# I The number zero

#### Dacu

Good morning,
How do we prove that the number zero is a negative or positive number?

#### jedishrfu

Mentor
Zero by definition is neither positive nor negative so there is no proof for it.

Here's a brief discussion on Zero:

Mathematics
0 is the integer immediately preceding 1. Zero is an even number, because it is divisible by 2 with no remainder. 0 is neither positive nor negative. By most definitions 0 is a natural number, and then the only natural number not to be positive. Zero is a number which quantifies a count or an amount of null size. In most cultures, 0 was identified before the idea of negative things, or quantities less than zero, was accepted.

The value, or number, zero is not the same as the digit zero, used in numeral systems using positional notation. Successive positions of digits have higher weights, so inside a numeral the digit zero is used to skip a position and give appropriate weights to the preceding and following digits. A zero digit is not always necessary in a positional number system, for example, in the number 02. In some instances, a leading zero may be used to distinguish a number.
from wikipedia:

https://en.wikipedia.org/wiki/0_(number)

• symbolipoint

#### Dacu

Why zero , in France , can be a negative or positive number?
My reasoning:
1) $$0>0$$ involves $$0=0+a$$ where $$a>0$$ and involves $$0+a>0+a$$ and $$a>a$$ and this is false.
2) $$0<0$$ involves $$0=0-a$$ where $$a>0$$ and involves $$0-a<0-a$$ and $$-a<-a$$ and this is false.
So the number zero can not be negative or positive.What was supposed to prove.My proof is correct?

#### Mark44

Mentor
Why zero , in France , can be a negative or positive number?
I doubt that this is correct. By definition of the terms negative and positive, zero is neither negative nor positive.
Dacu said:
My reasoning:
1) $$0>0$$ involves $$0=0+a$$ where $$a>0$$ and involves $$0+a>0+a$$ and $$a>a$$ and this is false.
You have started with a false assumption; i.e., that 0 is larger than itself. No finite number can be larger than itself, nor can it be smaller than itself.
Dacu said:
2) $$0<0$$ involves $$0=0-a$$ where $$a>0$$ and involves $$0-a<0-a$$ and $$-a<-a$$ and this is false.
So the number zero can not be negative or positive.What was supposed to prove.My proof is correct?
You're starting with a false assumption here, as well.

From wikipedia, https://simple.wikipedia.org/wiki/Positive_number
A positive number is a number that is bigger than zero.

The definition for negative number is similar.

#### jbriggs444

Homework Helper
I doubt that this is correct. By definition of the terms negative and positive, zero is neither negative nor positive.
By another definition, zero is both negative and positive. I was taught to use the terms "strictly positive" and "strictly negative" in order to avoid the resulting potential for ambiguity. On the other hand, that class was some forty years ago and fashions may have changed.

#### Ssnow

Gold Member
This fact

or this

doesn't convince me ...

#### Samy_A

Homework Helper
Why zero , in France , can be a negative or positive number?
(bolding mine)
Probably the confusion is just a matter of definition.
Wikipedia in English said:
The number 0 is neither positive nor negative
Wikipedia in French said:
Zéro est le seul nombre qui est à la fois réel, positif, négatif et imaginaire pur.
Translated, that says: "Zero is the only number that is real, positive, negative and purely imaginary".
So, it really depends if by "positive" one means > 0, as the English Wikipedia does, or ≥ 0, as the French Wikipedia does.

• pwsnafu

#### cmb

Can you not say here;

lim x; 1->0 [0+x^2] = +0
and
lim x; 1->0 [0-x^2] = -0

because for any non zero trial 'x' you can choose there is always a smaller value between the trial and zero as it tends to zero?

So if you create the specific algebraic conditions where you approach zero down to infinitesimals, then you can create a zero with a parity? If that is true, I think it would be strictly as a result of your definition.

#### Mark44

Mentor
lim x; 1->0 [0+x^2] = +0
and
lim x; 1->0 [0-x^2] = -0
What do you mean here? Something like this?
$$\lim_{x \to 0}x^2 = \lim_{x \to 0}-x^2 = 0$$
There's no difference between +0 and -0.
because for any non zero trial 'x' you can choose there is always a smaller value between the trial and zero as it tends to zero?
Well, of course -- that's what it means to be a limit of some function.
So if you create the specific algebraic conditions where you approach zero down to infinitesimals, then you can create a zero with a parity?
No.
If that is true, I think it would be strictly as a result of your definition.

#### WWGD

Gold Member
Why zero , in France , can be a negative or positive number?
My reasoning:
1) $$0>0$$ involves $$0=0+a$$ where $$a>0$$ and involves $$0+a>0+a$$ and $$a>a$$ and this is false.
2) $$0<0$$ involves $$0=0-a$$ where $$a>0$$ and involves $$0-a<0-a$$ and $$-a<-a$$ and this is false.
So the number zero can not be negative or positive.What was supposed to prove.My proof is correct?
What is your definition of the overall ordering given by '<´ ? How is the 0<0 you start with allowed within your ordering while the more general a<a is not?

#### cmb

What do you mean here? Something like this?
$$\lim_{x \to 0}x^2 = \lim_{x \to 0}-x^2 = 0$$
There's no difference between +0 and -0.
Well, of course -- that's what it means to be a limit of some function.
Well, there is no difference between 0.99999.... (reoccurring) and 1 but you can still write them differently.

I'm not arguing that is the case, but mathematics is a strange business, it is 'partially discovered' and 'partially invented', it can mean what we want it to mean.

I do not argue that +0 = -0 but the parity may indicate how it was arrived at.

If I say 0+x, where x is the smallest positive number possible, then it is a positive value *by definition*, and whatever x you say is the smallest you know, I will suggest a smaller x closer to zero, until the *value* of +0 = 0 .

It reminds me of a discussion many years ago in a maths class in which the answer of some geometric problem came out as a triangle of angles 180, 0 and 0. There was a brief discussion as to whether this was actually a triangle or merely a line segment. Yes they are the same thing, but they were derived in different ways and, I would argue, *therefore* different.

If you could create a 'mathematically perfect' copy of a Leonardo da Vinci hand drawing, would they be 'the same'? The difference would be how it was generated as an output.

#### jbriggs444

Homework Helper
If I say 0+x, where x is the smallest positive number possible
There is no such number in the reals.

In any case, a thread from 2016 is not the place to discuss this idea.

• PeroK

#### cmb

There is no such number in the reals.

In any case, a thread from 2016 is not the place to discuss this idea.
Again you are discussing human definitions.

If humans choose to define +0 as being a specific number that contains an indication of how it was arrived at (1->0 rather than -1->0), i.e, it also contains a component of its history, then there is no reason to be so sure about not allowing it, other than to put it into the context of modern 'accepted' mathematical practices.

I do not dissent that convention dictates 0 has no parity, I was just clarifying that it appears, to me at least, only to be an accepted convention.

(I am not sure why the age of the thread is germane, I was just looking for anything about the history of 'zero'.)

#### jbriggs444

Homework Helper
Again you are discussing human definitions.
Of course. That is why I added the caveat: "in the reals".

By adhering to convention, I can reason about numbers using theorems developed by expert mathematicians over many years. If you want to discard convention then you will need to re-invent all of the algebraic structure around your personal number system. The theorems developed by others may not hold within your system because their assumptions may have been violated.

• cmb

#### WWGD

Gold Member
Well, there is no difference between 0.99999.... (reoccurring) and 1 but you can still write them differently.

I'm not arguing that is the case, but mathematics is a strange business, it is 'partially discovered' and 'partially invented', it can mean what we want it to mean.

I do not argue that +0 = -0 but the parity may indicate how it was arrived at.

If I say 0+x, where x is the smallest positive number possible, then it is a positive value *by definition*, and whatever x you say is the smallest you know, I will suggest a smaller x closer to zero, until the *value* of +0 = 0 .

It reminds me of a discussion many years ago in a maths class in which the answer of some geometric problem came out as a triangle of angles 180, 0 and 0. There was a brief discussion as to whether this was actually a triangle or merely a line segment. Yes they are the same thing, but they were derived in different ways and, I would argue, *therefore* different.

If you could create a 'mathematically perfect' copy of a Leonardo da Vinci hand drawing, would they be 'the same'? The difference would be how it was generated as an output.
There is no "Smallest positive number" within the standard ordering of the Reals.

#### Mark44

Mentor
I'm not arguing that is the case, but mathematics is a strange business, it is 'partially discovered' and 'partially invented', it can mean what we want it to mean.
If we invent a brand new area of mathematics, then yes, but virtually no one will agree with you if you maintain something absurd, such as 2 + 1 = 5.
I do not argue that +0 = -0 but the parity may indicate how it was arrived at.
There is no parity. Maybe a more concrete example will help.
Suppose you have $100 in your checking account. If you write a check for$100, your account balance is $0. If you then write a check for$200, your balance will then be -$200. (For the sake of simplicity I'm ignoring the overdraft fee that many banks will charge you.) If you then deposit$200, your account balance will be $0 again. At neither time when your balance was$0 was there any hint that this balance occurred due to money being deposited or withdrawn.
If I say 0+x, where x is the smallest positive number possible, then it is a positive value *by definition*, and whatever x you say is the smallest you know, I will suggest a smaller x closer to zero, until the *value* of +0 = 0 .
As already noted a couple of times, there is no smallest positive number in the reals, nor is there a largest negative number.
If humans choose to define +0 as being a specific number that contains an indication of how it was arrived at (1->0 rather than -1->0), i.e, it also contains a component of its history, then there is no reason to be so sure about not allowing it, other than to put it into the context of modern 'accepted' mathematical practices.
You are trying to imbue a one-dimensional object (a real number) so that it has two attributes associated with it: its value, and some sort of operation that produced it. You would need two dimensions to represent that situation, but the real number line is only one-dimensional.

Going back to the question in the first post of this thread --
How do we prove that the number zero is a negative or positive number?
-- if we consider how floating point numbers are represented in computers, per the IEEE-754 Standard for Floating Point Numbers, there are different binary representations for +0.0 and -0.0 for both 32-bit single precision numbers and for 64-bit double precision numbers. In both representations, if the most significant bit is 1, the number is considered negative; if the most significant bit is 0, the number is considered positive. Here the parity is due completely to the scheme that computer scientists have devised to represent a finite collection of real numbers, and in now way conveys any idea about how the result was arrived at. Also, I should emphasize that I am not discussing mathematics here, but merely how things are done in computers.

#### symbolipoint

Homework Helper
Gold Member
Is it not sophisticated enough to just understand that zero is EVEN and that zero is also non-negative AND non-positive?

#### Quasimodo

Where on earth, -0 and +0 taken as limits would fit in the line of the reals? And what would be there in between?
0?

#### cmb

Where on earth, -0 and +0 taken as limits would fit in the line of the reals? And what would be there in between?
0?
I guess the answer to that would be "0"

I'd have imagined it would look like this;- Some identities;-
0 - (-0) = +0
-0 * -0 = +0
-0 * +0 = -0
-1 * -0 = +0
1* - 0 = -0
1 - (-0) = 1 + (+0)

It also helps you derive one parity for division by zero;-
0/(-0) = -(undefined)
+0/0 = +(undefined)

So
-0/0 <> +0/0

It reminds me a bit of the 'Aleph-0' arithmetic .. if I recall the chapter Martin Gardner wrote about it, they thought Gregor Cantor had lost the plot a little, too, when he first suggested there were bigger numbers than infinity, but it doesn't stop you doing maths on it!

Perhaps this might be expanded to be a bit like 'reciprocal Aleph' arithmetic, maybe?

Hey. I don't know..... I am just pointing out that maths is what we define it to be. So long as you state your axioms to start with, the next question is whether you can you do any sensible mathematics with it? I can write out a list of identities, which I think anyone would write out likewise, so I guess the answer is 'yes'.

... Don't forget ...

+0i and -0i

SQRT(0+0i) = +0+0i & -0-0i
CubeRoot (0+0i) = +0+0i, -½∙0+⅔∙0i & -½∙0-⅔∙0i

(sorry, that needs some 'root' signs there for root(2)/3 but my keyboard doesn't cut it.)

Don't take it too seriously, it's just maths geeks playing in the sand pit! ;)

Last edited:
• weirdoguy

#### WWGD

Gold Member
I guess the answer to that would be "0"

I'd have imagined it would look like this;-
View attachment 251979

Some identities;-
0 - (-0) = +0
-0 * -0 = +0
-0 * +0 = -0
-1 * -0 = +0
1* - 0 = -0
1 - (-0) = 1 + (+0)

It also helps you derive one parity for division by zero;-
0/(-0) = -(undefined)
+0/0 = +(undefined)

So
-0/0 <> +0/0

It reminds me a bit of the 'Aleph-0' arithmetic .. if I recall the chapter Martin Gardner wrote about it, they thought Gregor Cantor had lost the plot a little, too, when he first suggested there were bigger numbers than infinity, but it doesn't stop you doing maths on it!

Perhaps this might be expanded to be a bit like 'reciprocal Aleph' arithmetic, maybe?

Hey. I don't know..... I am just pointing out that maths is what we define it to be. So long as you state your axioms to start with, the next question is whether you can you do any sensible mathematics with it? I can write out a list of identities, which I think anyone would write out likewise, so I guess the answer is 'yes'.

... Don't forget ...

+0i and -0i

SQRT(0+0i) = +0+0i & -0-0i
CubeRoot (0+0i) = +0+0i, -⅓∙0+⅔∙0i & -⅓∙0-⅔∙0i

Don't take it too seriously, it's just maths geeks playing in the sand pit! ;)
But the definitions we use are the ones that have shown to be useful, consistent. If you want to create your own and make it useful , let alone consistent, and productive, more power to you.

#### cmb

I agree, I do respect the answer jbriggs444 gave in #14.

It was just I was asked the question if 0 was between the two. Seemed to me like there would be an answer to that.

• weirdoguy

#### WWGD

Gold Member
You will see how hard it is to come up with something new and productive. Low-hanging fruit has been picked.

• russ_watters and cmb

#### Mark44

Mentor
I guess the answer to that would be "0"

I'd have imagined it would look like this;-
View attachment 251979
Your drawing makes no sense at all. There are no such numbers as $-0.\overline{000}1$ and $+0.\overline{000}1$. A notation such as $0.\overline{14}$ means that the pattern '14' repeats endlessly in the decimal expansion. In other words, 0.1414141414... where the dots (ellipsis) means the pattern repeats endlessly. You can't have an endlessly repeating pattern of 0's that is followed by a 1 digit.
cmb said:
Some identities;-
0 - (-0) = +0
-0 * -0 = +0
-0 * +0 = -0
-1 * -0 = +0
1* - 0 = -0
1 - (-0) = 1 + (+0)
The only place these would be useful is in a number system that distinguished between +0 and -0, which the real numbers do not. I mentioned earlier that computer systems use as standard that has different floating point values for -0.0 and +0.0, but that has only to do with how numbers are stored in computers, and isn't mathematics.
cmb said:
It also helps you derive one parity for division by zero;-
0/(-0) = -(undefined)
+0/0 = +(undefined)
This is complete nonsense. Division by 0 is undefined. It makes no sense to attach a sign to something that isn't defined.
cmb said:
So
-0/0 <> +0/0
Again, nonsense.
cmb said:
It reminds me a bit of the 'Aleph-0' arithmetic .. if I recall the chapter Martin Gardner wrote about it, they thought Gregor Cantor had lost the plot a little, too, when he first suggested there were bigger numbers than infinity, but it doesn't stop you doing maths on it!
What you're espousing has nothing to do with cardinalities.
cmb said:
Perhaps this might be expanded to be a bit like 'reciprocal Aleph' arithmetic, maybe?

Hey. I don't know..... I am just pointing out that maths is what we define it to be. So long as you state your axioms to start with, the next question is whether you can you do any sensible mathematics with it? I can write out a list of identities, which I think anyone would write out likewise, so I guess the answer is 'yes'.

... Don't forget ...

+0i and -0i
Same number -- no difference
cmb said:
SQRT(0+0i) = +0+0i & -0-0i
CubeRoot (0+0i) = +0+0i, -½∙0+⅔∙0i & -½∙0-⅔∙0i
Again, no difference.

Last edited:
• PeroK and jbriggs444

#### Mark44

Mentor
It was just I was asked the question if 0 was between the two. Seemed to me like there would be an answer to that.
There is: -0 = 0 = +0.
End of story.

• weirdoguy