Is zero positive or negative ?

[Anonymous217]

If you approach zero on the real axis then it's puely real, although it's negative on the left and positive on the right.

If you approach zero on the imaginary axis, then it's purely imaginary, negative on the bottom and positive on top.

These are not the only ways to approach zero in the complex plane. If you approach zero in any other way then it is neither purely real, nor purely imaginary. You also lose the notion of positive or negative.

So, can we say that in the complex plane zero is both and neither but depends on how you approach zero?

That's being too vague on your definition. As I said, it depends moreover on how you define your set which contains 0. Any approach to zero in the complex plane approaches 0 = a + bi = 0 + 0i, which is 0 in the real and 0 in the complex. Or you could consider C as R^2, so 0 in C is just (0,0) in R^2. In R^2, (0,0) is "purely real". However in C, 0 is "both imaginary and real". So even if you're considering two isomorphic structures, what the element actually means depends on what the structure is.

This is your equivalent question: "What does the identity element represent in set-theoretic terms for any set containing it?" However, this can't be answered because first of all, it's not even possible to generalize what the identity element represents, since what it represents depends on the set you're discussing. In one set, the identity could be "purely real" and in another, the identity could be "purely bananas".

 

[agentredlum]

Does "it" in your sentence refer to the same thing? If so, zero is neither positive nor negative.Like Anonymous217 said, it depends on which zero you're talking about. Zero in the reals is different from zero in the complex numbers.
Both real and imaginary AND neither real nor imaginary?

That's being too vague on your definition. As I said, it depends moreover on how you define your set which contains 0. Any approach to zero in the complex plane approaches 0 = a + bi = 0 + 0i, which is 0 in the real and 0 in the complex. Or you could consider C as R^2, so 0 in C is just (0,0) in R^2. In R^2, (0,0) is "purely real". However in C, 0 is "both imaginary and real". So even if you're considering two isomorphic structures, what the element actually means depends on what the structure is.

This is your equivalent question: "What does the identity element represent in set-theoretic terms for any set containing it?" However, this can't be answered because first of all, it's not even possible to generalize what the identity element represents, since what it represents depends on the set you're discussing. In one set, the identity could be "purely real" and in another, the identity could be "purely bananas".

Well, let's just consider the real number line for now. If I want to discover whether zero is positive or negative, is it wrong to consider other numbers close to zero?

AFAIK zero is neither positive nor negative by definition. A definition that disregards the evidence.

Now let's consider the complex plane. Is it wrong to try to discover features of zero by considering features of other numbers in the neighborhood of zero?

Clearly not, as this is done all the time, numbers are explained by other numbers very close to them.

 

[SteveL27]

If you approach zero on the real axis then it's puely real, although it's negative on the left and positive on the right.

If you approach zero on the imaginary axis, then it's purely imaginary, negative on the bottom and positive on top.

These are not the only ways to approach zero in the complex plane. If you approach zero in any other way then it is neither purely real, nor purely imaginary. You also lose the notion of positive or negative.

So, can we say that in the complex plane zero is both and neither but depends on how you approach zero?

0 is real, because its imaginary part is zero; and its imaginary, because its real part is zero. Nothing to do with how you approach it. It's 0 + 0i regardless.

 

[agentredlum]

0 is real, because its imaginary part is zero; and its imaginary, because its real part is zero.

I don't have a problem with zero being both, the way I like to understand, it is because zero is at the intersection of the axes in the complex plane.

I'm just thinking here, if you approach zero on a straight line 45º to the positive real axis, then you stay away from both axes. Think of zooming in, the picture looks the same. So you can reach zero by AVOIDING all purely real and all purely imaginary numbers.

Of course there are many other ways to approach zero.

 

[Travis_King]

1: Zero is neither positive nor negative BY DEFINITION
2: Zero has both real and imaginary parts, just like all numbers. But like real integers, it's imaginary component is zero. The question is semantical and nonsensical.

 

[Anonymous217]

This is going to be my third time repeating it, but it depends on the set you're discussing. If you consider the complex numbers as your set, then ALL real numbers have both a real and imaginary part. That is, for x in R<C, x=x + 0i.

However, considering the reals as your set, then ALL real numbers have only a real part. That is, for x in R, x=x. That's it, and zero is only a specific case for x. It doesn't matter how you approach it; we're defining the set to be something, and so all elements are part of that 'something'. Shouldn't be too hard to understand..

 

[agentredlum]

1: Zero is neither positive nor negative BY DEFINITION
2: Zero has both real and imaginary parts, just like all numbers. But like real integers, it's imaginary component is zero. The question is semantical and nonsensical.

Ah-hmmmm...

x = -x has solution 0

substitute back

0 = -0 (result1)

x = -x has solution -0

-0 = -(-0)

-0 = 0

same as (result1)

-0 = 0 = +0

all 3 are equal

to say zero has no sign disregards evidence and is purely by convention, not by truth or proof.

 

[Travis_King]

Yea. What's -0 + -0?
What's +0 + (+0)?
There is no meaning to a sign as it applies to zero. Because zero is neither on the plus nor minus size of zero. It is zero.

The only time you would ever use a signed zero is to say "We are going to take this value as zero, but keep in mind that it is really an extremely small negative number rounded to zero. It's use is merely practical, and your "proof" merely demonstrates that you can use a signed zero without any real problems--basically because, well, IT IS ZERO lol

edit: and just so we are clear. By practical, I mean I understand that it has implications as far as math goes (i.e. 1/+0 should be +inf and 1/-0 should be -inf) but it is only because we are taking the signed zero as a simplistic way of representing impossibly small numbers on either side of the number line and want to maintain sign for computing purposes (say, after an underflow or similar situation)

 

[Jadaav]

Definitely a good question. As far as i recall, time is not quantized in the SM, but the gap between 23:59.99 and 00:00.01 would at least be quite huge

What's SM ?

Only if you believe time is continuous. Do we have any evidence for that?

This thread seems to be confusing math with physics.

Physics just relates everything that's happening in term of maths, so what's the problem.

For time to start or reset at midnight ( by saying ), shouldn't there be a big bang ?

This question now goes back to big bang or at the moment of creation. Time and everything else was produced at big bang so its going to be continuous Or else how would you explain time restarting at that moment ? in fact it wouldn't even be called a moment. With time stopping, everything stops with the notion of time also.

Everything is interlinked, so when 2 things coincide to become a proof. Each one proving the other one.

 

[agentredlum]

Yea. What's -0 + -0?
What's +0 + (+0)?

It is 0 - 0i + 0j - 0k...

It is at the intersection of all co-ordinate axes, whether they be Complex, Quaternions, Octonians, etc.

Hooray!

LOL

 

[SteveL27]

I found this. It's interesting.

An imaginary number is a number with a square that is negative.

http://en.wikipedia.org/wiki/Imaginary_number

By that definition, 0 is not imaginary. I would say that Wiki is wrong. I'd say 0 is imaginary because a) its real part is zero; and b) it lies on the imaginary axis.

Of course this is only a matter of semantics, like "Is 0 a natural number?" But still ... Wiki is usually correct when it comes to technical facts.

Curious to see what people think about this.

 

[agentredlum]

I found this. It's interesting.

An imaginary number is a number with a square that is negative.

http://en.wikipedia.org/wiki/Imaginary_number

By that definition, 0 is not imaginary. I would say that Wiki is wrong. I'd say 0 is imaginary because a) its real part is zero; and b) it lies on the imaginary axis.

Of course this is only a matter of semantics, like "Is 0 a natural number?" But still ... Wiki is usually correct when it comes to technical facts.

Curious to see what people think about this.

The wiki article works if 0 = -0

then (0)^2 = 0

by transitive property (0)^2 = -0

So zero is imaginary because it's square is negative.

LOLOL.

 

[Mark44]

The wiki article works if 0 = -0

then (0)^2 = 0

by transitive property (0)^2 = -0

So zero is imaginary because it's square is negative.

-0, +0, and 0 all represent the same number, which is neither negative nor positive. Merely attaching a sign to 0 does NOT make it either negative or positive.

 

[Mark44]

Ah-hmmmm...

x = -x has solution 0

substitute back

0 = -0 (result1)

x = -x has solution -0

-0 = -(-0)

-0 = 0

same as (result1)

-0 = 0 = +0

all 3 are equal

to say zero has no sign disregards evidence and is purely by convention, not by truth or proof.

No. To say that zero has a sign ignores what you have shown here. Since -0 = 0 = +0, that should convince you that a sign on zero is irrelevant.

Since negative numbers are always smaller than positive numbers, your equation above should read -0 < 0 < +0. I hope that you will agree that this inequality is nonsense.

 

[agentredlum]

No. To say that zero has a sign ignores what you have shown here. Since -0 = 0 = +0, that should convince you that a sign on zero is irrelevant.

Since negative numbers are always smaller than positive numbers, your equation above should read -0 < 0 < +0. I hope that you will agree that this inequality is nonsense.

Yes, the inequality is nonsense. Zero is an exception. It is the only real number equal to it's negatve. You claim negative numbers are always smaller than positive numbers. If zero is not positive or negative, how does your argument above work?

I never said -0 is less than 0 don't know why you think I did...

 

[Mark44]

Yes, the inequality is nonsense. Zero is an exception. It is the only real number equal to it's negatve. You claim negative numbers are always smaller than positive numbers.

And this is not an idle claim. Every negative number is smaller than any positive number. A look at the number line should convince you of this.

If zero is not positive or negative, how does your argument above work?

What argument are you talking about?

I never said -0 is less than 0 don't know why you think I did...

Because you are saying that 0 is negative.

by transitive property (0)^2 = -0
So zero is imaginary because it's square is negative.

When you attach a sign to zero, you are implying that there is some difference between -0, 0, and +0. In some contexts (particularly in floating point number representations in computer science) there are differences, but in mathematics these all represent the same number.

 

[agentredlum]

And this is not an idle claim. Every negative number is smaller than any positive number. A look at the number line should convince you of this.
What argument are you talking about?

Because you are saying that 0 is negative.


When you attach a sign to zero, you are implying that there is some difference between -0, 0, and +0. In some contexts (particularly in floating point number representations in computer science) there are differences, but in mathematics these all represent the same number.

If I thought there was a difference I would not write -0 = 0 = +0

You do not want to accept the idea that a positive number can be equal to a negative number. For real numbers this happens only once, in the case of zero so it does not have any undesirable consequences for other real numbers. Why is the idea so offensive?

 

[micromass]

Why is the idea so offensive?

Because it's not true by definition.
What would be a benefit if 0 was considered positive and negative?? Is there a benefit?

 

[agentredlum]

Because it's not true by definition.
What would be a benifit if 0 was considered positive and negative?? Is there a benifit?

It makes the wiki article work!

See posts 61 and 62

LOL

 

[micromass]

It makes the wiki article work!

See posts 61 and 62

LOL

No it doesn't. The wiki article implies that 0 is not purely imaginary. I'm pretty sure that is indeed the standard definition.

 

[agentredlum]

No it doesn't. The wiki article implies that 0 is not purely imaginary. I'm pretty sure that is indeed the standard definition.

Well according to your post #11 I was starting to believe in you. Unless you disregard definitions only when it suits your argument. Now you are seeking shelter behind the definitions again.

I believe there is good reason to put the definition aside for a little while and explore consequences. I believe there is good reason to approach zero in different ways in the complex plane.

Example: If we approach 0 using a curve x^(2n +1) for large integer n then close to zero the curve 'hugs' the real axis. From the bottom on the left, from the top on the right.

If we use x^(2n) for large integer n then the curve 'hugs' the real axis from the top on both sides.

If we use y = x then it avoids both axes as I mentioned in a previous post.

 

[Mark44]

I thought there was a difference I would not write -0 = 0 = +0

Since we both agree there is no difference, and all three are equal, then what is the point of attaching a sign?

You do not want to accept the idea that a positive number can be equal to a negative number.

Correct, I do not accept this assertion. The positive numbers are to the right of zero; the negative numbers are to the left of zero. These are two disjoint sets, so there is no number that is in both sets. Therefore, there is no positive number that is equal to any negative number.

For real numbers this happens only once

No, it doesn't happen at all.

, in the case of zero so it does not have any undesirable consequences for other real numbers. Why is the idea so offensive?

I wouldn't call this idea offensive, but would describe it as nonsensical, for the reason that it goes against the definitions of "positive" and "negative."

 

[agentredlum]

Since we both agree there is no difference, and all three are equal, then what is the point of attaching a sign?
Correct, I do not accept this assertion. The positive numbers are to the right of zero; the negative numbers are to the left of zero. These are two disjoint sets, so there is no number that is in both sets. Therefore, there is no positive number that is equal to any negative number.

Well, then you have to put zero in a set all by itself. Why is it better to create a new set instead of putting it in both sets already there?

 

[Studiot]

What is wrong with the idea that zero is simultaneously a boundary point for both sets (the set of all negative numbers and the set of all positive numbers)?

All other numbers in each set are interior points.

Zero is then required to make this possible.

 

[Mark44]

I believe there is good reason to put the definition aside for a little while and explore consequences. I believe there is good reason to approach zero in different ways in the complex plane.

Example: If we approach 0 using a curve x^(2n +1) for large integer n then close to zero the curve 'hugs' the real axis. From the bottom on the left, from the top on the right.

If we use x^(2n) for large integer n then the curve 'hugs' the real axis from the top on both sides.

If we use y = x then it avoids both axes as I mentioned in a previous post.

These examples are irrelevant in a discussion of whether the real number zero is positive or negative or whether the complex number 0 + 0i is purely real, purely imaginary, or whatever.

What you say about the graphs of y = x^{2n + 1} and y = x²ⁿ, for large n, is true, but so what? You seem to be confusing the ideas of limits with how zero is defined.

 

[agentredlum]

What is wrong with the idea that zero is simultaneously a boundary point for both sets (the set of all negative numbers and the set of all positive numbers)?

All other numbers in each set are interior points.

Zero is then required to make this possible.

I love your argument, why didn't I think of that?

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

I think this link supports your argument.

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

 

[Studiot]

Well to look at it another way

There are three types of points

(1)Interior points where any neighborhood contains only members of the set.

(2)Exterior points where any neighbourhood contains no members of the set.

(3)Boundary points where any neighbourhood contains both members and non members of the set.

Zero satisfies (3)

 

[SteveL27]

No it doesn't. The wiki article implies that 0 is not purely imaginary. I'm pretty sure that is indeed the standard definition.

Are you saying that by the standard definition, 0 is not imaginary? That's interesting ... again, it's only semantic ... but if 0 is not imaginary, what's it doing on the imaginary axis?

The same Wiki article I linked earlier says that the [entire] vertical axis is the imaginary axis. So according to Wikipedia, 0 lies on the imaginary axis but is not imaginary.

Would you regard that as generally agreed upon? In other words if you cornered a half dozen colleagues at faculty tea and asked them if 0 is an imaginary number, what would they say? (Besides, "Micromass, go get us some tea and stop asking silly questions!" )

 

[micromass]

Are you saying that by the standard definition, 0 is not imaginary? That's interesting ... again, it's only semantic ... but if 0 is not imaginary, what's it doing on the imaginary axis?

The same Wiki article I linked earlier says that the [entire] vertical axis is the imaginary axis. So according to Wikipedia, 0 lies on the imaginary axis but is not imaginary.

Would you regard that as generally agreed upon? In other words if you cornered a half dozen colleagues at faculty tea and asked them if 0 is an imaginary number, what would they say? (Besides, "Micromass, go get us some tea and stop asking silly questions!" )

I think most would say that 0 is imaginary. I know I would say that. But I've seen books where they state "let x be purely imaginary or zero, ...". So I honestly don't know what the standard definition is, but I guess that 0 is not purely imaginary...

 

[Studiot]

I have to admit to being pretty disinterested in 'the Wiki article'

Surely the nature of 0 depends partly upon the set you are working with?

The integers, rationals, reals etc as do the positive and negative derived sets all include a unique element (0) such that

A + (0) = A for all A

To obtain such a property in the complex domain (where there is no ordering property) you must use (0,0)

 

[agentredlum]

Well to look at it another way

There are three types of points

(1)Interior points where any neighborhood contains only members of the set.

(2)Exterior points where any neighbourhood contains no members of the set.

(3)Boundary points where any neighbourhood contains both members and non members of the set.

Zero satisfies (3)

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.

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.

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?

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.

 

[micromass]

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.

Uuuh, if you believe zero to be positive, then the complement of the positives will not contain zero. If you believe zero to be negative, then the complement of the negatives will not contain zero.

This is silly.

 

[agentredlum]

Uuuh, if you believe zero to be positive, then the complement of the positives will not contain zero. If you believe zero to be negative, then the complement of the negatives will not contain zero.

This is silly.

Why did you ignore the word 'both'? an argument made not only by myself but other posters as well.

 

[micromass]

Why did you ignore the word 'both'? an argument made not only by myself but other posters as well.

I did not ignore the word both. If zero is both negative and positive, then zero is positive, no?

 

[agentredlum]

I did not ignore the word both. If zero is both negative and positive, then zero is positive, no?

yes, of course

 

[micromass]

yes, of course

So, what is wrong with my post then?

Uuuh, if you believe zero to be positive, then the complement of the positives will not contain zero. If you believe zero to be negative, then the complement of the negatives will not contain zero.

This is silly.

 

[agentredlum]

So, what is wrong with my post then?

You assume that zero will not go in with the negatives, but it must because it is negative also. At the same time, not taken as separate cases.

 

[micromass]

You assume that zero will not go in with the negatives, but it must because it is negative also. At the same time, not taken as separate cases.

I did not say that. I said: zero is positive, so will not be contained in the complement of the positive numbers. Nothing about negative numbers so far.

It's just because you want the complement of positive numbers to be the negative numbers that there is a contradiction. But in reality, there is no contradiction at all...

 

[agentredlum]

I did not say that. I said: zero is positive, so will not be contained in the complement of the positive numbers. Nothing about negative numbers so far.

It's just because you want the complement of positive numbers to be the negative numbers that there is a contradiction. But in reality, there is no contradiction at all...

Now you're going around in circles.

The assumption is that zero is both, you are assuming zero is ONLY positive.

 

[micromass]

Now you're going around in circles.

The assumption is that zero is both, you are assuming zero is ONLY positive.

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.

 

[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.