn to nonstandard analysis to fix what you have.

Let x and y be hyperreal numbers. Denote their infinitesimal closeness by writing x==y. x==y means |x-y|<r for all positive real numbers r.

The main fact your assuming by writing 1/infinity=0 is that IF x and y are real numbers then x==y if and only if x=y in which case |x-y|=0<r for all positive real numbers r.

The second thing you're doing is dividing non real numbers. While in standard real analysis, this makes about as much sense as dividing apples into oranges (both of which are nonreal numbers), you can kind of do this in nonstandard analysis.

Let x be a hyperreal number. x is called unlimited if |y|<|x| for all real numbers y. Then instead of saying 1/infinity=0 here's the correct statement:

1/x==0 for all unlimited x.

If x is unlimited then 1/x is an infinitesimal. An infinitesimal y is a hyperreal number such that y==0.

x^-y = 1/x^y
x*0 = 0
x^infinity = infinity
The second result holds if x is limited and we can say that a noninfinitesimal x raised to an unlimited power is unlimited.

Then do this:
1 - 0.9 = 1 - 9*10^-1 = .1 = 10^-1
1 - 0.09 = 1 - 9*10^-2 = .01 = 10^-2
skip ahead...
0.999~ = 1 - 9 * 10^-infinity = 1 - 9 * 1/10^infinity = 1 - 1/infinity = 1 - 0 = 1
Technically, 0.9999....==1 because 1/x==0 for unlimited x, not =0. But this is ok since for two real numbers x and y, x==y if and only if x=y. Therefore, 0.9...=1.

0.9... is a real number because it is the least upper bound of the set {0.9,0.99,0.999,0.9999,...} which is bounded above by 1. This is part of the definiton of real number.

[Aphex_Twin]

2) If 1 is greater than .999..., then by how much, exactly, is it greater? After all, 1 is greater than .9 by .1. It is greater than .99 by .01. It is greater than .99999999999999999999 by .00000000000000000001. And so on, and so on. However, if the number of nines is infinite, then the number of zeros preceding the 1 will be infinite, and 0.000... is obviously equal to 0, which means that 1 is greater than .999... by exactly 0.

If you're taking the discussion to the "Surreal number set" then there can be a "one" after an infinity of "zeros".

[Hurkyl]

First off, there are "too many" surreal numbers to fit into a set. :tongue:


It's very easy to describe sequences that have infinitely many of one thing, and then another; you just need to index your sequence by an infinite ordinal that is not a limit ordinal. For example, the ordinal \omega + 1 consists of a copy of the natural numbers adjoined with a single element. (\omega) that comes after the natural numbers. You could then write the sequence \{a_n\} where a_n = 0 for finite n, and a_{\omega} = 1.

This sequence, however, is not a decimal. By definition, decimals are indexed by the limit ordinal \omega, the natural numbers, all of whose elements are finite. There is no place in a decimal that has infinitely many predecessors.

End broken record.

[hypnagogue]

Thanks for the response Hurkyl.

I believe this thread has run its course. 0.999... = 1 is simply a mathematical fact, and much effort has gone into this thread to provide various rigorous and intuitive support of this fact. There is no need to continue recycling old debates on what is certainly a completely settled issue.