0.9999....(recurring) = 1?

Bob · 2012-06-22 18:13:58

hi Stefy,

I think you will have to define 'exist' in the context of numbers.

Isaac Azimov made a good case for the 'existance' of complex numbers that contained the challenge "Pick up that piece of chalk; now hand me half a piece of chalk."

I can have 3 apples, or write a 3 on a piece of paper, I can hold up a piece of wood carved in the shape of the symbol 3, but I cannot hold a 3 itself. It's just an abstract concept.

So what do you mean by "It doesn't exist"?

Bob

anonimnystefy · 2012-06-22 23:29:22

What I mean is that we cannot use the concept of 0.999... in mathematics. We know that every rational number has a unique decimal representation, so we cannot have both 1.000... And 0.999...

bobbym · 2012-06-23 00:08:06

Hi;

We know that every rational number has a unique decimal representation

That is not correct.

Top of page 3:

http://www.math.cornell.edu/~kahn/countingreals07.pdf

or:

http://mas.lvc.edu/~lyons/pubs/mathreasoning2.pdf

Section 6.4

Lastly, in "Introduction to Mathematical Proofs," by Charles Roberts in section 7.3 we have,

However some rational numbers have 2 different decimal expansions.

Your statement should be, "Every irrational number has a unique and non terminating decimal expansion." 1 is not irrational.

anonimnystefy · 2012-06-23 00:23:13

It seems that guy is stating the exact opposite thing from me. I said that 0.999... cannot exist because every number has a unique decimal representation and he says that not every number has a unique decimal representation because 0.999... exists.

I have one question. How does existence/non-existence of 0.999... affect mathematics. Is it like the issue of an infinity between aleph 0 and aleph 1?

bobbym · 2012-06-23 00:27:13

Hi;

You are going to have to define exists.

anonimnystefy · 2012-06-23 00:37:03

And why is that?

bobbym · 2012-06-23 00:40:53

To me .999999999... is this

That is really all you need from a practical viewpoint.

anonimnystefy · 2012-06-23 00:51:36

No, I asked you why you think 0.999... has to exist? "exists" is an already defined concept.

bobbym · 2012-06-23 00:54:52

I do not think it has to exist. I just can find no reason why I can not write that sum in post #983. Or why I can not sum it to 1.

anonimnystefy · 2012-06-23 01:03:36

You can. But that doesn't mean you can represent it as 0.999... .

bobbym · 2012-06-23 01:07:02

.9+.09+.009+.0009+.00009+...+

That is the same as .9999... If you agree I can then you have allowed for .9999...

anonimnystefy · 2012-06-23 01:12:29

Okay. I was trying to see the problem from the eyes of my prof. It would seem that he is wrong.

bobbym · 2012-06-23 01:13:41

Are you saying your professor does not think .999999... = 1?

anonimnystefy · 2012-06-23 01:17:01

He thinks that 0.999... doesn't exist or something like that.

bobbym · 2012-06-23 01:20:37

Here is what I say. There are many skeletons in the mathematics closet. As a discretist I tell you about them everyday. But I have never given the validity of .999... = 1 a thought. It just seems too obvious. There are more important concepts to be argued than this one. In my opinion your prof. is wasting his time.

anonimnystefy · 2012-06-23 01:24:54

No, he isn't obsessed with the 0.999...=1 discussion. It is just his opinion that 0.999... doesn't exist.

I am just having a discussion on Facebook about this problem.

Last edited by anonimnystefy (2012-06-23 01:25:11)

bobbym · 2012-06-23 01:30:19

It is just his opinion that 0.999... doesn't exist.

Opinions are sort of like bedbugs. They can bite ya.

anonimnystefy · 2012-06-23 01:33:35

I know.

bobbym · 2012-06-23 02:03:23

Hi anonimnystefy;

You are on facebook?

anonimnystefy · 2012-06-23 02:37:49

Yes, I am.

Stangerzv · 2012-06-23 13:15:44

0.9999999.... or 3x (1/3) doesn't exist in our human limited understanding. It is like partitioning a unit into 3. The divided units (1/3) only exist in the mathematics and not in the real world. This is what we call irrational numbers, same with the diagonal length of a square with a unit sides. We can see these irrational numbers in our life even though they actually don't exist on their own. Can you measure 1/3 or square root of 2? Square root of 2 does exist in a square but can you measure it? Never ever because they don't exist in the real world. In other words, irrational numbers only exist as complementary to others but when you take them out on their own, they do exist as a form which can't be quantify exactly into decimal system because they are endless into infinity. This is why you can never say 0.99999.....=1 because somewhere in the infinity there is a residue left when 1-0.9999....it is endless.

anonimnystefy · 2012-06-23 13:45:45

You remind me of George Y. Mathematics isn't dependent on the real world. Just because we cannot see something doesn't mean it is not there.

Bob · 2012-06-23 20:05:05

hi all,

I was out gardening at my Mum's yesterday so missed most of this discussion.

My point about defining 'exist' is a continuation of the idea in Stangerzv's post

We can see these irrational numbers in our life even though they actually don't exist on their own.

I don't think you can 'see' any number in this sense. You can no more 'see' a 3 than you can 'see' √3.

All numbers are just abstract concepts:

Dictionary.com wrote:

'exist'
5. philosophy a.to be actual rather than merely possible
b.to be a member of the domain of some theory, an element of some possible world, etc

Humans have invented various numbers for our convenience and mathematicians have tried to make those inventions consistent and logical. If you can call √-1 a number then you can call 0.9999999............ a number. Then you can investiagte its properties.

One property of 0.99999999999999999........... is that it behaves the same as 1. You might think that is odd but if you don't allow that then a lot of the other rules of numbers that mathematicians have constructed fall apart. So it's rather like defining x^0 to be 1. You cannot actually multiply x by itself 0 times but the rules of powers work nicely if you use this definition. Maths is full of stuff like that. You can join the crowd on this one or define your own rules. Just make sure they are consistent and logical.

Bob

anonimnystefy · 2012-06-23 23:55:32

Did you acrually read all posts from yesterday and today?

Bob · 2012-06-24 00:46:29

Yes I did. You have said your prof says it doesn't exist.

Either you misunderstood; or he was trying to provoke critical thinking (succeeded too); or his statement is easy to disprove.

You do this by writing 0.99999 recurring on a piece of paper and say "Look! There it is. I see, therefore it is."

It has as much right to be considered to exist as 'pi', or 'e' or ∞

What is of greater interest surely, is, how does this number behave? And all the evidence suggests it behaves just like 1.

Mathematics isn't 'the real world'; it attempts to model the real world. And to do this we have to make up idealised concepts. Here's a list of a few:

points with zero dimensions
infinite planes
frictonless pulleys
angles that can be measured with perfect accuracy (have you ever tried to draw a triangle, measure its angles and check what they add up to?)
decimals with an infinite number of digits
hyperspace
the multiverse

Do any of these 'exist'?

Bob

Math Is Fun Forum

#976 2012-06-22 18:13:58

Re: 0.9999....(recurring) = 1?

#977 2012-06-22 23:29:22

Re: 0.9999....(recurring) = 1?

#978 2012-06-23 00:08:06

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#979 2012-06-23 00:23:13

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#980 2012-06-23 00:27:13

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#981 2012-06-23 00:37:03

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#982 2012-06-23 00:40:53

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#983 2012-06-23 00:51:36

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#984 2012-06-23 00:54:52

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#985 2012-06-23 01:03:36

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#986 2012-06-23 01:07:02

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#987 2012-06-23 01:12:29

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#988 2012-06-23 01:13:41

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#989 2012-06-23 01:17:01

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#990 2012-06-23 01:20:37

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#991 2012-06-23 01:24:54

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#992 2012-06-23 01:30:19

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#993 2012-06-23 01:33:35

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#994 2012-06-23 02:03:23

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#995 2012-06-23 02:37:49

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#996 2012-06-23 13:15:44

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#997 2012-06-23 13:45:45

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#998 2012-06-23 20:05:05

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#999 2012-06-23 23:55:32

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#1000 2012-06-24 00:46:29

Re: 0.9999....(recurring) = 1?

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