## Is 0 real?

For the discussion of math. Duh.

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abeautifulsomething
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### Is 0 real?

Right, so i got bored in my maths class today and to annoy my friend next to me i just asked him whether 0 exists.
This kinda progressed into a full blown debate, and despite the teachers efforts to tell us to shut up and work, we continued out discussion.
So does 0 exist?

Numbers are all relative to language and i can say that i have 1 pencil or 2 pencils. As a concept, 0 can exist and we can prove this through equations though surely as a quantity it can't. If i have 0 of something, i dont have anything therefore its not there in the first place.
Another way of explaining this i think is that if an object has a length, width and depth of all 0 then it is not there. And we dont descirbe millions of things infront of us as having 0cm sides jsut because we can see the room on the other side.

I think first of all we probably have to determine whether 'nothing,' ( as in the 'nothing' inside a vacuum) is 0.
Because if i have 0 of something then there is nothing there.
Also the universe came from nothing and expands into nothing so what is it. Thats another question though so we'll stick to 0 as a quantity

Between -1 and 1 there is an infinate amount of quantities however would it just bypass 0?

Basically, i have no idea and have thoroughly confused myself.
Any help!?

1010101010001010

Generic Goon
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### Re: Is 0 real?

There are limitations on numbers that can be used for length. You also can't have negative lengths. That doesn't mean that negative numbers don't exist. For f(x)=1/(x2-1), X can't be one, but 1 exists. Length just has similar restrictions.

As for the number of pencils, what you have just shown is that 0 is not a counting number, a subset of real numbers, so something can be real but not "countable"

mike-l
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### Re: Is 0 real?

Numbers can be more than just quantities. They can, for example, be distances. Pick a spot, say where you are standing, a direction, say in front of you, and a length, say your arm's length. If you hold your hand out it's 1, if someone stands another arm's length away, they are 2, etc. Zero certainly exists in the context, it's you.

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Yesila
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### Re: Is 0 real?

You're pondering is following quite a historic trail.

The issues you raise are some of the reasons why "zero" came to be much much after the counting numbers of 1,2,3, etc...

The Nothing that Is: A Natural History of Zero by Robert Kaplan is a nice little history of how zero came about. And Wikipedia has a nice entry on zero too that outlines some of the points in the book I mentioned.

antonfire
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### Re: Is 0 real?

Since somebody is going to say it at some point, I'll just get it out of the way. 0 is the only number that is both real and imaginary.
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### Re: Is 0 real?

0 is the integer between 1 and −1. In most systems, 0 was identified before the idea of 'negative integers' was accepted. Zero is an even number.[2]

Zero is a number which quantifies a count or an amount of null size; that is, if the number of your brothers is zero, that means the same thing as having no brothers, and if something has a weight of zero, it has no weight. If the difference between the number of pieces in two piles is zero, it means the two piles have an equal number of pieces. Before counting starts, the result can be assumed to be zero; that is the number of items counted before you count the first item and counting the first item brings the result to one. And if there are no items to be counted, zero remains the final result.

While mathematicians accept zero as a number, some non-mathematicians would say that zero is not a number, arguing that one cannot have zero of something (for example, 'zero oranges'). Others hold that if one has a bank balance of zero, one has a specific quantity of money in that account, namely none.

From Wikipedia. http://en.wikipedia.org/wiki/0_%28number%29

I'd also add that all numbers are benchmarks from a standard. You have one pencil. This means you have one pencil compared to the default situation of having 0 pencils. Not to mention all the other values that can exist as a 0. Distance, temperature (on most scales), charge.
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tennis7192
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### Re: Is 0 real?

Take the integral of 0. It's C. Now to get a constant means that 0 must exist because you can't just get something out of nothing.

archgoon
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### Re: Is 0 real?

tennis7192 wrote:Take the integral of 0. It's C. Now to get a constant means that 0 must exist because you can't just get something out of nothing.

You didn't get something out of nothing (and why on earth shouldn't you?). C is determined by the initial conditions. For all you know it's also zero.
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semaphore
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### Re: Is 0 real?

antonfire wrote:Since somebody is going to say it at some point, I'll just get it out of the way. 0 is the only number that is both real and imaginary.

Huh. Isn't it true for any number x that x = x + 0.i and does that not mean that every number is both real and imaginary?

In a + ib, a is the real part and ib is the imaginary part, but isn't a + ib an imaginary number as a whole?

I might be splitting hairs but I also wanted to get that out of the way.

-AT

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jestingrabbit
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### Re: Is 0 real?

semaphore wrote:
antonfire wrote:Since somebody is going to say it at some point, I'll just get it out of the way. 0 is the only number that is both real and imaginary.

Huh. Isn't it true for any number x that x = x + 0.i and does that not mean that every number is both real and imaginary?

No. A complex number is real if its imaginary part is 0, and imaginary if its real part is 0. You described a real number, and noted it had an imaginary part of 0.

semaphore wrote:In a + ib, a is the real part and ib is the imaginary part, but isn't a + ib an imaginary number as a whole?

No. Its a complex number.

semaphore wrote:I might be splitting hairs but I also wanted to get that out of the way.

These are good questions imo.

@OP:

"Married to the Sea" hits the nail on the head.

And Antonfire, I don't want to see that joke again, you hear?
Last edited by jestingrabbit on Fri Feb 01, 2008 5:52 am UTC, edited 1 time in total.
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Peripatetic
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### Re: Is 0 real?

The problem with asking, "Does zero exist?" is that numbers are adjectives, not nouns. It's like asking, "Does big exist?" The question doesn't make sense. Now, numbers are adjectives that are so rigidly defined that we can prove that if X has some property that can be described by numbers, then that property must follow all of the mathematical rules we've discovered/invented. So, as others have said, it's all a matter of context.

This reminds me of an op-ed piece I read in the LA Times some years ago: http://crispinsartwell.com/math.htm

Muhahahahaz
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### Re: Is 0 real?

By definition, 0 E R.

(Zero is an element of the Real numbers.)

QED.

Morndas
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### Re: Is 0 real?

^ found the forum, saw the thread, couldn't resist registering....

0 is a real rumber such that a + 0 = a

it is the identity under addition.

that's what we were taught in analysis...

Ranorith
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### Re: Is 0 real?

My Real and Complex Analysis lecturer started off the semester with a quote which may or may not be relevant:

"Complex numbers are not complex. Imaginary numbers are not imaginary. And Real numbers are not real."

Token
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### Re: Is 0 real?

Ranorith wrote:My Real and Complex Analysis lecturer started off the semester with a quote which may or may not be relevant:

"Complex numbers are not complex. Imaginary numbers are not imaginary. And Real numbers are not real."

It's not relevant, but I like it.
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shrdlu
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### Re: Is 0 real?

Peripatetic wrote:The problem with asking, "Does zero exist?" is that numbers are adjectives, not nouns.

Obviously, numbers are verbs

the tree
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### Re: Is 0 real?

0 exists (or doesn't) in the same way that 3 exists (or doesn't). In a very platonic sense, yes 0 exists since it so very well defined, just like 3. It doesn't exist in the same physical reality that tomatoes exist in, but neither does any number that you'd care to name.

The first(ish) chapter of The Road To Reality has a good description of platonic existence.

For the silly definition of 'real' which I'm pretty sure the OP wasn't using, you can define the reals as urm.. "the complete ordered field" and thus it requires 0 as a member. Kew Eee Dee.

abeautifulsomething
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### Re: Is 0 real?

Woah, i'm glad i've started such a big debate!

0 is a real rumber such that a + 0 = a
it is the identity under addition.

If a + 0 = a then 0 need not exist as that hasn't changed anything. It is like saying a=a

In response to the post above me, all numbers and language are relative and all that matters is that people understand what other people are saying.

With the number 3, i can prove that what we take to mean 3, is what I am holding. However I cannot prove that the concept of 0 is a quantity (define by google as: "how much there is of something") as there isn't any amount of anything!

Cheers for all the responses, look forward to hearing more points of view.

hyperion
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### Re: Is 0 real?

abeautifulsomething wrote:Woah, i'm glad i've started such a big debate!

0 is a real rumber such that a + 0 = a
it is the identity under addition.

If a + 0 = a then 0 need not exist as that hasn't changed anything. It is like saying a=a

0 is the additive identity just like 1 is the multiplicative identity. Let me adjust what you just said:
"If a * 1 = a then 1 need not exist as that hasn't changed anything. It is like saying a=a"

You can indeed have 0 of something. How many apples are locked inside my computer case? Zero. Your argument is not a very good one . Negative numbers are also real, but can you have negative length? Negative apples?
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abeautifulsomething
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### Re: Is 0 real?

Good point.

I don't mean to argue one side or the other, hence why i started this discussion i just don't understand some of the posts so try to challenge them

Peripatetic
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### Re: Is 0 real?

HYPERiON wrote:Negative apples?

Sure, an apple made of antimatter. 1 apple + 1 anti-apple = 0 apples. Therefore, 1 anti-apple = -1 apples.

Morndas
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### Re: Is 0 real?

hmmm, it's a good one

I think the existence of Zero is proven moreby the reason we have zero, if that makes sense.

it's like the Empty Set. in normal terms, if you showed someone an empty set, they'd say something like "It has nothing in it... what?" But it's purpose (among other things) is to give a relative value for a non-empty set.

How can we say that something is true? one way is to say that it's not false. we can say that a number is non-zero by proving that it isn't zero; I.e. if we add it to something else we get a different number; it isn't an identity under addition. One could say that that shows a need for zero, and therefore proves a need for it's existence.

Once we have that, my Original argument is enough: 0 is a number such that if you add it to any other number:

a + 0 = a.

the tree
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### Re: Is 0 real?

abeautifulsomething wrote:Woah, i'm glad i've started such a big debate!
This what happens when you let a drop of philosophy into a sea of logic.

abeautifulsomething wrote:
0 is a real rumber such that a + 0 = a
it is the identity under addition.

If a + 0 = a then 0 need not exist as that hasn't changed anything. It is like saying a=a
No. An identity element does make a difference. If I were to ask you how many ways you could arrange the furniture in your room, and you left off the option of leaving everything where it is, then the number you would give me would be 1 too small.

abeautifulsomething wrote:In response to the post above me, all numbers and language are relative and all that matters is that people understand what other people are saying.
Not quite, a lot of the time we spend building rigorous constructions is to make sure that we know what we're talking about ourselves.

Mathematical rigour is part of a search for a sort universal truth, particularly something that would be true whether we knew it or not. A lot of those truths are to do with platonic existence, such as how a perfect sphere might behave. Some things definitely don't exist even in the platonic realm; the set of all non self containing sets comes straight to mind. Zero, however doesn't lead to such paradoxes so in the platonic sense, we're fairly certain that it can(/does -- these are basically the same thing in this context) exist.

abeautifulsomething wrote:With the number 3, i can prove that what we take to mean 3, is what I am holding.
Sorry, what? I'm not sure what you're holding. But I'll take a shot in the dark and say it's not the same as what I'm holding.
threeducks.jpg (30.37 KiB) Viewed 9933 times

If you look at the attached picture, you'll see that the contents of my hand does have a certain three-ness about it, but I am certainly not holding 'three' any more than I am holding the concept of duckness.
abeautifulsomething wrote:However I cannot prove that the concept of 0 is a quantity (define by google as: "how much there is of something") as there isn't any amount of anything!
This is certainly true, 0 sucks for describing a quantity. But numbers are a helluva lot more versatile than just describing quantities

garypenn
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### Re: Is 0 real?

the tree offers much good insight in this discussion.

What doesn't seem to have been mentioned is that we take "zero" to exist axiomatically. So basically, we assume it exists.

We do this is many different ways or contexts. The set theorists take an axiom that there exists a set containing no elements. When a set theorist constructs the natural numbers he does the following:

0 = { } (the empty set - axiomatically taken to exist)
1 = {0} = {{ }} (this set exists by an axiom of set theory called the pair axiom. It is the set {0, 0} = {0})
2 = {0, 1} = {{ }, {{ }}}
and so on...

Or you can construct the natural numbers using the Peano axioms, of which the first one is "there is a number 0".

Many people have mentioned the identity element of an algebraic structure. Without axiomatically taking such structures to have such an element, we can't do any algebra--i.e. we can't solve equations. For example, these two axioms appear when coming up with many different algebras (groups, rings, fields, vector spaces, etc):

There exists an identity element e such that for any a in the structure a + e = e + a = a. (If you want to represent your operation using multiplication you can say a*e = e*a = a.)

For each element a in the structure there is an inverse element -a such that a + (-a) = (-a) + a = e. (Multiplicatively this is a*a-1 = a-1*a = e)

When we talk about natural numbers, integers, rationals, reals, etc, the e described above is 0. In other words, 0 is crucial in order for us to solve equations.

We can keep constructing "zero" and analogous ideas, but the point is that we have to assume that such a concept exists. It's somewhat disappointing at first, but we can't make any deductions without taking assumptions. Such a deduction is inherently flawed. When it comes down to it, there is no "fact" we can be absolutely certain of. Perhaps, in some sort of Cartesian way, I could say that the only thing I can be fully certain of is that I have consciousness.

If you choose not to take this assumption (that "zero" exists) then you need to abandon all mathematics--and related fields--that involve the assumption that "zero" exists. This is a great deal of mathematics, and I would argue that it's a bad idea to do such a thing.

Ended
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### Re: Is 0 real?

abeautifulsomething wrote:With the number 3, i can prove that what we take to mean 3, is what I am holding. However I cannot prove that the concept of 0 is a quantity <snip> as there isn't any amount of anything!

I think this is the heart of the matter. Your argument here is logically flawed (more precisely, it begs the question). Consider the analogous argument:

'With the colour red, I can prove that what we take to mean 'red', is what I am seeing. (There is a definite wavelength of light, ~700nm, that is reaching my eyes.) However I cannot prove that the concept of 'black' is a colour, as there isn't any amount of anything reaching my eyes.'

Sure, black isn't a colour, if you define something as 'coloured' iff it reflects light to your eyes. But this is an arbitrary definition, and one which is not very useful. It makes sense to talk about 'my black sweater'.

Sure, 0 isn't a number, if you define something as a 'number' iff it represent an amount of apples which I can hold. But this is an arbitrary defintion, and one which is not very useful. It makes sense to say 'I have no apples'.

As an aside: What's red and invisible?
Spoiler:
No tomatoes.
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Nimz
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### Re: Is 0 real?

Peripatetic wrote:
HYPERiON wrote:Negative apples?

Sure, an apple made of antimatter. 1 apple + 1 anti-apple = 0 apples. Therefore, 1 anti-apple = -1 apples.

You may have zero apples, but you also have a helluva lotta energy!

garypenn wrote:We can keep constructing "zero" and analogous ideas, but the point is that we have to assume that such a concept exists. It's somewhat disappointing at first, but we can't make any deductions without taking assumptions. Such a deduction is inherently flawed. When it comes down to it, there is no "fact" we can be absolutely certain of. Perhaps, in some sort of Cartesian way, I could say that the only thing I can be fully certain of is that I have consciousness.

If you accept if/then reasoning, then given the "if" is true, you can be certain the "then" is true. Mathematics doesn't necessarily require you to accept its premises, but you can be certain that the conclusions follow necessarily from them.

Of course, that's IF you accept IF/then reasoning. Otherwise you could get caught up in an infinite regression to show that method of reasoning really works. If only I remembered what that link was, I might point it out. Oh well.

And an alternative answer to the invisible red ninja,
Spoiler:
no apples
LOWA

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### Re: Is 0 real?

Nimz wrote:And an alternative answer to the invisible red ninja,
Spoiler:
no apples

D'oh! I fail at continuity.
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abeautifulsomething
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### Re: Is 0 real?

Touché all of you

Thanks for all these answers and explaining that 0 is real. :0

Cheers

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### Re: Is 0 real?

garypenn wrote:Many people have mentioned the identity element of an algebraic structure. Without axiomatically taking such structures to have such an element, we can't do any algebra--i.e. we can't solve equations. For example, these two axioms appear when coming up with many different algebras (groups, rings, fields, vector spaces, etc):

There exists an identity element e such that for any a in the structure a + e = e + a = a. (If you want to represent your operation using multiplication you can say a*e = e*a = a.)

For each element a in the structure there is an inverse element -a such that a + (-a) = (-a) + a = e. (Multiplicatively this is a*a-1 = a-1*a = e)

Of course, depending on your structure and operation, the second axiom may not be present - e.g. in a ring, not all elements may have a multiplicative inverse.
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### Re: Is 0 real?

To the OP: You may want to read the discussion there : viewtopic.php?p=100280 and maybe also viewtopic.php?f=17&t=4800 .
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Izzhov
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### Re: Is 0 real?

I do believe that this discussion is almost entirely semantics.

We aren't going to get anywhere without rigorous definitions of "zero" and "real."

Peripatetic
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### Re: Is 0 real?

Izzhov wrote:I do believe that this discussion is almost entirely semantics.

We aren't going to get anywhere without rigorous definitions of "zero" and "real."

But, then we'll actually have to come to a conclusion, and that's just not fun. I'd prefer we stick to rigidly defined areas of doubt and uncertainty.

I should have been a philosopher.

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### Re: Is 0 real?

I feel that I should make some comment at this point.

...but I won't.
Is this a wok that you've shoved down my throat, or are you just pleased to see me?

Izzhov
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### Re: Is 0 real?

majikthise wrote:I feel that I should make some comment at this point.

...but I won't.

Brilliant! That man is a genius! A GENIUS I SAY!

Joemaniax
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### Re: Is 0 real?

I was goggling around for the string "does 0 exists". I was happy to find this "debate" from quite recent (just 10 days ago last post.)
So I feel that I should come punching in a little with my view on the matter.

Does zero exists? [Part 1]
Starting with the first rider of the apocalypse, the demigod WHY:
Why was I goggling this sentence? I believe the answer of this is a key for opening one more door in an infinite corridor filled with doors that finally lead to the ultimate truth.
If 0 aka nothing aka null does exists, then we are stuck to not ever finding it. The core of mathematic presents us this simple 1-st degree equation:
x*y=z if you already know y,z but you need the x, then x=z/y where y is not 0... If 0 exists then we must take into the account that y can be 0 and thus make us deal with the paradox and overview the postulation that y can not be 0.
we can have 0 multiplied by 0... we can have 0 multiplying by anything and the result is 0. what happens when you "reverse engineer" the equation and you divide by 0? you get an infinity of answers.
0*0=0 --> 0=0/0
1*0=0 --> 1=0/0
2*0=0 --> 2=0/0
...
oo*0=0 --> oo=0/0 (oo I'll use as infinite sign)

what exactly happens when you divide 0 by zero? If 0 exist then it is dividable. right?
... so on and so forth. here is the basic mathematical 0 paradox. If 0 exist then what is the result of 0/0 ? is it 0? or 42? or infinity?
from what I can figure I get to the conclusion that zero can be equal to infinity, or better said that infinity is 0, null, nothing...
this conclusion is common sense not valid, therefore, as I "think" that I exist and thus not equal to 0 I postulate that 0 can not/does not/will not exists.
check out soon for part 2

antonfire
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### Re: Is 0 real?

Could we not have a part 2 please?
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### Re: Is 0 real?

Izzhov wrote:I do believe that this discussion is almost entirely semantics.

We aren't going to get anywhere without rigorous definitions of "zero" and "real."

And "exists."

(and maybe "to have" for the oft-mentioned idea that we can't "have" [or hold] 0 pieces of your favorite fruit)
Kabann wrote:Aw hell, as far as I'm concerned the world started in late 1967. Everything else is just semantics and busy-work.

phr34k
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### Re: Is 0 real?

garypenn wrote:Many people have mentioned the identity element of an algebraic structure. Without axiomatically taking such structures to have such an element, we can't do any algebra--i.e. we can't solve equations.

But... Z+ has no inverses whatsoever, no additive identity, and we can do algebra very well on it.

Cancelation law: x+y=z+y implies x=z. Proof: assume x\not=z, by def. of + we're done.

(of course one could easilly just build Z and Q and R from Z+, which is how we're trained to do algerba: in terms of R. The fact Z[sup]+[/b] is a subset allows us to talk about it in more sophisticated ways.

The point, however, is different. 0 obviously exists by virtue of us talking successfully about it: there is this common idea for which the symbol 0 stands, if there wasn't, we wouldn't know what we meant. Now the question is: is it real? It's as real as dragons, physics theories or (gasp, flame-shield), God. Dragons aren't real by a conventional definition, but they are still meaningful to us, unless we lack a certain cultural ground. Other than that, the discussion is some pretty hardcore philosophy, of which I know not enough to engage usefully with.

Joemaniax
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### Re: Is 0 real?

[Part 2]
This second part of my posts is dedicated to the case that 0, in fact, exist. It's a contra argument of what I said above.

I approached this subject starting from the quantum mechanics point of view and results about the quantum world.
So, the Zeno paradox says something like this: u want to run 100 meters to win a race. After 50 meters you are half way to the finish line. You run half of the remaining distance and you still have half more to go. After half more distance you still half of that to go. So on and so forth you will always have some half of distance to travel and thus have no physical way to actually reach that finish line. But yet you are able to defy logic and you manage to cross it...
How was it possible? The answer is simple and relays in the fact that at a certain point the space (the distance) can not be divided anymore. This distance is from what I understood the Plank's constant (we'll not get dipper into that... something with 35 fraction zeros). This non dividable distance is called a quanta of space.

So far so good, now we all have an idea of what quantum physics is about. Trust me... there's a hole lot more mind hurting things to find out in this area... and I only need to mention that this quanta of space has mass - it's a substance, a primordial matter ; I always believed this but only recently while reading about qm I found out that I was actually right and that there was scientists saying it not just a crazy little me

So here we are picturing the universe as a 3D network of little "cubs of space" in which a particle of something, like a photon, teleports from one quanta to the next (that's right... space is tiny little worm holes and things are teleporting around ).

What's all this have to do with 0?
well... in any, Lego like , 3D system u need a reference point in order to calculate distances. If we'll ever be able to determine and localize precisely those quanta then we can postulate that this one is quanta 0 and that is the direction x, that to the left of q0 is -x above is z, below is -z and similar to y...
Of course we can choose q0 to be anywhere and it's all a matter of everyone agreeing that there is coordinate (0,0,0).
(don't get me wrong, it's pointless to reference the space by counting the quantas since in one mm you have billion of billion... of them.)

So this is how 0 brings "light" to the space.

Now, I mentioned all this quantum physics stuff for the following reason: for all the matter in the universe there's automatically created anti-matter. Everything with a plus has, somewhere, the corespondent minus. The minus stuff is the black matter, or the black hole material if you like. Oh, and I should also mention that they are saying that the universe is exponentially expanding...
And all the scientists have got to the conclusion that the sum of all those + and - is equal to 0.
Equal to 0...
Let me rephrase this... the universe itself, the infinity of the hole universe, the sum of all the electrons, photons, protons... is equal, eventually, with 0.
The infinite is 0.
0 is the infinite.
everything is nothing.
nothing is everything.
So... my friends, there you have it... 0 exists and it's quite an important little sucker.

- I hope I described this issue accurate enough to reality, there's still lots of things I don't understand about QM -

Oh, if you missed the part where this post is a contra argument of the previous one I should mention that quantum physic laws are applied only at the quantum level (tiny bitty stuff), at a certain point of size there's like a barrier when the things start acting according to the Newtonian laws.

So there we have it, you live in 2 worlds the one you know, feel and see - where 0 exists as an imaginary creation of our minds, and the other world which you can not sense, feel nor see (yet), but where 0 exists as the god of everything and nothing.

Joemaniax
Posts: 7
Joined: Fri Feb 22, 2008 5:05 am UTC

### Re: Is 0 real?

antonfire wrote:Could we not have a part 2 please?

muhaha
is the part with the demigod thing that freaked you?

well... as an off topic there are 4 demigods (I like the similarity with the apocalypse thing ) and one god in the world. They are the ones with the keys of the secrets of the universe, life and anything else.

The 4 demigods (no particular):
WHY is the key holder of destiny and the reason of everything. You will know WHY when you will know HOW.
HOW is the key holder of methods and answer to everything. You will know HOW if you know WHEN.
WHEN is the key holder of time and oracle of anything. You will know WHEN if you know WHERE.
WHERE is the key holder of space and substance of anything. You will know WHERE is you know WHY.

if you manage to know WHY, HOW, WHERE and WHEN then you will finally reach WHAT.
WHAT is the god of everything. He holds the secret of the essence of anything and the outcome of everything. If you will truly know WHAT then you'll know all there is to know and you'll feel very depressed for having nothing else to do because you already know everything...