The Importance of Zero: Uncovering its Significance

[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?

 

[strid]

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

I'll have to confess that that is a little problemtaic... but for the moment, let just throw the result of x-x of the line of numbers... if something equal nothing, it doesn't exist on the line... at least for the moment... will think if I can get a better solution...

 

[Moo Of Doom]

Wow! You just "threw" 0 onto the line of numbers. Good Job!

 

[matt grime]

So, you're going to allow subtraction of numbers unless they're equal. How is this any less philosophically dubious than disallowing 1/0?

 

[strid]

Wow! You just "threw" 0 onto the line of numbers. Good Job!

I "threw" zero off, not onto, the line of numbers...

I'll let zero exist as a concept but not as a number. So when you subtract tvo equal numbers you get the concept zero, which is nothing. Similarly, if someone asks how many points exist on a line, the asnwer will be the concept infinity. So neiter infinity or zero is a "quantitiy"...

so my line of numbers would be something like this...


continue to infinitely negative big... -100... -50... -5... -1... -0,5... -0,1... -0,01... -0,0001 ... continue to negative infinitely smalll... ... ...continue to infinitely small... 0,0001... 0,01... 0,1... 0,5... 1... 5... 50... 100... continiue to infinitely big

weird to write it on computer without a timeline, but I hope you get the picture...

 

[matt grime]

But you're still not being consistent. You want the result of all subtractions (and additions) to be strid defeind, which you also claim you want to be a "strid number", so why isn't x-x=0 a strid number? And why if you're allowed to say that x-y is a binary operation, except when x=y, are we not allowed to state that x/y is a well defined binary operation except when y=0? You're just being completely inconsistent.

 

[katlpablo]

so can anyone come up with a place where the zero is good,,..

you mean one like this?

n^-1 = 1/n

n^0 = 1

n^1 = n



or


n^0 =1

 

[strid]

you mean one like this?

n^-1 = 1/n

n^0 = 1

n^1 = n



or


n^0 =1

sorry to disappoint you but n^0=1 isn't true for n=0...

0^0 is undefined and according to me (and many other) just the same as 0/0...

 

[Gokul43201]

I'm sure he meant, n \not{=}~ 0.

This thread has run its course, wot ?