The Importance of Zero: Uncovering its Significance

[arildno]

I've been totallly misinterpreted in this topic, which might partly be because of my unclear statements, but I still insist on the fact that the infinity is not a number but a concept. My point from the beginning was that 0 is as much number as infinity, and if now you guys are saying that infinity IS a number than, for you 0 is of course a number as well... but for those of us that think that infinity is not a number (there are many of us) the zero becomes quite interesting...

I might fbe criticesed for this analogy but it is sort of like this:
There isn't a number infinity just as there isn't a temperature less than 300K. It just doesn't exist (how we now may define exist ...

On the contrary, we understand you perfectly well.
You are clinging to your own personal fantasies as to what numbers OUGHT to be, and, because fantasies are fuzzy, warm and cozy, you want to live with them, rather than learn how to think by means of rigourous logical systems, which you fear because they seem strange, cold and hard to you.
You are locked in emotionalism, that's all there is to it.
It is not difficult to understand you at all.
After all, your condition is quite prevalent in the human race..

And, you seem to have missed out something: Everyone here agrees that infinity is NOT, for examples: a natural number, integer, rational number or real number.
The fact that there are lots of number systems in which infinity cannot be regarded as a number does not make it impossible to comstruct legitimate number systems in which infinity IS a number.
It is really not anything more special than that "most" fractions cannot be considered as natural numbers, but ARE rational and real numbers.

 

[strid]

You opened the door : the temperature in the room where I'm typing this right now is less than 300K (it is ~295K).

Now let's get you to define "number", wot ?

sorry.. i mistyped... of course i meant either less than -300' C or less than 0K...


Will try to answer on what I'm critized on right now...

My definition on number (in this context) is a quantity... It can be anything quantitive including lengths and other such stuff...

My first reason to think that 0 was not a "number" was that I saw it as much number as infinity... It is sort of like that the numbers 0,0000...1 to 10^9999... are possible to exist in a totally different way than zero an infinity...
Its like that zero is infinitely small while infinity is infinitely big..,,

And the stuff with that you can't divide with zero, is that you can divide by all other "numbers"... so the fact that you can't divide vy zero makes it somewaht different from other numbers...

 

[matt grime]

In that case you're definition of "numbers" is completely different from any mathematical one. So you can quit worrying about mathematics. The problem isn't mathematics it is yours.

Incidentally, water freezes at 0 degrees C, so zero exists there as a measurement.

 

[hypermorphism]

so the fact that you can't divide vy zero makes it somewaht different from other numbers...

This is only true if your system does not define division by zero. For example, taking the square root of a number that is not a perfect square is impossible in the rationals, making those numbers different from other numbers. You claim we should then take these so-called "numbers" out of the system, instead of finding a meaningful extension of our system. The latter choice brings new vistas of mathematics, while the former choice is a step backwards. Your personal problem with zero is echoed by others' problems with other aspects of other systems. Some may not want any numbers other than 1, because it makes no sense to define a new number other than a whole object. You may argue against this, but I'm sure you can see that your arguments will be just as futile as ours are to your belief.

 

[strid]

This is only true if your system does not define division by zero.

is there any system where it is defined??

 

[matt grime]

Dear God do you not read the posts here? The extended real numbers, the extended complex plane, both allow you to define 1/0 (though nto 0/0 for obvious issues with continuity).

 

[Alkatran]

Dear God do you not read the posts here? The extended real numbers, the extended complex plane, both allow you to define 1/0 (though nto 0/0 for obvious issues with continuity).

Hold on, they do? Don't you need limits for it to make any sense?

 

[Hurkyl]

Nope. However, you have to be careful with them; ordinary arithmetical facts like x + 1 != x don't always hold in these systems.

 

[Alkatran]

Nope. However, you have to be careful with them; ordinary arithmetical facts like x + 1 != x don't always hold in these systems.

I'm assuming that you're talking about +- infinity (or in the case of the complexe numbers, complexe infinity)?

 

[Icebreaker]

I think the OP may have just read this book, which over-hypes the importance of zero from a historical perspective.

 

[strid]

ok... if now 1/0 is defined... than what is the differnce between 1/0 and 2/0? are they equal or what?

 

[Moo Of Doom]

If $$\frac{1}{0}=\infty$$, then you could say $$\frac{2}{0}=\frac{2*1}{2*0}=\frac{2}{2}*\frac{1}{0}=1*\infty=\infty$$

Heh.

 

[strid]

If $$\frac{1}{0}=\infty$$, then you could say $$\frac{2}{0}=\frac{2*1}{2*0}=\frac{2}{2}*\frac{1}{0}=1*\infty=\infty$$

Heh.

yeah right...

you can also say that

1/0 = inf.
2/0= 2 *(1/0) =2*inf.


or...
2/0= 2/ (4*0) =0,5* (1/0)= 0,5 * inf.

As you see you can quite many answers... :)

 

[matt grime]

Again, Strid, you have not in stated in which system you are talking about. Why don't you actually do that?

In Cu{\infty} 1/0=2/0=\infty.

This is "by continuity".

You do understand that things in mathematis essentially follow from the definitions and not your real life intuition?

 

[arildno]

yeah right...

you can also say that

1/0 = inf.
2/0= 2 *(1/0) =2*inf.


or...
2/0= 2/ (4*0) =0,5* (1/0)= 0,5 * inf.

As you see you can quite many answers... :)

And why don't you think that we may have 2*inf=inf and 0.5*inf=inf?

 

[strid]

you seem to missed the sarcasm again.. i answered Moo of Doom... read his post before criticing mine...

 

[arildno]

"criticing"... what a delightful new word in English!
Where did you find it?

 

[matt grime]

Strid, why don't you sit down and write out the lits of rules that your "numbers" must satisfy. Then attempt to show if there is or isn#t a model of this system.

Because the "numbers" in mathematics are axiomatic constructs. Stop trying to use your "intution" on them. We have axioms, we know they are not self contradictory since we can produce a model of them. And we can deduce results about them. Notice, we deduce things, we don't make wild and unmotivated guesses that we insiste must be true even after it has been carefully explained to us why this guess is wrong.

 

[arildno]

Just to help you along a bit with that list of rules we're waiting for, strid:

Do you want the following rules to apply to your numbers:
1) Whenever I add two numbers, I'll get a number back.
2) Whenever I multiply two numbers, I'll get a number back.

Will your system have these two rules, for example?

 

[hypermorphism]

Also, here are two examples of lists of rules (axioms) that you are (hopefully) already familiar with:
A ring (specifically, a ring which is an integral domain), a model of which is the set of integers under addition and multiplication.
A field, a model of which is the real numbers under addition and multiplication.

 

[Tom Something]

zero is very interesting

i agree that its use in math is often to make things "work" as you put it. when the derivative of a formula is zero, that tells you something. you need to "plug in" zero to see when it happens.

when zero comes out as an aswer, it takes the form of a word more than anything else. it could be one of many words:
no, not, none, never, stopped, constant, initial (position, velocity, whatever your flavor). it could even mean "yes".

it's value lies in it's use as a tool, because in use it has no value.

i would rocommend posing this question in a philosophy or english forum, just for fun.

 

[strid]

Just to help you along a bit with that list of rules we're waiting for, strid:

Do you want the following rules to apply to your numbers:
1) Whenever I add two numbers, I'll get a number back.
2) Whenever I multiply two numbers, I'll get a number back.

Will your system have these two rules, for example?

Tanks for the beginnning and I will add on 2 other points that just fit your list well...

3) Whenever I subtract two numbers, I'll get a number back.
4) Whenever I divide two numbers, I'll get a number back.

Seems logical to have these 2 added... and then... Zero doesn't fit the defintition of number anymore...

 

[matt grime]

Well, it does and it doesn't. Since we can subtract x from x we get 0, if x is a number so must zero be. And thus we must be able to divide by zero. Thus *you* must be careful not to be inconsistent, since these are *your* defintions of Strid's Numbers.

 

[CRGreathouse]

Tanks for the beginnning and I will add on 2 other points that just fit your list well...

3) Whenever I subtract two numbers, I'll get a number back.
4) Whenever I divide two numbers, I'll get a number back.

Seems logical to have these 2 added... and then... Zero doesn't fit the defintition of number anymore...

So you drop 0 from the set of numbers, and then by point (3) -- or, really, by point (1) -- nothing's a number, since $$a+(-a)=0$$ and 0 isn't a number any more. This leaves you with the null set!

Edit: I started to post before Matt Grime, and he wrote just about the same thing I did, only slightly more eloquently.

 

[Icebreaker]

Tanks for the beginnning and I will add on 2 other points that just fit your list well...

3) Whenever I subtract two numbers, I'll get a number back.
4) Whenever I divide two numbers, I'll get a number back.

Seems logical to have these 2 added... and then... Zero doesn't fit the defintition of number anymore...

Actually if you allow rational numbers and negative numbers, you don't need those two.

 

[strid]

Well, it does and it doesn't. Since we can subtract x from x we get 0, if x is a number so must zero be. And thus we must be able to divide by zero. Thus *you* must be careful not to be inconsistent, since these are *your* defintions of Strid's Numbers.

yeah.. missed that one... didnt think very much on that as the 2 first rules were written by someone else...

let me rephrase those rules...

1) Whenever I add two numbers, I'll get a defined answer.
2) Whenever I multiply two numbers, I'll get a defined answer.
3) Whenever I subtract two numbers, I'll get a defined answer.
4) Whenever I divide two numbers, I'll get a defined answer.

EDIT: This also means that complex numbers and irrationals numbers are included in the difinition... please point out if theses rules excludes any number (except zero if you want to have that)

 

[matt grime]

So, you've got a set S, of "strid numbers", and you're defining binary operations +,-,*, and / on them from SxS to the "defined answers". Now, you do not state what a defined answer is, so who knows what on Earth you're talking about. In what set are you talking about. You do not include irrationals or complexes in the definition at all. In fact, all you're doing is seemingly specifying the "non-zero elements of a field, or division ring", though as we don't know what a "defined answer" is we cannot possibly say for sure.

 

[CRGreathouse]

Does this accurately describe your position?

1) Whenever I add two numbers, I'll get a defined answer.
2) Whenever I multiply two numbers, I'll get a defined answer.
3) Whenever I subtract two numbers, I'll get a defined answer.
4) Whenever I divide two numbers, I'll get a defined answer.

EDIT: This also means that complex numbers and irrationals numbers are included in the difinition... please point out if theses rules excludes any number (except zero if you want to have that)

Let $$\mathfrak{S}$$ be the set of Strid numbers and $$\mathfrak{D}$$ be the set of Strid-defined numbers.

For $$s_1,s_2\in\mathfrak{S}$$:

1. $$s_1+s_2\in\mathfrak{D}$$
2. $$s_1\cdot s_2\in\mathfrak{D}$$
3. $$s_1-s_2\in\mathfrak{D}$$
4. $$s_1\div s_2\in\mathfrak{D}$$

Strid's Conjecture: $$\mathfrak{S}=\mathbb{C}\backslash0$$, $$\mathfrak{D}=\mathbb{C}$$ is consistent.

 

[matt grime]

Yes, and it works for any ring too where S is subset of the set of units and D is the ring, so all strid has done is give (some of) the axioms of a ring, assuming the reading of "defined" is as you say (and that is how i'd read it too).

Of course, there's nothing there that requires the operation + is commutative, and that + and * are associative, or that distribution holds. In fact there is nothing to suggest + and * ought to even be addition and multiplication and so on. Ie we do not know that a+b-b=a, or that z/z=1, or even if there is a mutlipicative identity.



Other examples include S the set of nxn invertible matrices and D the set of all matrices.

 

[CRGreathouse]

Of course, there's nothing there that requires the operation + is commutative, and that + and * are associative, or that distribution holds. In fact there is nothing to suggest + and * ought to even be addition and multiplication and so on. Ie we do not know that a+b-b=a, or that z/z=1, or even if there is a mutlipicative identity.

You're absolutely right about that, and really I should have either added that explicitly or left off the conjecture. I meant to express that the four Strid operations mapped 1-to-1 with the same operations in $$\mathbb{C}$$. Otherwise it's pretty simple to make the conjecture true for arbitrary $$\mathfrak{S},\mathfrak{D}$$ with constant functions.

What's really funny for me is that, taking this process to the logical extreme, we have the conjecture as "$$\mathfrak{S}\cup0$$ is a ring", which really defeats Strid's purpose.

Oh, and I like your point on units... it would work with $$\mathfrak{S}=\{1\}$$, $$\mathfrak{D}=\mathbb{Q}$$.

 

[matt grime]

Of course, without any restrictions on the interaction of the operations +,* etc, then I'm also free to declare that S=Q, and D=Qu{T}, where T is some symbol such that x/0 is defined to be T for all x (including 0) As I don't need to define an arithmetic involving T this is ok. Of course we run into problems such as what is a*b*c (note i'll pretend x*y is the same as x/y) when b*c isn't a strid number (and hence what is a*(b*c)?) but a*b is, so that (a*b)*c is allowed, even though it is "strid defined" only.

 

[strid]

when I said that it has to be defined I don't mean that you can just insert any variable as an answer. The answer we get should be on th line of numbers (including the line of complex numbers)... I hope anyone doesn't sugget creating a weird line to fit x/0...

 

[HallsofIvy]

1) Whenever I add two numbers, I'll get a defined answer.
2) Whenever I multiply two numbers, I'll get a defined answer.
3) Whenever I subtract two numbers, I'll get a defined answer.
4) Whenever I divide two numbers, I'll get a defined answer.

If, in addition, you want addition and multiplication to have "nice" properties, i.e. commutative, associative, distributive, then you want a "field" in which every member has a multiplicative inverse. It is easy to prove that the only such field contains only a single member: 0 is the only number and 0+0= 0, 0*0= 0 are the only possible operations.

 

[matt grime]

when I said that it has to be defined I don't mean that you can just insert any variable as an answer. The answer we get should be on th line of numbers (including the line of complex numbers)... I hope anyone doesn't sugget creating a weird line to fit x/0...

And exactly how does that stop the "problem" you've created? x-x=0 should be on the "line of numbers".

 

[arildno]

And what, BTW, do you mean with the "line of numbers", strid?
Is that something deep and inexplicable?
And, while you're at it, what is a "weird" line?
Is it one true line and many untrue lines?