Does zero have a 'value'

[drkitten]

aren't they all the same?

Of course not. What's the difference between "my bank account is zero" and "I have no bank account"? (Well, for one thing, I can't make a deposit into a non-existent account, but I can into one with a zero balance. If I have overdraft protection, I can even write a check on an account with zero balance.

Similarly, there's a difference between "no temperature" and "a temperature of zero." One is simply cold, while the other suggests that the thermometer is broken.

 

[69dodge]

Um, I think i got it wrong, the 0* e=0 must be an axiomatic requirement for F to be a field, otherwise I'd be able to string-substitute and "1 * e = 1" would be equally legitimate.

You were right the first time. If 0 is the additive identity, and if F satisfies a bunch of field axioms that you didn't mention explicity, then it follows that 0 multiplied by anything is 0.

0e = 0e [anything equals itself]
(0 + 0)e = 0e [0 is the additive identity, so 0 + 0 = 0]
0e + 0e = 0e [multiplication is transitive over addition]
0e + 0e - 0e = 0e - 0e [additive inverses exist; cancelling works]
0e = 0

NB that this theorem follows without stating the relationship between + and * [...]

This part is not quite right. Transitivity relates addition and multiplication, and is used in the proof.

What's important about 0 is that it's the additive identity. You could call the additive identity "1" if you wanted to confuse everyone, but it wouldn't really change anything fundamental. (You'd have to call the multiplicative identity something else, in that case, unless you want your field to have only a single element. Using standard notation, if 0 = 1, then for any element e, we have e = 1e = 0e = 0. So there's only one element: 0.)

 

[DrMatt]

Mathematician here (if having a degree qualifies me)... I think you've touched on definition rather than value--though two important definitions, one in abstract algebra and the other in set theory.

In a previous life I studied something called Denotational Semantics, and a mathematical definition of the NUMBER denoted by 0 is by definition the value of the NUMERAL 0 itself.

 

[DrMatt]

You were right the first time. If 0 is the additive identity, and if F satisfies a bunch of field axioms that you didn't mention explicity, then it follows that 0 multiplied by anything is 0.

0e = 0e [anything equals itself]
(0 + 0)e = 0e [0 is the additive identity, so 0 + 0 = 0]
0e + 0e = 0e [multiplication is transitive over addition]
0e + 0e - 0e = 0e - 0e [additive inverses exist; cancelling works]
0e = 0

This part is not quite right. Transitivity relates addition and multiplication, and is used in the proof.

What's important about 0 is that it's the additive identity. You could call the additive identity "1" if you wanted to confuse everyone, but it wouldn't really change anything fundamental. (You'd have to call the multiplicative identity something else, in that case, unless you want your field to have only a single element. Using standard notation, if 0 = 1, then for any element e, we have e = 1e = 0e = 0. So there's only one element: 0.)

Okay, that's sounding famliar, the ol' neurons have been away from it for a while, I should pick up a few books to replace the ones my ex hocked, so I can check the books and refresh my memory every so often. There's room on the shelves next to the score of Gurrelieder.

 

[gnome]

In a previous life I studied something called Denotational Semantics, and a mathematical definition of the NUMBER denoted by 0 is by definition the value of the NUMERAL 0 itself.

I'm not sure I agree... a "value" I think is a measure, what the number is used for... it's definition is what the number is defined as in mathematical theory, whether or not it is used for measuring.

 

[luvhumility]

thus, we still have seven (at least) 7 unsolved math conjectures fully described at the clay math institutes web site. You get a $mil for each one you can solve. I think the P v NP problem possibly dabbles in a similar conjecture? I am not sure. "0" in our minds is undefineable in the physical sence. Mathmatically usefull yes... although realistically, I have never seen anything in the physical realm that is or stayed at "0". If anyone finds a "zero" someplace I will take them all (since zero is of no value to most of us) then I will add up all these "zeros" and I will have somthing from nothing. hmmm.....

 

[Alkatran]

thus, we still have seven (at least) 7 unsolved math conjectures fully described at the clay math institutes web site. You get a $mil for each one you can solve. I think the P v NP problem possibly dabbles in a similar conjecture? I am not sure. "0" in our minds is undefineable in the physical sence. Mathmatically usefull yes... although realistically, I have never seen anything in the physical realm that is or stayed at "0". If anyone finds a "zero" someplace I will take them all (since zero is of no value to most of us) then I will add up all these "zeros" and I will have somthing from nothing. hmmm.....

Zero has just as much meaning in our minds as one. Let's say that we only have the couting numbers {1, 2, 3...}

Now all our numbers have a number before them, except 1. Also, this system isn't closed under subtraction. So we add zero, then the negative numbers. (then the rationals, then the reals, then the complex, then quaternions if we aren't satisfied with the complex's being algebraicly closed)

 

[69dodge]

I think the P v NP problem possibly dabbles in a similar conjecture? I am not sure.

What makes you think that?

I know what P and NP are, and I don't see how this thread is related to the question of whether they're equal.

 

[gnome]

thus, we still have seven (at least) 7 unsolved math conjectures fully described at the clay math institutes web site. You get a $mil for each one you can solve. I think the P v NP problem possibly dabbles in a similar conjecture? I am not sure. "0" in our minds is undefineable in the physical sence. Mathmatically usefull yes... although realistically, I have never seen anything in the physical realm that is or stayed at "0". If anyone finds a "zero" someplace I will take them all (since zero is of no value to most of us) then I will add up all these "zeros" and I will have somthing from nothing. hmmm.....

Well, no, if you add up zeros you still have zeros. Here, have zero BMW's, personal jet planes, and attractive fembots. I have monitored carefully, and the number of those that I have has stayed at 0 consistently.

 

[luvhumility]

Gnome,

I get it basically mathmatically, yes it is "zero" in pure math terms but what I am trying to say is that as math applies to the physical world It does not wash. I have never seen anything reach a real 0.000000... in physical / electrical experements. Sure the meter "reads" zero or 0.000000, but two years later some measurment instruments accuracy is increased and what I thought was zero two years ago is now really 0.000000001 or similar. SO I Revise my experement and use "better" electrical grounds for example or use a more thermally stable environment and then Now I get a real zero? well, JUST FOR NOW! Until a better measuring tool come along. I just do not follow this "0" concept in "real" measurement!

anyzip, here is a vacuum/zero blerb I got somewhere on the net:
The relation between the virtual particles and the vacuum is an essentially dynamic relation; the vacuum is truly a "living Void," pulsating in endless rhythms of creation and destruction. The discovery of the dynamic quality of the vacuum is seen by many physicists as one of the most important findings of modern physics. From the role as an empty container of the physical phenomena, the void has emerged as a dynamic quantity of utmost importance. The results of modern physics thus seem to confirm the words ofthe Chinese sage Chang Tsai: "When one knows that the Great Void is full of*Ch'i,* one realizes that there is no such thing as nothingness."
We think of vacuum as zero, nothing, empty, nada, nichts, zilch. If we have 7 rocks and we take away 7 rocks, then we have no rocks, zero rocks.
Later we learn that this is not always so. If we have 7 pieces of matter and 7 pieces of anti-matter, when combined we have zero, although we actually have 14 pieces. You may say that they canceled out each other and you now have nothing. Think of what you are saying. Are you saying that 14 things can be combined and cease to exist and become nothing? Then isn't it conversely possible that "nothing" can yield or be divided into 14 "real""things."
In algebra we add seven plus-ones to seven minus-ones and get a sum total of zero, again in spite of the fact that there are actually 14 numbers.
Although zero is the absence of just a *single*thing, it can be the presence of *multiple* things.
"If force CONSISTS OF mass welded to acceleration, it cannot exist in the absence of mass. Hence force does not exist in vacuum... Therefore neither magnetic field nor gravitational field exist in vacuum. The greatest error in physics has been the assignment of a force as a cause, and thus assigning it to the vacuum. The force is an *EFFECT,* and it never exists in a vacuum, a priori."

 

[luvhumility]

excuse the sloppy spelling above, but maybe that explains my point better?

 

[Alkatran]

excuse the sloppy spelling above, but maybe that explains my point better?

No, because 'value' is defined mathematically, not 'physically'. It doesn't matter if we never measure it. We'll never measure infinity, but it still has a 'value' (in the extended reals, anyways).

If you have two equal charges, the voltage between them is 0.

 

[luvhumility]

Alkatran,

thanks for the reply! this is fun conjecture. I hope this reply is not too "verbose" but I like to throw around new Ideas so I can learn new things...

you wrote:
No, because 'value' is defined mathematically, not 'physically'. It doesn't matter if we never measure it. We'll never measure infinity, but it still has a 'value' (in the extended reals, anyways).

I agree with the added qualifier that the value is understood to be a "static" absolute value and only a small slice or frame of measurement in time. The real world is truly dynamic and "real values" are always in flux. I seem to wrestle with this static v dynamic problem.

Here is a simple def of absolute value:
Absolute value

NOUN:

1. The numerical value of a real number without regard to its sign. For example, the absolute value of -4 (written -4) is 4. Also called numerical value.
2. The modulus of a complex number, equal to the square root of the sum of the squares of the real and imaginary parts of the number.

note: No 2. is commonly applied in electronics as an RMS value because there is an “averaging” function needed due to numerous complex variations of the "real" part of the signal or system being measured. Nothing I have seen in real measurement is comprised of a pure sinusoidal or any other pure or absolute set of waveforms. Averaging assumptions are almost always made in the measurements. I do not prefer "assumptions" in science! History shows us that assumptions almost always cause trouble in application of the sciences.

If you have two equal charges, the voltage between them is 0.
Yes, if you qualify that it is only at that exact time of measurement and only if they are truly "equal". I doubt this can be true in the real world. At least not for long! maybe for only 10-google seconds if at all!
Heisenberg uncertainty principle may come into play here?

In The results of a Nobel prize winning laser interferometer experiment attempting to reach a temp of 0 Deg K, they got very close to stopped motion or 0 Deg K, but not exactly “0.000…” Deg K, and only for an instant in time. So this somewhat proves 0.000000... anything is probably just "imaginary"?

Though, I do understand and agree in the pure imaginary numerical math sense. although again isn’t "0" undefined in mathematics as applied to physics? That's what every math teacher, scientist or PHD I have ever worked with has told me. Am I incorrect here?

I disagree regarding "It doesn't matter if we never measure it."
Measurement, quantification and qualification of our physical world are the means of how better measure and understand the worlds physical system of patterns and how we interact with them.

Math’s true purpose when properly applied is to quantify and relate patterns of the physical world around us. This discussion / argument about definitions of "0" and real measurements has been going on for century’s between physicists and mathematicians. They never seem to agree, I wonder WHY!!? We certainly are not done figuring this out yet! I am sure we can agree on that point! (I hope).

I somewhat agree with "We'll never measure infinity, but it still has a 'value' (in the extended reals, anyways)." This I suppose is because it is in a constant state of change and we can only hope to measure small frames or parts of it at any one time. Pretty big and small place of little/big pieces out there......all at the same time

 

[drkitten]

I get it basically mathmatically, yes it is "zero" in pure math terms but what I am trying to say is that as math applies to the physical world It does not wash. I have never seen anything reach a real 0.000000... in physical / electrical experements.

You're not looking hard enough, then. How many capacitors are there in a purely resistive circuit? How many LEDs are on when the battery is disconnected? For that matter, how many beers are left in the refrigerator after you, your wife, and the four people you invited over to watch the game have all helped themselves?

You may never have seen a pure 0.00000.... but I suspect that you've never seen a pure 1.00000....
on any real-valued measuring instrument, either. That's not a limitation of the number system, but of our ability to build and engineer accurate components and accurate measuring equipment.

Though, I do understand and agree in the pure imaginary numerical math sense. although again isn’t "0" undefined in mathematics as applied to physics? That's what every math teacher, scientist or PHD I have ever worked with has told me. Am I incorrect here?

Well, either you are, or they are. I don't have enough information to determine which.

If you really think that 0 is undefined w.r.t. physics, go count the beers in the fridge again.

 

[Alkatran]

Mathematics is not physics. You do not have to measure a number for the concept to exist.

Of course if you keep picking values that are random in nature you can go on about how the value jumps around. Just remember: if you have no apples you have 0 apples.

Imaginary numbers aren't "real", but they're just as "real" as the Real numbers and we use them all the time.

 

[luvhumility]

You're not looking hard enough, then. How many capacitors are there in a purely resistive circuit? How many LEDs are on when the battery is disconnected? For that matter, how many beers are left in the refrigerator after you, your wife, and the four people you invited over to watch the game have all helped themselves? = 0, true!

You may never have seen a pure 0.00000.... but I suspect that you've never seen a pure 1.00000.... I agree...


on any real-valued measuring instrument, either. That's not a limitation of the number system, but of our ability to build and engineer accurate components and accurate measuring equipment. I agree...

I will keep "looking" in my "free" time for "0..."?? tnx...

 

[DevilsAdvocate]

In programming languages, there is a difference at least between zero, null, and nothing.

Nothing means that you don't have anyhting.
Null means that you have something, but it has no value.
Zero means that you have somehting, it has a value, and the value is zero.

So to get to non-zero, you must first have something (IS NOTHING = FALSE) then have a value (IS NULL = FALSE), then not have zero (IS ZERO = FALSE).

Then and only then do you have something with a value that is not zero.

 

[SGT]

Um, I think i got it wrong, the 0* e=0 must be an axiomatic requirement for F to be a field, otherwise I'd be able to string-substitute and "1 * e = 1" would be equally legitimate.

The missing axioms are:

For any elements a, b and c in the field:
a * (b+c) = (b+c) * a = a*b + a*c

For every element e in the field, there is an element -e and an element 1/e such that:
e + (-e) = 0
e * (1/e) = 1

The theorem is:
0 * e = [a + (-a)] * e = a*e + (-a)*e = ae + (-ae) = 0

 

[gnome]

In programming languages, there is a difference at least between zero, null, and nothing.

Nothing means that you don't have anyhting.
Null means that you have something, but it has no value.
Zero means that you have somehting, it has a value, and the value is zero.

So to get to non-zero, you must first have something (IS NOTHING = FALSE) then have a value (IS NULL = FALSE), then not have zero (IS ZERO = FALSE).

Then and only then do you have something with a value that is not zero.

I'm with you on Null and Zero (for example, from a database standpoint)... but I'm not clear on what it means for a variable to be "Nothing". Can you give a concrete example?

 

[drkitten]

I'm with you on Null and Zero (for example, from a database standpoint)... but I'm not clear on what it means for a variable to be "Nothing". Can you give a concrete example?

What's the value, in C++, of a variable that has not been declared?