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 probability, 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