Is 0.999 Repeating 1? Debate & Opinions

  • Thread starter killerinstinct
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In summary, the debate over whether or not 0.999999 repeating is equal to 1 is a very common one, with many people arguing that they are not equal. However, in the normal real number system, they are in fact equal and this can be easily proven. The expression .000...1 is not valid in the set of real numbers and the last little bit is essentially zero by definition. Despite there being many different ways to represent points on the real number line, 0.999999 repeating and 1 are equivalent in the cauchy sequences of rationals modulo convergence that define the real number system.
  • #1
killerinstinct
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Very common debate: is 0.999999 repeating 1?
Opinions?
 
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  • #2
didn't realize that there was anther exactly the same question in logics thread.
 
  • #3
killerinstinct said:
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. :smile: 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.
 
  • #4
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.
 
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  • #5
Well I'm 99.99999...% certain it's equal to 1 :D
 
  • #6
This has to be the most asked question on this forum.
 
  • #7
JonF said:
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.
 
  • #8
.999… not equal to 1? That’s kiddy stuff, just watch me argue that .3333… is not equal to 1/3
 
  • #9
Corollary to JCSD's theorem:

every bulletin board etc attracts an idiot, a troll, or possibly both.
 
  • #10
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.
 
  • #11
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 [tex] \frac 1
\infty [/tex]. 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.
 
  • #12
Integral said:
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.
 
  • #13
Eyes can trick your mind.
 
  • #14
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.
 
  • #15
Grizzlycomet said:
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:

[tex]\sum_{n=1}^{\infty} \frac{9}{10^n} = 1[/tex]
 
  • #16
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.
 
  • #17
Integral said:
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 :)
 
  • #18
Read http://home.comcast.net/~rossgr1/Math/one.PDF [Broken] 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.
 
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  • #19
Thak you for the link, Integral. I think I'm on my way to getting it now.
 
  • #20
Integral said:
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 [tex] \frac 1
\infty [/tex]. 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?
 
  • #21
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.
 
  • #22
Bob,
Yes, I am referring to different bases, as well as other forms of representation.
 
  • #23
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.
 
  • #24
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.
 
  • #25
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.
 
  • #26
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...
 
  • #27
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
 
  • #28
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.
 
  • #29
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.
 
  • #30
here we go again
 
  • #31
TENYEARS said:
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...
 
  • #32
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...?
 
  • #33
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.
 
  • #34
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
 
  • #35
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.
 
<h2>1. Is 0.999 Repeating 1 really equal to 1?</h2><p>Yes, it is a mathematical fact that 0.999 repeating is equal to 1. This can be proven using various mathematical methods, such as algebra or calculus.</p><h2>2. Why does 0.999 Repeating 1 equal 1?</h2><p>0.999 repeating is a decimal representation of the number 1. In the decimal system, there are infinite numbers between 0 and 1, so it is possible to represent 1 as 0.999 repeating.</p><h2>3. Can you provide an example to show that 0.999 Repeating 1 is equal to 1?</h2><p>One example is to convert 1/3 into a decimal, which is 0.333 repeating. When multiplied by 3, it becomes 0.999 repeating, which is equal to 1.</p><h2>4. How is the concept of 0.999 Repeating 1 used in real life?</h2><p>In mathematics and science, 0.999 repeating is often used as a shorthand notation for the number 1. It is also used in various calculations and equations, such as in calculus and number theory.</p><h2>5. Is there any controversy surrounding the concept of 0.999 Repeating 1?</h2><p>Yes, there are some who argue that 0.999 repeating is not exactly equal to 1, but this is a minority viewpoint and is not supported by mathematical evidence. In most cases, it is accepted that 0.999 repeating is equal to 1.</p>

1. Is 0.999 Repeating 1 really equal to 1?

Yes, it is a mathematical fact that 0.999 repeating is equal to 1. This can be proven using various mathematical methods, such as algebra or calculus.

2. Why does 0.999 Repeating 1 equal 1?

0.999 repeating is a decimal representation of the number 1. In the decimal system, there are infinite numbers between 0 and 1, so it is possible to represent 1 as 0.999 repeating.

3. Can you provide an example to show that 0.999 Repeating 1 is equal to 1?

One example is to convert 1/3 into a decimal, which is 0.333 repeating. When multiplied by 3, it becomes 0.999 repeating, which is equal to 1.

4. How is the concept of 0.999 Repeating 1 used in real life?

In mathematics and science, 0.999 repeating is often used as a shorthand notation for the number 1. It is also used in various calculations and equations, such as in calculus and number theory.

5. Is there any controversy surrounding the concept of 0.999 Repeating 1?

Yes, there are some who argue that 0.999 repeating is not exactly equal to 1, but this is a minority viewpoint and is not supported by mathematical evidence. In most cases, it is accepted that 0.999 repeating is equal to 1.

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