Is zero positive or negative ?

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SUMMARY

Zero is defined as neither a positive nor a negative whole number, and it is classified as an even number due to the existence of an integer x such that 0 = 2x. In mathematical contexts, zero can be represented as both +0 and -0, which are treated as the same number in arithmetic but can have different implications in calculus and signed number representations in computing. Furthermore, zero is not considered a prime number, as it does not meet the necessary criteria for primality.

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  • #61
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.
 
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  • #62
SteveL27 said:
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.
 
  • #63
agentredlum said:
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.
 
  • #64
agentredlum said:
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.
 
  • #65
Mark44 said:
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...
 
  • #66
agentredlum said:
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.
agentredlum said:
If zero is not positive or negative, how does your argument above work?
What argument are you talking about?
agentredlum said:
I never said -0 is less than 0 don't know why you think I did...
Because you are saying that 0 is negative.
agentredlum said:
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.
 
  • #67
Mark44 said:
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?
 
  • #68
agentredlum said:
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?
 
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  • #69
micromass said:
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

:smile:
 
  • #70
agentredlum said:
It makes the wiki article work!

See posts 61 and 62

LOL

:smile:

No it doesn't. The wiki article implies that 0 is not purely imaginary. I'm pretty sure that is indeed the standard definition.
 
  • #71
micromass said:
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.
 
  • #72
agentredlum said:
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?
agentredlum said:
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.
agentredlum said:
For real numbers this happens only once
No, it doesn't happen at all.
agentredlum said:
, 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."
 
  • #73
Mark44 said:
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?
 
  • #74
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.
 
  • #75
agentredlum said:
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 = x2n + 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.
 
  • #76
Studiot said:
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)
 
  • #77
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)
 
  • #78
micromass said:
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!" :smile:)
 
  • #79
SteveL27 said:
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!" :smile:)

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...
 
  • #80
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)
 
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  • #81
Studiot said:
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.
 
  • #82
agentredlum said:
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.
 
  • #83
micromass said:
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.
 
  • #84
agentredlum said:
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?
 
  • #85
micromass said:
I did not ignore the word both. If zero is both negative and positive, then zero is positive, no?

yes, of course
 
  • #86
agentredlum said:
yes, of course

So, what is wrong with my post then?

micromass said:
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.
 
  • #87
micromass said:
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.
 
  • #88
agentredlum said:
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...
 
  • #89
micromass said:
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.
 
  • #90
agentredlum said:
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.
 

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