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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi Calligar;

Welcome to the forum. That is exactly the point.

3.333333333... = 3 + 1 / 3

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

hi guys

i found this topic on wikipedia,and it has some pretty good arguments.

look at it this way: 0,5 is just another way of representing 1/2,and by the same logic 0,3333... recurring is just another way of representing as 1/3.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

I have seen mathematicians that hate set theory. Mathematicians that hate continuous math. Mathematicians who hate pure math, ones who hate applied math and computers. Ones who hate Topology and Logic, almost every type. But I have never seen anyone advancing while he/she held on to the idea that .9999999... ≠ 1

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**reconsideryouranswer****Member**- Registered: 2011-05-11
- Posts: 172

anonimnystefy wrote:

look at it this way: 0,5 is just another way of representing 1/2,and by the same logic 0,3333... recurring is just another way of representing as 1/3.

Also, don't forget to include that

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

hi rya

yes that is correct,but my point is that we have multiple ways of representing the same number.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**Calligar****Member**- Registered: 2011-09-24
- Posts: 234

Hi, this is more to bobbym...

I'm not saying that 0.3 recurring can't equal 1/3, I'm only saying it doesn't have to, however I realise this idea will likely be debated and/or dismissed by assuming I'm automatically wrong.

1/3 = 0.3 recurring because that is the only way to express it in decimal form, but that doesn't mean 0.3 recurring = 1/3, reason is because what if I simply wanted to use the number 0.3 recurring NOT intending it to be 1/3, as I see this as arguably possible. But in turn, something that might get confusing about it, both of those 0.3 recurring numbers would in fact be 2 different numbers, without a way to express the difference in decimal form. You can do this with anything really, not just 1/3. Also, I would argue that 0.5 ≠ 0.49 (with recurring 9s) to reconsideryouranswer, because they are both decimals that are not the same number. One last thing to bobbym, the idea of 0.9 recrurring ≠ 1 has in a sense been used in mathematics, just not in that same way. Let me give you an example: 1 - 0.9 recurring = 1/∞, or an infinitesimal, instead of 0. Although according to the normal rules, this is arguably 0, infinitesimals are still used, therefore I wouldn't dismiss the idea.

To be fair though, if this is controversial, I will not argue this further even though I can. I am just trying to make my point clearer was all...

Life isnt a simple Math: there are always other variables. -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end. -Aristotle

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi;

First of all this has been argued many times before by the theoretical type of mathematicians and they can explain it better than I can.

When you say .333333333... = 1 / 3 what you are really saying is

That equals 1 / 3. Same thing with .9999999... If you really can not accept that then think of math like a formalist does. That it is a game and games have rules. We do not debate why the knight moves as it does in chess.

Rule 1) .999999999... = 1

You can now accept this and move on.

however I realise this idea will likely be debated and/or dismissed by assuming I'm automatically wrong.

Although that sometimes happens in mathematics it is much rarer than in other fields like biology or psychology for instance. In math we prove things, there are dozens of proofs about this. I have seen many arguments that

.9999999999... ≠ 1 but I have never seen any proof of that.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

hi Calligar

just tell me the difference between 0.3 reccuring and 0.3 reccuring? then i will accept almost anything you say (not really but anyway,...).

also,an infinitesimal is just a term,but none can really be found.you cannot find any numbers greater than zero that are less than any other number.

1/infinity is not defined!

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,562

What about 9.999999999999...89999999... See the problem, Ricky would have said this number is

impossible because you can't append on the end of the 9.99999999999999... , but why not append

after infinite digits, if you can imagine it, it should be true, so therefore, there are many numbers

that all seem to be equal to one another, but have different digits after the infinite decimal place,

which most agree doesn't exist, but I say it does, otherwise the outerspace would not go in two directions

or more in 3-D, it would just go one way. In order to have shapes in the real world you need infinity to

have ending points and boundaries where new shapes can exist in the same room. For example, all

of the furniture in your room you may think is just 2 feet or 3 feet wide, but imagine again if there is

no smallest unit of measure, if this could be true then there everything we see that has shapes and

therefore a beginning and an end if measured, is also infinite in size when you shink yourself to a

very small size an look at it. There is no end to how small things can be and no end to the fact that

these infinite shapes have contours and therefore 9.999....643...999.... can exist!! (this is not the belief

of most math people, however: (disclaimer))

**igloo** **myrtilles** **fourmis**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

i like persistence,as long as it doesn't turn into stubbornness.we didn't get there yet,but we may very soon.

i also like your point of view,but if infinity had boundaries it wouldn't be infinity.

if you go through the digits of 9.9999...899999999...,you would never get to the 8.in order for this number to exist you would need a finite number of 9s before the 8.

and btw even if 9.9999...899999...exists it isn't the same as 9.9999... so you haven't disproved that 9.9999....=10

Here lies the reader who will never open this book. He is forever dead.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi John;

I do not think you can do that. When you write:

.99999999999... the three dots means continue on forever with nines. If you put an 8 later than the original statement is violated.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**Calligar****Member**- Registered: 2011-09-24
- Posts: 234

Okay, to bobbym...

Rule 1) .999999999... = 1

You can now accept this and move on.

It is true math has rules, but math is directly logical. In order to do math, you start with the rules, and using those rules, you work your way through it logically. Adding rules later just to justify things aren't....correct persay; if they directly defy logic because of it, then that doesn't make them correct. However, since there are rules against this, and its controversial as I said earlier, I think this will be my last argumentative post about that. I still don't agree with it though, but thank you.

To anonimnystefy,

What I was explaining earlier was that 0.3 recurring and 0.3 recurring were different because of how they are represented.

1/3 = 0.3 recurring because that is the only way to express it in decimal form, but that doesn't mean 0.3 recurring = 1/3, reason is because what if I simply wanted to use the number 0.3 recurring NOT intending it to be 1/3, as I see this as arguably possible. But in turn, something that might get confusing about it, both of those 0.3 recurring numbers would in fact be 2 different numbers, without a way to express the difference in decimal form.

just tell me the difference between 0.3 reccuring and 0.3 reccuring? then i will accept almost anything you say (not really but anyway,...).

One is showing 0.3 recurring, recurring 3s go on forever, the way it can be expressed accurately in decimal form, however the other 0.3 recurring per-say was equal to 1/3, as in exactly equal, which can not be expressed accurately as a decimal form, nor any other way. Second, I'd rather you not accept everything I'd say, because I'm neither perfect nor right about everything.

Third, an infinitesimal can be found, just not using limiting numbers.

also,an infinitesimal is just a term,but none can really be found.you cannot find any numbers greater than zero that are less than any other number.

1/infinity is not defined!

This is why I first posted in the first place, to express that 3.3 recurring + 6.6 recurring = 9.9 recurring, not 10, because 9.9 recurring is 1/∞ away from 10, as they are not limited. It is therefore not making it the same number UNLESS it was originally intended that it was 3 1/3 + 6 2/3 = 10 expressed as a decimal, which in that case is correct. However, I will not argue this any further, due to the fact that there are apparently rules in math that go against this according to bobbym, which whether I disagree with it or not, it is stating I'm wrong....

*Last edited by Calligar (2011-10-10 08:59:14)*

Life isnt a simple Math: there are always other variables. -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end. -Aristotle

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

hi

Ive been watching this discussion for some time, without comment, but feel its time to throw in my thoughts.

Numbers, all numbers, are just abstract concepts. You cannot hold a three in your hand. You can have three apples or write a squiggle on the page to represent three, but three doesnt exist except in the mind of a mathematically inclined person.

Weve spotted that you can have consistent rules for these abstract concepts. For example, if Ive got three apples and someone gives me another four apples then Ive got seven apples. If Ive got three goats and I buy four more goats, then, when I count them up, I find Ive got seven goats.

So, in certain circumstances, its ok to say 3 + 4 = 7.

But, notice this rule doesnt always work. If I walk 3 miles and then walk 4 miles, I'm not necessarily 7 miles away from where I started.

Someone once critcised Isaac Azimov for using an "imaginary" number (square root of minus one.) But Azimov responded by arguing that all numbers are imaginary (ie. made up figments of our imagination). When the critic protested that he knew some numbers were completely real, like a half for example, Azimov pointed to the blackboard (this tale is an old one) and asked him to hand over half a stick of chalk!

The point is, you make the rules for the situation. If the rule works then its worth knowing. Otherwise its likely to be consigned to the great rubbish bin of ideas that didn't get off the drawing board.

So, for instance, when we make the rule for powers:

we find it works ok with all the other stuff we know about powers.

Then when someone poses the question I wonder if we can find a sensible meaning for:

the answer is, yes, we can. It makes good sense to let it have the value 1 because this is consistent with the rule of powers:

Now to get back to those infinitely recurring decimals. They are just another abstract form of number. I know I cannot write all the digits, but I can still think about how the number behaves as if I could. I know that if I try to work out 1 divided by 3, Ill keep getting a remainder, and so if I keep on dividing Ill end up with 0.33333333r.

So, its entirely consistent to treat 1/3 and 0.3333333333r as if they were the same. It does lead on, as others have shown, to results like ½ = 0.5 = 0.49999999999r but so what? The rules that allow this are quite consistent and allow us to teat all terminating decimals as if they were recurring. eg. 0.37 = 0.36999999999999r. That then allows us to explore the concept of number density as in the reals are more dense on the number line than the rationals.

And theres nothing wrong with exploring mathematically interesting concepts. Number theory, and especially all the theorems involving primes, were once thought to be the province of the most pure of pure mathematicians. But now this theory is essential for internet trading.

And consider all those uses for complex numbers. Not so imaginary now, are they?

So dont worry if some mathematical ideas seem a little strange when you first meet them. You just need to let your imagination run with the ideas and be prepared to see where it all leads.

Infinity, underground maps, non Euclidean geometry .... all are possible in the world of mathematics. And theres definitely a continent where

If youve read this far, thank you for your patience.

Bob

*Last edited by bob bundy (2011-10-10 11:23:08)*

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**Au101****Member**- Registered: 2010-12-01
- Posts: 286

bob bundy wrote:

If youve read this far, thank you for your patience.

I don't wish to appear sycophantic or anything, but actually I think you made some pretty good points which the over-zealous (I'm certainly thinking of my younger self when I say this) should think long and hard about, before treating mathematics as something which it is not.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,535

Very nicely explained, bob, well done.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

T. Planiha wrote:

Nothing unreal exists!

Jean Dieudonné wrote:

The number 3 exists, because I can immediately see 3 apples. The number 10^10 is an abstraction. It is ridiculous to say I know what 10^10 is. It only has the meaning I give it through the axiomatic system.

Dr. Mifune wrote:

I am not a madman, Mwhahahahahah...

Darn! I am going to get the last word in on one of these .999999999999...., 7.77777777777...,6.6666666666...., 3.333333333333...., 0.00000000000..., ∞.∞∞∞∞∞∞∞∞∞∞..., discussions. I will achieve victory by writing a lengthy rebuttal in the old style. No one will be able to get through it and I will win.

Begin rant:

This kind of stuff is only a bother to the topologists, analysts and logicians who worry about infinity all the time. They suffer with Tarski's paradox, with who cuts whose hair, catalogs of catalogs that do not list themselves, or do and they need two different frameworks to keep set theory afloat. Do we have the axiom of choice today or should we leave it out? Is the continuum hypothesis, true, false? No it is undecidable. Bah, humbug!

To us computational boys this is fine:

Compute 10 / 3 to 16 digits.

3.333 333 333 333 333

Compute 10 / 3 to 25 digits.

3.333 333 333 333 333 333 333 333

Compute 10 / 3 to 31 digits.

3.333 333 333 333 333 333 333 333 333 333

Compute 10 / 3 to hundreds of digits.

3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333

Notice the lack of ... on the end of any of them. Fellas, that is what 10 / 3 is if that is all you have time or space for to compute. Remember T. Planiha.

We do not think about what is on the end of that...

To the discretists infinity is the largest number that can currently fit in your computer. See the work of David Deutsch or Doron Zeilberger. Look up DZ's vid on transfinite numbers. If you see it you are in for a shock. A quantum computer type said I was too full of infinity and those 3 dots to understand it. Now, I have been called full of a lot things but never infinity. Those are fighting words where I come from so he is dead but he may have been right.

Doron Zeilberger wrote:

Euclid ruined mathematics.

In the book "1984," Orwell sought to banish thoughtcrime by reducing the language to the point where incorrect thought was impossible.

The discretists ( begins with Kummer ) seek to eliminate the controversy in mathematics by eliminating the need for infinity, continuous math, or even an axiomatic system.

End rant:

(my guess is they will just pass it up and ignore it)

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**reconsideryouranswer****Member**- Registered: 2011-05-11
- Posts: 172

bob bundy wrote:

but three doesnt exist except in the mind of a mathematically inclined person.

It need not be a person. Certain animals have a concept of numbers and counting.

And numbers, (counting numbers, for instance) have their own existence independent

of human thought. Before there were humans to think about numbers,

there were always places where there was one thing, two things, etc.

bob bundy wrote:

So, in certain circumstances, its ok to say 3 + 4 = 7.

But, notice this rule doesnt always work. If I walk 3 miles and then walk 4 miles,

I'm not necessarily 7 miles away from where I started.

This is not a correct example of "a rule that doesn't always work,"

as you put it. You're comparing apples and oranges. Those movements equal

a total of 7 miles walked. The rule works. Mixing in where someone ended

up relative to the starting point is changing the subject.

bob bundy wrote:

So, for instance, when we make the rule for powers:

we find it works ok with all the other stuff we know about powers.

Then when someone poses the question I wonder if we can find a sensible meaning for:

the answer is, yes, we can. It makes good sense to let it have the value 1 because this

is consistent with the rule of powers:

You have made a blanket statement. It is not so where x = 0.

Under that method,

So 0^0 could equal 1,

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi;

reconsideryouranswer wrote:

but it could equal other values as well.

It can be indeterminate, undefined or 1. Those are the ones that are consistent and make sense with all branches of math.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

to reconsideryouranswer

This is not a correct example of "a rule that doesn't always work,"

as you put it. You're comparing apples and oranges. Those movements equal

a total of 7 miles walked. The rule works. Mixing in where someone ended

up relative to the starting point is changing the subject.

I was trying to simplify the idea that the rules for, say, vectors, are not the rules for arithmetic. You have supported my argument by demonstrating that one needs to be clear of the circumstances before applying a blanket rule . And ...

You have made a blanket statement. It is not so where x = 0.

Under that method,

<bits of code lost here>

So 0^0 could equal 1,

<bits of code lost here>

Here again you have shown that mathematicians may select their own rules to fit what they want to use them for.

You could make an axiom that 1 = 2. Plus all the usual rules.

It wouldn't be long before you found that the whole number system collapses down to a single number, 1, and a single binary operation, say, times.

Thus you would end up with 1 x 1 = 1. True but somewhat limited in application.

That's my point. You can make up the rules and explore where it leads. If it has use then others will want to use it too.

It need not be a person. Certain animals have a concept of numbers and counting.

And numbers, (counting numbers, for instance) have their own existence independent

of human thought. Before there were humans to think about numbers,

there were always places where there was one thing, two things, etc.

That's very interesting. If it's true, I don't think it changes my argument; just substitute 'some animals including humans' for 'person'. But which animals did you have in mind?

If all humans (and the animals of the above paragraph) were to disappear from the Universe there would still be one thing and two things etc. Yes, but where is the concept of 'three'. You need an intelligence to conceive a 'three'. I still claim it has no independent existence.

Bob

*Last edited by bob bundy (2011-10-10 19:52:49)*

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**reconsideryouranswer****Member**- Registered: 2011-05-11
- Posts: 172

bob bundy wrote:

.

It need not be a person. Certain animals have a concept of numbers and counting.

And numbers, (counting numbers, for instance) have their own existence independent

of human thought. Before there were humans to think about numbers,

there were always places where there was one thing, two things, etc.That's very interesting. If it's true, I don't think it changes my argument;

But which animals did you have in mind?

just substitute 'some animals including humans' for 'person'.

Certain chimpanzees

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**Calligar****Member**- Registered: 2011-09-24
- Posts: 234

Okay, I have spoken to my brother about this in great detail (whom I consider a math genius). He was explaining to me that this has been argued numerous times before in history, and ultimately from what it is considered now, I was wrong about this. On top of that, he corrected me about infinitesimals saying that was the accepted way how to represent that, but that doesn't make what I say related to one, as infinity is mostly only used in calculus and higher. As an example, he showed me an indefinite integral problem. Now, my brother did state something interesting. Logically, I can argue this, because he understands the way I'm looking at this, and can understand where I am coming from. However, that according to the rules that already exist in math to prevent misunderstandings and things that people don't understand well enough to be definite on, that I am wrong, and that in order to be right, the rules would need to be changed on that. But just as Bobbym had said in an earlier post, there were in fact rules in place that my brother actually showed and explained to me why it was like that. So to conclude all of this, I admit defeat. I was told I was wrong here, only to be proven it later by my brother. It's like my brother said, I can argue this logically, but that wouldn't make it correct in math. So, let me say this: 3.¯3+6.¯6=10 because 9.¯9 DOES equal 10, unlike what I was arguing before.

Life isnt a simple Math: there are always other variables. -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end. -Aristotle

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi Calligar;

Very good! Glad your brother helped.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Au101 wrote:

(I'm certainly thinking of my younger self when I say this)

This reminds me of "By his bootstraps" and "The man who folded himself"...

Here lies the reader who will never open this book. He is forever dead.

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,555

10.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

I give it a 9.99999999999...

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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