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**Anthony.R.Brown****Banned**- Registered: 2006-11-16
- Posts: 516

Done it!

1/3 can Equal 1 again! providing it is done as part of the original Calculation! as in below,

A = (1/3 x 3) = 1

But as soon as you make it a separate Calculation! it losses it's precise value! as in below,

A = 1/3 (which now becomes 0.33333.....)

B = 3 x A

So B Now equals 0.99999999.......................

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**Dross****Member**- Registered: 2006-08-24
- Posts: 325

Anthony.R.Brown wrote:

B = 3 x A

So B Now equals 0.99999999.......................

... but since A = 1/3 (as you cited earlier in the same post), your logic leads us to the conclusion that:

B = 3 x A = 3 x 1/3 = 1

(again, this was given by you yourself...)

So we now have B = 1 *and* B = 0.9999999...

How is one to do anything other than conclude that 1 = 0.99999...?

Bad speling makes me [sic]

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**Patrick****Real Member**- Registered: 2006-02-24
- Posts: 1,005

I fail to see how you go from A = 1/3 * 3 to A = 1/3...

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

They slipped up in variable naming. But the idea is still true.

A = 3 × (1/3) = 1

B = (1/3) = 0.333...

A = 3B = 3 × (0.333...) = ?

Doing the calculation separately does indeed give a different answer ... *if you are a calculator!*

But that is because of the finite nature of the computations. We can think beyond that and see the result.

In fact, writing computer programs drives you up against this all the time ... it is often important to perform calculations in the correct order to get a good result.

3 × (0.333...) = 3/3 = 1

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

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**Toast****Real Member**- Registered: 2006-10-08
- Posts: 1,321

Changing your mind on 0.999... = 1 yet Anthony?

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

What does one-ninth in decimal, that being point one one one one forever really mean??

Can you really go on forever??

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

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Can you really go on forever??

Yes, the axiom of infinity (ZF set theory) guarantees it.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**Dross****Member**- Registered: 2006-08-24
- Posts: 325

Ricky wrote:

Yes, the axiom of infinity (ZF set theory) guarantees it.

Firstly, that axiom tells us that an infinite set exists - what says that we are dealing with it in this particular case?

Secondly, I question the truth of the axiom of infinity! (though not neccessarily because I think it's false)

First, how can you simply just *assume* such a set exists? Madness if I do say so! And the axiom of infinity relies on the existence of the empty set - now, define the term "set" without making reference to it containing elements*. If you cannot, does the term "the empty set" have any actual meaning?

* - and an element by any other name is still an element!

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**Anthony.R.Brown****Banned**- Registered: 2006-11-16
- Posts: 516

To Dross

Quote:

"... but since A = 1/3 (as you cited earlier in the same post), your logic leads us to the conclusion that:

B = 3 x A = 3 x 1/3 = 1"

A.R.B

Your mistake is in the above! as I said B is now part of a different Calculation! and A has lost its precise Value! the answer is below!

B = 3 x A (which now is 0.999... because A has become 0.333..)<> (1/3) because this = 1

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**Anthony.R.Brown****Banned**- Registered: 2006-11-16
- Posts: 516

So there we have it!

When someone says Infinite 0.9 is <> 1 they are correct!

Because Infinite 0.9 is a Value that is not related to 1/3 by the very fact,that 1/3 is not a precise Value,it is only a precise Value when used in the Calculation (1/3 x 3) = 1

the values within the Brackects () cannot be seen or known or wrote down!

But 1/3 = 0.333... we can see outside the Brackets as a true Value! which leads us to....

Infinite 0.9 = 3 x 0.333...... which as we know = 0.999999999...........

I rest my case!!!

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**Patrick****Real Member**- Registered: 2006-02-24
- Posts: 1,005

Well, I just learned something there Anthony! What I learned is, that I should never even consider reading any of your posts again. You don't listen and you don't make sense.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Firstly, that axiom tells us that an infinite set exists - what says that we are dealing with it in this particular case?

The axiom of infinity tells us than an infinite set exists. All a number is, is a certain set. Thus, a collection of infinite numbers exist. This answers the question: "Can you really go on forever??" with a resounding "yes".

First, how can you simply just assume such a set exists? Madness if I do say so! And the axiom of infinity relies on the existence of the empty set - now, define the term "set" without making reference to it containing elements*. If you cannot, does the term "the empty set" have any actual meaning?

It's an axiom, that's how. Axioms are assumptions. Also, we define the empty set S such that for all x, x is not in S. Did I say element? Did I make any qualification on what x is? No. x can be anything. Literally, anything. It could be my dog, or your left shoe. Doesn't matter, it's not in S. And the existence of such an empty set is an axiom as well, another assumption.

You prove that ZF set theory is not consistent, then you can say we can't have that as axiom. Also, I will eat my hat. Of course, on the other hand, ZF set theory has not been proven consistent either, and it is impossible to do so. You can't use ZF set theory to prove the consistency of ZF set theory.

Also, one of the misconceptions is the term "element". An element is a set and a set is an element of something (thanks to a few other axioms, specifically axiom of the power set and axiom of pairing). There is no difference between a set and an element. Its just that some sets are elements of other sets.

Edit:

Also, we never define what a set is. We define what is **a** set, and make a couple of restrictions on what can be a set (Axiom of regularity), and detail how sets may be combined or split up to make new sets, but we never say what a set is. There is no need to.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**Anthony.R.Brown****Banned**- Registered: 2006-11-16
- Posts: 516

ToPatrick

Quote: "I fail to see how you go from A = 1/3 * 3 to A = 1/3..."

A.R.B

A = (1/3 x 3) = 1 So A now equals 1 again! A = (1/3 x 3) = 1 etc..........

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**Anthony.R.Brown****Banned**- Registered: 2006-11-16
- Posts: 516

CONCLUSION!

A = (1/3 x 3) = 1 "A never equals 0.3... or 0.9.... in this Calculation within Brackets!"

B = 1/3

B "now equals 0.3......" B x 3 = 0.999999999............Infinite 0.9 has now been found! which will

always be < 1

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**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

there is no difference between 1/3 and 0.333...., they are different representations of the same number, one in fractional form, and one in expanded decimal form, so how can you say that they are equal, but that suddenly, multiplying by 3 makes them inequal?

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

I think he is saying that 0.333... is an inaccurate way of representing (1/3).

But I am saying it is only inaccurate if you have a limited number of decimal places, such as a computer or calculator would.

But the "..." notation we are using here implies an infinite number of decimal places, and if you think about *never ending* decimal places, then 0.333... = 1/3, which I think is the point of disagreement.

So, ARB, does 0.333... = 1/3?

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

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

... and just to make things more interesting ...

The decimal representation of (1/3) is 0.333...

The ternary (base 3) representation of (1/3) is 0.1

The nonary (base 9) representation of (1/3) is 0.3

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

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

I'll have to check out this ZF set theory someday when I'm prepared.

Thanks for mentioning it.

What are the usual prerequisite courses that are useful to learn prior to ZF set theory?

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

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

A good basic knowledge of logic and naive set theory (a set is a group of things, subsets, elements of a set, what the null set is, union, intersection, etc) is really all thats required. But may I recommend you go very slowly through the axioms, as it is easy to get very quickly confused.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**Dross****Member**- Registered: 2006-08-24
- Posts: 325

Ricky wrote:

The axiom of infinity tells us than an infinite set exists. All a number is, is a certain set. Thus, a collection of infinite numbers exist*. This answers the question: "Can you really go on forever??" with a resounding "yes".

Yes, the axiom of infinity tells us that there exists *an* infinitely large collection of numbers - but please clarify for me what exactly tells us that we are dealing with such a collection *in this particular case*.

Obviously, we are not dealing with such a collection when we consider the set {1, 2, 3}. What I'm sort of saying is that I know we *can* go on forever (if the right conditions are met), but *do* we go on forever, with this particular situation?

Ricky wrote:

It's an axiom, that's how. Axioms are assumptions. Also, we define the empty set S such that for all x, x is not in S. Did I say element? Did I make any qualification on what x is? No. x can be anything. Literally, anything. It could be my dog, or your left shoe. Doesn't matter, it's not in S. And the existence of such an empty set is an axiom as well, another assumption.

I'm well aware that axioms are assumptions - I'm not asking for how you get to the assumptions, I'm simply trying to reject thm on the grounds of absurdity - I'm saying it's simply not true! Some people (most people) would reject the axiom that I am the only thing which exists in the universe on the grounds of absurdity, for example.

Ricky wrote:

You prove that ZF set theory is not consistent, then you can say we can't have that as an axiom. Also, I will eat my hat.

I'll join you on the hat thing. I reckon it won't be so bad with the right accompaniments - is it red or white wine with clothing?

Bad speling makes me [sic]

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**Anthony.R.Brown****Banned**- Registered: 2006-11-16
- Posts: 516

To luca-deltodesco, and MathsIsFun Administrator,

Lets make this more clear for you Both!

1/3 and 0.3333....etc... are different Numbers!

1/3 is a Number no one can write down! all we know is that its a third of a whole number!

0.3333..etc is a Number that is exactly what MathsIsFun Administrator Quoted:

"I think he is saying that 0.333... is an inaccurate way of representing (1/3)."

So providing we all agree 0.333..etc is inaccurate then we are getting somewhere!

B = 1/3 which now becomes 0.333..etc because its not enclosed in Brackets ()

Then B x 3 must equal 0.99999....etc as in Infinite 0.9 it does not have to be accurate! just

Infinite! i.e will always have the 0.1 missing!.........

A = (1/3) = "a third of a whole number!"

B = 1/3 = "The nearest decimal place Number i.e 0.3..."

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**Toast****Real Member**- Registered: 2006-10-08
- Posts: 1,321

But... there is nothing missing. You can't say that the smallest possible number is missing because there is no such thing. The concept of infinity eliminates it. Infinity is so infinitely enormous that it, by brute force, vanquishes the smallest possible number.

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

Well, Infinity is ... ummm ... What is Infinity?

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**Anthony.R.Brown****Banned**- Registered: 2006-11-16
- Posts: 516

To Toast" is this a name?"

Quote:"But... there is nothing missing. You can't say that the smallest possible number is missing because there is no such thing. "

A.R.B

Only you said the smallest possible number is missing!

Decimal 0.9999999999........remains 0.9

A = (1/3 x 3) = 1

B = 1/3 = 0.333.....

B x 3 = 0.99999999999999..................

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**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

you keep repeating that, and repeatedly, make absolutely no sense at all

/*quoting you*/

[1] ----> A = (1/3 x 3) = 1

[2] ----> B = 1/3 = 0.333.....

[3] ----> B x 3 = 0.99999999999999..................

[4] ----> When someone says Infinite 0.9 is <> 1 they are correct!

let me show you how you are making no sense at all.

you say that 1/3 × 3 = 1, you state that at [1]

you then say that 1/3 = 0.333...., you state that at [2], so you are saying that 1/3 and 0.333.... are exactly the same, they are equal

and yet, you then go onto say that 0.333....×3 = 0.999...., you state this at [3], and yet, you relentlessaly argue, that 0.999... doesn't equal one, [4]

if 1/3 and 0.333.... are exactly the same, equal, which is what you stated here, then 1/3 × 3 and 0.333... × 3, are also equal, by the axioms of the most basic math, but yet, you deny that they are the same, you can't even get it together yourself to make a non contradicting conclusion.

*Last edited by luca-deltodesco (2007-01-11 01:45:06)*

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