Is zero positive or negative ?

[agentredlum]

I did not.

Let me break this argument down. Tell me where you disagree

0 is both positive and negative
==> 0 is positive
==> 0 is not contained in the complement of the positive numbers.

Let me make it simple for you. If zero is both positive and negative you must put it in both sets. If it is neither positive nor negative, then it winds up in both sets anyway.

Forget about analyzing HALF a statement, it doesn't work that way.

 

[micromass]

Let me make it simple for you. If zero is both positive and negative you must put it in both sets. If it is neither positive nor negative, then it winds up in both sets anyway.

Forget about analyzing HALF a statement, it doesn't work that way.

Ok, if it doesn't work that way, then please tell me where I've gone wrong??

 

[agentredlum]

Ok, if it doesn't work that way, then please tell me where I've gone wrong??

0 is both positive and negative
==> 0 is positive
==> 0 is not contained in the complement of the positive numbers.

o-k in line 1 you define zero as both positive and negative

line 2 is vague because one cannot tell whether you mean only positive or you are considering the 2 properties separately

line 3 contradicts line 1 directly regardless of line 2

 

[Mark44]

I agree with everything here and can find no fault.

Consider the real number line.

Suppose I believe that zero is neither positive nor negative. Then someone can ask 'what is the complement of the set of positive numbers?' Then I got to put zero in with the negatives.

The complement of the set of positive numbers is the nonpositive numbers, {x | x $\leq$ 0}, which is the union of the negative numbers and zero. Including zero with the set of negative numbers doesn't mean that zero is negative.

Suppose again that I believe zero is neither positive nor negative and someone asks 'what is the complement of the set of negative numbers?' Then I got to put zero in with the positives.

The complement of the set of negative numbers is the nonnegative numbers, {x | x $\geq$ 0}, which is the union of the positive numbers and zero. Just as before, including zero with the set of negative numbers doesn't mean that zero is positive.

So even though I chose to exclude zero, it wound up in both sets anyway contradicting my belief.

Is there something wrong with this line of thought?

Yes.
If I buy a bag of apples and a banana at the store, and the checker puts the banana in with the apples, that doesn't mean that the banana has somehow turned into an apple. All it means is that the bag has apples and a banana in it.

If I believe that zero is both positive and negative, then the complement of the positives contains zero and the complement of the negatives contains zero avoiding a contradiction.

The only contradictions I see have to do with your flawed understanding of the meanings of the terms positive and negative.

 

[agentredlum]

The complement of the set of positive numbers is the nonpositive numbers, {x | x $\leq$ 0}, which is the union of the negative numbers and zero. Including zero with the set of negative numbers doesn't mean that zero is negative.The complement of the set of negative numbers is the nonnegative numbers, {x | x $\geq$ 0}, which is the union of the positive numbers and zero. Just as before, including zero with the set of negative numbers doesn't mean that zero is positive.Yes.
If I buy a bag of apples and a banana at the store, and the checker puts the banana in with the apples, that doesn't mean that the banana has somehow turned into an apple. All it means is that the bag has apples and a banana in it.

Good points, but the last comment is not appreciated by me. I am not the only one supporting that position and I think studiot made an excellent case for it. You found the flaw in my argument overlooked by me. Well done!

You should also send the checker to have his head examined, cause he's mixing apples and bananas.LOL

 

[Mark44]

Good points, but the last comment is not appreciated by me. I am not the only one supporting that position and I think studiot made an excellent case for it.

I think you are misunderstanding what studiot said. No one else in this thread is seriously arguing (contra the accepted definitions) that zero is both positive and negative.

You found the flaw in my argument overlooked by me. Well done!

 

[Studiot]

{x | x ≤ 0},

I have also been at pains to point out that you cannot use this argument or definition in the complex domain.

 

[Amergin]

I was always led to understand that zero was devised by Hindu mathematicians as a place filler and borrowed by the western traders and scientists to create our number system using Arabic numerals and getting rid of the virtually impossible to use Roman system that would not usefully allow computation. With the development of set theory it came to denote the empty set. All this huffing and puffing about positive or negative, odd or even, is just trying to count the angels on the point of a needle. Why does it have to have or need such properties? Consider it simply as a place filler in our number sysytem and as denoting not 'nothingness' but the absence of elements in a well defined set, both attributes which seem to me to be identical.
BTW if you have served in the armed forces you will know that they do not have a time of 12 midnight. It is either 23.59 or 00.01.

 

[agentredlum]

This was the first sentence of my argument.

'Consider the real number line.'

BTW Mark44 'explanation' works ONLY if you accept the definition that zero has no sign. If, for whatever reason, one decides to question that definition, his clever argument doesn't make sense.

Heres what he did...

He took an apple, decided to define it as a banana, then he put it in a bag with other apples and concluded that he now had a bag full of apples and 1 banana. In fact the bag contains only apples. His definition of a particular apple is irrelevant. It is what it is.

The fact still remains that

-0 = 0 = +0

For real numbers, any of the above symbols work.

 

[micromass]

0 is both positive and negative
==> 0 is positive
==> 0 is not contained in the complement of the positive numbers.

o-k in line 1 you define zero as both positive and negative

line 2 is vague because one cannot tell whether you mean only positive or you are considering the 2 properties separately

I'm considering the 2 properties separetely. You don't really want to argue that "P AND Q ==> P" is false, do you??

line 3 contradicts line 1 directly regardless of line 2

Well, it only contradicts line 1 in your world. As of now, you have still not told me where exactly I've gone wrong here.

 

[agentredlum]

You don't really want to argue that "P AND Q ==> P" is false, do you?

No, this is a tautology. Please post the entire argument then we can discuss it.

Hey, I'm a bit of a rebel, I may question usefulness of certain definitions, but I would never argue against a tautology, that would be silly!

 

[micromass]

No, this is a tautology. Please post the entire argument then we can discuss it.

I already posted the argument:

0 is positive and is negative
==> 0 is positive (by the rule "P AND Q ==> P" )
==> 0 is not in the complement of positive numbers (since for each set A, it holds $A\cap A^c=\emptyset$)

 

[agentredlum]

I already posted the argument:

0 is positive and is negative
==> 0 is positive (by the rule "P AND Q ==> P" )
==> 0 is not in the complement of positive numbers (since for each set A, it holds $A\cap A^c=\emptyset$)

line 3 contradicts line 1 in all worlds because if zero is negative you must put it in the set of negative numbers, so it would be in the complement of the positive numbers.

 

[micromass]

line 3 contradicts line 1 in all worlds because if zero is negative you must put it in the set of negative numbers, so it would be in the complement of the positive numbers.

I don't care. Just tell me where I have gone wrong.

 

[agentredlum]

I don't care. Just tell me where I have gone wrong.

Line 3 does not follow from line 2 if trichotomy property fails for 0.

Where you went wrong is you assumed that trichotomy property holds for zero.

-0 = 0 = +0

What is the point if you don't care?

Trichotomy property holds for all real numbers but for zero it is ill defined because one can write the above equation and it has meaning.

example: -1 = 1 = +1

clearly nonsense

-0 = 0 = +0

not clearly nonsense.

 

[micromass]

OK, so what you're saying is that the law

A∩A^c=∅fails. Right? Because that's what I used in (2)==> (3).

So, let me prove that, tell me where I went wrong:

By contradiction: Take x in A∩A^c
==> x is in A AND x is in A^c
==> x is in A AND x is not in A
Contradiction
So there does not exist x∈A∩A^c. So A∩A^c=∅.

So, where did I go wrong here?

 

[agentredlum]

OK, so what you're saying is that the law

$$A\cap A^c=\emptyset$$

fails. Right? Because that's what I used in (2)==> (3).

So, let me prove that, tell me where I went wrong:

By contradiction: Take x in $A\cap A^c$
==> x is in A AND x is in $A^c$
==> x is in A AND x is not in A
Contradiction
So there does not exist $x\in A\cap A^c$. So $A\cap A^c=\emptyset$.

So, where did I go wrong here?

I am having some trouble decoding Tex

Does the first line say 'take x in A or A complement'?

It would help if you can post it in words.

 

[micromass]

My previous post is now TeX-free.

 

[agentredlum]

My previous post is now TeX-free.

Thank you for Tex-free. I really mean that.

A intersection Ac = emty set

Not according to this, if x is considered the boundary

The boundary of a set is closed. The boundary of a set is the boundary of the complement of the set

Found here

EDIT: I posted the wrong link. Here is the right link

http://en.m.wikipedia.org/wiki/Boundary_(topology)

http://en.m.wikipedia.org/wiki/Equivalence_relation

this last link is helpfull in analyzing -0 = 0 = +0 under reflexive, symmetric, and transitive.

 

[micromass]

Thank you for Tex-free. I really mean that.

A intersection Ac = emty set

Not according to this, if x is considered the boundary

The boundary of a set is closed. The boundary of a set is the boundary of the complement of the set

Found here

http://en.m.wikipedia.org/wiki/Equivalence_relation

Boundary has nothing to do with this. Points in the boundary either lie in A or A^c, not in both. So boundary points are no contradiction to A intersection Ac = emptyset.

I observe that you did not point out the flaw in my proof...

 

[agentredlum]

Boundary has nothing to do with this. Points in the boundary either lie in A or A^c, not in both. So boundary points are no contradiction to A intersection Ac = emptyset.

I observe that you did not point out the flaw in my proof...

what does this mean in YOUR world?

The boundary of a set is closed. The boundary of a set is the boundary of the complement of the set

 

[micromass]

The boundary of a set is closed. The boundary of a set is the boundary of the complement of the set

So??

Care to explain what was wrong in my proof of A intersection Ac=empty??

 

[Mark44]

This was the first sentence of my argument.

'Consider the real number line.'

BTW Mark44 'explanation' works ONLY if you accept the definition that zero has no sign. If, for whatever reason, one decides to question that definition, his clever argument doesn't make sense.

With all due respect, this is a specious argument. No serious mathematician believes that zero has a sign. The definition for positive in my dictionary (American Heritage Dictionary of the English Language) has this definition:

12. Mathematics. Pertaining to or designating: a. A quantity greater than zero."

The definition for negative is a "quantity less than zero."

If we can't agree on the definitions of basic terms, then further discussion is futile.

Heres what he did...

He took an apple, decided to define it as a banana, then he put it in a bag with other apples and concluded that he now had a bag full of apples and 1 banana. In fact the bag contains only apples. His definition of a particular apple is irrelevant. It is what it is.

You are mischaracterizing what I said in post #94, which I'll add here verbatim.

If I buy a bag of apples and a banana at the store, and the checker puts the banana in with the apples, that doesn't mean that the banana has somehow turned into an apple. All it means is that the bag has apples and a banana in it.

I have a bag of apples. I have a banana. The checker rings these items up, and puts the banana in with the apples. The bag contains several apples and one banana. I don't know how to say it more simply than that.

The fact still remains that

-0 = 0 = +0

For real numbers, any of the above symbols work.

The question about whether zero is positive or negative (by implication from the question, we are talking about real numbers only) has been answered long ago in this thread. There is no point in going on and on about how the terms "positive" and "negative" are defined, so I am locking the thread.