Is 0.999 Repeating 1? Debate & Opinions

[killerinstinct]

Very common debate: is 0.999999 repeating 1?
Opinions?

 

[killerinstinct]

didn't realize that there was anther exactly the same question in logics thread.

 

[Bob3141592]

Very common debate: is 0.999999 repeating 1?
Opinions?

Yes, it is, as long as we're talking about the normal real number system. I can't see any difference. Except for the typography.

What is the difference, meaning subtract .999... from 1.000... The difference would be 0.000...1. But it's not valid to put something after the "..." That's asking what comes after infinity, which isn't a valid question. The expression .000...1 is a typographic error, and not something that is even defined in the set of real numbers.

 

[matt grime]

It is not an opinion that they are equal, it is a very easy provable fact and only cranks who don't understand the way mathematics work insist they are different after it has been patiently explained to them.

We mean base ten, work out what the infinite sum 0.999... is, if that doesn't convince you then you need to look up the definitions you don't understand in the phrase:

they represent the same equivalence class in the cauchy sequences of rationals modulo convergence that define the real number system.

 

[jcsd]

Well I'm 99.99999...% certain it's equal to 1 :D

 

[JonF]

This has to be the most asked question on this forum.

 

[jcsd]

This has to be the most asked question on this forum.

JonF I have never been on any mesage board where there have not been arguments about this and that includes non-maths/sci boards.

In fact I now propose jcsd's theorum:

On any bulletin board, no matter the subject area of that board, sooner rather than later someone will argue that 0.99.. is not equal to one.

 

[JonF]

.999… not equal to 1? That’s kiddy stuff, just watch me argue that .3333… is not equal to 1/3

 

[matt grime]

Corollary to JCSD's theorem:

every bulletin board etc attracts an idiot, a troll, or possibly both.

 

[Grizzlycomet]

Why should .9999... equal 1 and not .9999...? Trying to get from .9999... to 1 is just like trying to accelerate your spaceship to the speed of light. You keep getting closer, but you can't get that last little bit.

 

[Integral]

I knew it.^~ A last little bit poster would have to show up. Do we try to explain it to him?

That "last little bit" is \frac 1 <br /> \infty. By the definition of infinity, that last little bit is zero. So essentially this is true by definition, but beyond that it is completely consistent and provable in many different manners. There is no law that says each point on the real number line must have a unique representation. In fact just the opposite is true, every point on the real number line has many (perhaps an infinite) number of different ways to represent it.

 

[Grizzlycomet]

I knew it.^~ A last little bit poster would have to show up. Do we try to explain it to him?

I must say I dislike these kind of comments. I am a 16 yr old student who has not taken a lot of math, certainly not on the subject of infinity. I fail to see why you would judge me as "a last little bit poster" or whatnot, for simply voicing a (to me) logical view. Though these thing may be obvious to you, that is not so for everyone. I find that your post without the two first lines would have been completely satisfactory.

 

[hello3719]

Eyes can trick your mind.

 

[Integral]

My apologies, having been involved in this same discussion on several different forums over the last 2 or 3 years I do not recall anyone ever saying "oh I see" so perhaps am a bit cyncial about the whole issue.

 

[jcsd]

I must say I dislike these kind of comments. I am a 16 yr old student who has not taken a lot of math, certainly not on the subject of infinity. I fail to see why you would judge me as "a last little bit poster" or whatnot, for simply voicing a (to me) logical view. Though these thing may be obvious to you, that is not so for everyone. I find that your post without the two first lines would have been completely satisfactory.

Just follow this:

let x = 0.9999... =>

10x = 9.99999...

10x - x = 9x = 9 =>

x = 1


All we are really saying is:

\sum_{n=1}^{\infty} \frac{9}{10^n} = 1

 

[ahrkron]

I'm sorry for the harsh response to your question, Grizzlycomet. We generally try to not be hard on people because of the questions they ask; the problem is that this particular topic is visited way too often by people trying to push their "new math", "theory of infinity" and whatnot, instead of trying first to understand how standard math deals with the issue.

 

[Grizzlycomet]

My apologies, having been involved in this same discussion on several different forums over the last 2 or 3 years I do not recall anyone ever saying "oh I see" so perhaps am a bit cyncial about the whole issue.

Thank you, apology accepted :) I understand that you may have seen this question many times, thus growing very tired of it. Your explanation was in itself good :)

 

[Integral]

Read http://home.comcast.net/~rossgr1/Math/one.PDF page. There are 2 proofs, the first simply uses the sum of an infinite series formula. The second is my version of how a Mathematician approaches the problem. I feel that it also gives a very intuitive feel for why equality holds.

 

[Grizzlycomet]

Thak you for the link, Integral. I think I'm on my way to getting it now.

 

[Bob3141592]

I knew it.^~ A last little bit poster would have to show up. Do we try to explain it to him?

That "last little bit" is \frac 1 <br /> \infty. By the definition of infinity, that last little bit is zero. So essentially this is true by definition, but beyond that it is completely consistent and provable in many different manners. There is no law that says each point on the real number line must have a unique representation. In fact just the opposite is true, every point on the real number line has many (perhaps an infinite) number of different ways to represent it.[/QUOTE]

Whoa there! In the emphasis added section above, I don't understand what you're getting at. Are you referring to different bases? If we stick with only a single base, don't all irrational numbers have one and only one representation? Rational numbers can have two representations, a "finite" form "a.bcdef000..." where the infinite series of zeroes at the end are ignored, and "abcde(f-1)999..." Much like the inverse of the question of this thread. I'm hazy about this, but I vaguely remember that the representation of some numbers are unigue, and the representation of others are not unigue but have exactly two forms, as being critical to some fundamental proof Cantor used in developing his theories.

The only thing I'm sure about is that I didn't understand it at the time, and had to go on to other things. Regretably, I never got back to it. Does anyone know more clearly what this was? As it was fundamental, might that serve as a good argument when this topic rears it's cursed ugly head again, as it surely will?

 

[Hurkyl]

If we stick with only a single base,

But there's nothing that says we should stick to a single base, or even that we should use radix notation!

What you are saying, though, is correct. For a given base n, any real number that can be represented as a terminating n-ary expansion has exactly two n-ary representations, and all other real numbers have a unique n-ary representation.

 

[Integral]

Bob,
Yes, I am referring to different bases, as well as other forms of representation.

 

[TENYEARS]

It's value is relative to it's use. If you are doing 99% of things on this planet you may want a few extra places. If you are lanching a rocket you may want to carry out the decimal places 20 positions. If you want to talk in terms of reality, you may want to think again. You could fill this universe with a goo goo plex for every sub atomic particle and place each upon the back of the other forming the power of. It looks like a large number, incomprehensible and yet as one expands into infinity, it is just a dot then less. So if you are thinking in terms of infinite probablity, this would be very important. That would mean you are already thinking outside of the box, and yet, what I see is most playing both ends against the middle because it is not understanding which creates the answer to the question but belief which stops short of true logic and true understanding.

 

[Didd]

Good!
0.99999... when written in fraction form it will be 1. So 1 is a fraction form of 0.999999...
Need more explanation, post a reply.

 

[matt grime]

The issue arises because people think that the real numbers are actually decimals. This is understandable as no one teaches the proper definition of them at high school, which is again understandable as that would be an impossibly difficult task, unless one lived in Russia or Hungary (mathematical analysis joke).
Hopefully this doesn't cause propblems, just as using naive set theory doesn't cause any problems, most of the time. In a good sense there are people who are quick enough to realize that there are problems with decimals and naive sets and if we are lucky they are doing so at a time when they are being taught be people who are au fait with this and can explain, or at least give broad hints as to, how one properly negates these issues. Sadly there are those who don't accept these explanations and insist that the real numbers represented by 0.99... and 1 are distinct. They fail to appreciate the meanings of the words represent, real, and the symbolism of the expansion, and no amount of reasoning can put some of them right. That is why you get these RTFM replies to this question.
Always the mathematical answers explain what the definitions are, why these two things are equal and ought to offer alternate systems or examples to elucidate this matter.

The simplest case I can think of where two different symbols represent the same object is in modulo arithmetic, or if you prefer group theory. This embodies entirely the idea of working with different representatives of the same equivalence class, and usually students don't see that as morally repugnant, yet they seem to find something inherently wrong with that idea in terms of decimals because it contradicts an irrationally held belief.

 

[MiGUi]

I like the demo in the last post... I posted that in another thread:

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

Then 1 = 0.99...

 

[Physics is Phun]

I look at it as 1/9 = .11111 repeating 2/9 = .2222222... ... 8/9=.888888888 therefore 9/9 should be .9999999999... but 9/9 is obvously 1 and that does it for me to show that .99999... = 1

 

[gravenewworld]

Infinity is a funny concept, there are many strange results when working with infinity such as .9 repeating=1 and the proof that there are different sizes of infinity. Infinity is one of the most interesting subjects in math, I recommend you read up on it.Check out Cantor's proof on different sizes of infinity you will find it interesting. The best thing about it is that you don't need high level math to understand it. It is an easy read for any high school student.

 

[TENYEARS]

It is a real number, but it will never be complete. It is not 1. One is be, but be is not .99... You will always be chasing your tail.

 

[hello3719]

here we go again

 

[HallsofIvy]

It is a real number, but it will never be complete. It is not 1. One is be, but be is not .99... You will always be chasing your tail.

All you are telling us here is that you do not know what a real number is. Didn't you read the previous responses in this thread? There are a variety of (equivalent) definitions of real number. They all give the result that 0.999... is exactly the same as 1.00...

 

[jcsd]

TENYEARS, there is always another real number (infact there's always an infinite number of real numbers between any two real numbers, the proof is trivial) between any two real numbers, what's inbetween 1 and 0.9999...?

 

[robert Ihnot]

This question is a lot more important than is getting credit for here. Cantor was very concerned with this in in drawing up a list of rational numbers without duplicates. Remember that 1/2 + 1/4 + 1/8 +++ = 1. So that using base 2, which seems the simplest from a theoretical standpoint, then .111111111...=1.

 

[MiGUi]

What you suggest, TENYEARS, is wrong because you think in "0.99..." and "1" as two different numbers, but they are not. There are two different forms to write the same. You may not discuss that "8/4" and "2" are the same thing, and with "1" and "0.99..." occurs the same, because if not, if that two numbers were different, then aritmethic would not work properly.

Writting 0.99... may let you think (wrong) that the <i>next</i> real number is 1. But that thing is not correct, because in the real field, between two numbers there are infinity different numbers.


Bye
MiGUi

 

[TENYEARS]

I was going to change the post right after I posted, but did not. Throw out the word real. .99999... is not equal to 1. Case point end of discussion. It will have relavance to you some day. Apparently just not now.

 

[ahrkron]

I'm sorry, TENYEARS, but if you want to argue for a system in which 0.999... is not equal to 1.000..., you would need to do so in the TD forum, since their equality is a solid piece of math; it is not a matter that can "become relevant" at some point for someone, but an identity that follows from definitions and formal logic.

 

[HallsofIvy]

No, ahrkon, TENYEARS does not want to argue anything, He just likes to hear himself babble. I would say that I suspect TENYEARS is his age, but I don't want other ten year olds complaining that I am dissing them.

 

[jcsd]

TENYEARS takes ome advice: when people have nothing to say of any import, or indeed meaning, they usually keep quiet.

 

[TENYEARS]

Because ideas are accepted as fact by the masses, does it make it so? What defines logic but the logic before it which is based upon the agreement of a base worked forward. Because we have decided to accept something as such, is it such? It is such because you were taught so is it not.

It is an accepted value, I have accepted it as such a value in caluclations, it is rounded and creates one, but is it one? No it tries to be but never quite reaches. The issue can be avoided in large calculations by using factionary math, but when rounding must occur, it is what it is.

You can sweep it under the carpet, but that is why so many do not understand. Because you do not question and you accept. It is one by whos authority? This authority is certainly no greater than mine or yours unless you make it so by not seeing.

 

[jcsd]

Because ideas are accepted as fact by the masses, does it make it so? What defines logic but the logic before it which is based upon the agreement of a base worked forward. Because we have decided to accept something as such, is it such? It is such because you were taught so is it not.

It is an accepted value, I have accepted it as such a value in caluclations, it is rounded and creates one, but is it one? No it tries to be but never quite reaches. The issue can be avoided in large calculations by using factionary math, but when rounding must occur, it is what it is.

You can sweep it under the carpet, but that is why so many do not understand. Because you do not question and you accept. It is one by whos authority? This authority is certainly no greater than mine or yours unless you make it so by not seeing.

Meaningless waffle.

I don't need to accept anyone's authority on it I've seen many proofs. There's no need to appeal to authority when it can be shown with logical arguments, if you throw logic out the window you shouldn't expect to get answers of any menaing.

It's not about 'rounding up', it's basic maths.

 

[matt grime]

Because ideas are accepted as fact by the masses, does it make it so? What defines logic but the logic before it which is based upon the agreement of a base worked forward. Because we have decided to accept something as such, is it such? It is such because you were taught so is it not.

It is an accepted value, I have accepted it as such a value in caluclations, it is rounded and creates one, but is it one? No it tries to be but never quite reaches. The issue can be avoided in large calculations by using factionary math, but when rounding must occur, it is what it is.

You can sweep it under the carpet, but that is why so many do not understand. Because you do not question and you accept. It is one by whos authority? This authority is certainly no greater than mine or yours unless you make it so by not seeing.

to show you're not a crank explain from first principles the construction of the real numbers in your favourite way and explain how it is that 0.999... and 1 represent different real numbers (base 10 expansions).

if as i suspect you've no idea what the definitions of the real numbers are, then this could be an interesting exercise for you. HINT they are not decimal expansions.

 

[quartodeciman]

I am interested why TENYEARS insists:

"Throw out the word real".

 

[ahrkron]

TENYEARS,

It is nice that you express your feelings about the meaning of authority, and the need for people to question what they are taught. However, they have nothing todo whatsoever with the issue at hand. The reason is that, in math, it all starts with an agreed upon set of definitions. There is no dogma as to how "things should be". Once one understands that, it is clear that all conclusions are based on the definitions used.

In this particular case, the fact that 0.999...=1.000... is an inevitable consequence of the definitions used for the objects called "real numbers". You are free to use or not use that representational tool (the "real numbers"), but you cannot change the fact that, given the definitions of the system, the stated property follows.

A slightly different matter is the application of math concepts to physical science, but also in that case there is not much room for people to "express their freedom" or "question authority", since in that case the only possible adequacy test is given by experiment; again, this is so only because the agreed-upon definition of physical science.

If you want to use different definitions and methods, you are certainly free to do so, but then you need to realize (and you also need to be clear about it, so others don't get confused) that your statements do not refer to "math", "physics", or "real numbers", but other, distinct, disciplines and objects.

 

[Zorodius]

... infact there's always an infinite number of real numbers between any two real numbers, the proof is trivial ...

If it wouldn't be a bother, could I see that proof?

 

[Hurkyl]

Suppose a < b.

Then, a + a < a + b (add a to both sides)
Also, a + b < b + b (add b to both sides)
Divide both inequalities by 2 to get:

a < (a + b)/2 < b

In this way, given any two distinct numbers in an ordered field, such as the real numbers, we can show there is another number between those two.

Similarly,
a < (2a + b)/3 < b
a < (3a + b)/4 < b
a < (4a + b)/5 < b

and so on.


Or, if you prefer, you can structure the proof like this:

There must be a c such that a < c < b,
then, there must be a d such that a < d < c (< b),
then there must be an e such that a < e < d (< b),
and so on.

 

[HallsofIvy]

Let a, b be any two numbers, a< b. Then, adding a to both sides, a+b< 2b so (a+b)/2< b. Also, adding b to both sides of a< b, 2a< a+ b so a< (a+b)/2. That is, (a+b)/2 is a real number strictly between a and b. Repeat to show that (3a+b)/4 is a real number strictly between a and (a+b)/2 and use induction to show that ((2ⁿ-1)a+ b)/2ⁿ) is a real number strictly between a and ((2^n-1-1)+b)/2^n-1. Since the inequality is strict, these are all distinct real numbers. That is, there exist an infinite number of distinct real numbers strictly between a and b.

Although it is a bit harder, the proof can be extended to show that between any two real numbers there is a rational number and between any two real numbers there is an irrational number.

 

[Integral]

Because ideas are accepted as fact by the masses, does it make it so? What defines logic but the logic before it which is based upon the agreement of a base worked forward. Because we have decided to accept something as such, is it such? It is such because you were taught so is it not.

It is an accepted value, I have accepted it as such a value in caluclations, it is rounded and creates one, but is it one? No it tries to be but never quite reaches. The issue can be avoided in large calculations by using factionary math, but when rounding must occur, it is what it is.

You can sweep it under the carpet, but that is why so many do not understand. Because you do not question and you accept. It is one by whos authority? This authority is certainly no greater than mine or yours unless you make it so by not seeing.

This is a pretty funny argument! According to all the evidence I have seen (several web polls) The masses believe that 1<> .999... , so by your own argument it seems that you should believe the opposite. Yet you follow the masses like a blind sheep. Sound logic and knowledge of the Real Numbers lead to a conclusion contrary to the believe of the Masses. Perhaps rather then being a blind sheep you should educate yourself and learn to draw conclusions based on knowledge and logic rather then intuition.

 

[TENYEARS]

What proof you would like to come up with is irrelevant. .99999999... <> 1.
You would have to project it into infinity and it would approch 1 but never be 1. You may accept it as such, but it is not correct. 1/3 * 3/1 will yeild one, but .333333333333... * 3 will not yeild one. We accept it as one, but it is not one. One is infinite and complete. The approximation is not. Like I said, I do not expect you to understand. I am done with expectation. If you are trying to project infinite probability, you will always stop short with .999999... you must stop somewhere to project, if not you will always be running.

 

[ahrkron]

What proof you would like to come up with is irrelevant. .99999999... <> 1.

As I said, you should be explicit in the fact that you are not talking math any more. If proof is irrelevant, you are definitely talking about something other than math.

You would have to project it into infinity and it would approch 1 but never be 1.

This sounds like philosophy or pseudoscience.

1/3 * 3/1 will yeild one, but .333333333333... * 3 will not yeild one.

So, in the language you are using, 1/3 <> 0.3333...?

This is starting to go in circles.

 

[master_coda]

.999… not equal to 1? That’s kiddy stuff, just watch me argue that .3333… is not equal to 1/3

Wow, and I thought that was just a joke.