Does X times 0 always equal 0 or are there exceptions?

[bahamagreen]

Not sure if the correct term is indefinite or undefined... I mean something like an infinite series that does not sum to a particular value, like this:

X = 1 - 1 + 1 - 1 + 1 - 1 + 1...

where pending the placement of parentheses one can infer multiple answers for X

So, is it proper to say

X times 0 equals 0 is true in spite of X being indefinite/undefined (or not well formed because X is an infinite series)?

Another way to ask this would be, are there any exceptions to X times 0 = 0 other than X having a division by 0 ?

 

[jbriggs444]

It is undefined.

The rule that zero times anything is zero applies only to objects that are part of the numbering system. Undefined values are not part of the numbering system.

Actually the problem is worse than that. It is not proper to take an undefined value and call it X. It is a little-stated rule of mathematical discourse that before you can talk about an entity in any way or give it a label like X, you must first have an assurance that the value actually exists.

 

[bahamagreen]

Thanks,

Just to be clear, if at least one side of an equality representation is undefined, the equality representation itself is considered false, not "undefined". That is, all equality representations are either true or false. Is that correct?

Edit--- Just to be more clear, let me tell why I'm asking about this.

Since I can write 2=3 the equality (what I'm calling the representation of equality) actually seems to me to be a claim of equality that can be true or false.

When you use the addition operation to add the same thing to each side of an equality, a true equality remains true and a false equality remains false. Undoing or reversing the operation also preserves the true/false of the equality.

When you use the multiplication operation to multiply each side of an equality by other than zero, a true equality remains true and a false remains false, and the reversal also preserves the true/false of the equality. But when you multiply both sides by zero the resulting equality becomes true no matter what its previous true/false state, and this operation is not reversible because division by zero is not allowed.

It is this "order-asymmetric latching to true" aspect of multiplication by zero that I was thinking about and wanted to be sure that a representation of equality must always be either true or false without the possibility of it being considered undefined (was not sure of the convention).

 

[CrazyNinja]

X = 1 - 1 + 1 - 1 + 1 - 1 + 1...

By the way, just as a matter of curiosity, did you know that this value is 1/2?

 

[pbuk]

Just to be clear, if at least one side of an equality representation is undefined, the equality representation itself is considered false, not "undefined". That is, all equality representations are either true or false. Is that correct?.

No, and @jbriggs444's post explains exactly why.

 

[pbuk]

By the way, just as a matter of curiosity, did you know that this value is 1/2?

No it isn't, and this is off topic so please start a new thread if you want to discuss why (or just look at the many other threads on Ramanujan summation and other non-standard summations).

 

[Pythagorean]

It is undefined.

The rule that zero times anything is zero applies only to objects that are part of the numbering system. Undefined values are not part of the numbering system.

Actually the problem is worse than that. It is not proper to take an undefined value and call it X. It is a little-stated rule of mathematical discourse that before you can talk about a entity in any way or give it a label like X, you must first have an assurance that the value actually exists.

How do you prove whether something exists or not without the label? Aren't you already begging the question, then, as soon as you go about trying to prove existence?

 

[pbuk]

How do you prove whether something exists or not without the label? Aren't you already begging the question, then, as soon as you go about trying to prove existence?

That is precisely the point: if something doesn't exist you can't assign a label to it. Why would you want to try to prove that something exists that you know doesn't exist?

 

[Pythagorean]

That is precisely the point: if something doesn't exist you can't assign a label to it. Why would you want to try to prove that something exists that you know doesn't exist?

If you didn't know, a priori, if it existed, but you theorize it would be useful if it did. You'd want to prove it's existence before applying it to your problem.

I guess the specific question is, can you at correctly measure existence when the thing you're measuring doesn't exist? Could that lead, somehow, to false positive?

 

[Mark44]

X times 0 equals 0 is true in spite of X being indefinite/undefined (or not well formed because X is an infinite series)?

No, not true.
Many limit expressions are of the indeterminate form ##[\infty \cdot 0]##. A limit of this form can evaluate to a real number, ##\infty##, or ##-\infty##.
Here are some simple examples:
##\lim_{x \to \infty} x^2 \cdot \frac 1 x## -- The second factor approaches 0, but the limit value is ##\infty##.
##\lim_{x \to \infty} x \cdot \frac 2 {x - 1}## -- the second factor approaches 0, but the limit value is 2.
##\lim_{x \to \infty} x \cdot \frac 1 {(x - 1)^2}## -- the second factor approaches 0, but the limit value is 0.

Another way to ask this would be, are there any exceptions to X times 0 = 0 other than X having a division by 0 ?

See my first two examples above.

 

[Ssnow]

Yes, it is undefined because ##X## is undefined ...

 

[bahamagreen]

Still not clear to me.

Please distinguish the defined/undefined attribute of the terms of an equality from the true/false attribute of the equality itself -

With the presence of an undefined term, it seems to me the equality must be false, not undefined.

 

[pbuk]

When we write {formula1} = {formula2} we are asserting that the value of {formula1} equals the value of {formula2}. In order for that assertion to be either true or false both {formula1} and {formula2} must have a value. That is how we define equality.

It may seem to you that we could define equality differently: you might like to play with your definition and see what you can prove (clue: the hard task would be finding something you couldn't prove).

 

[ad infinitum]

Still not clear to me.

Please distinguish the defined/undefined attribute of the terms of an equality from the true/false attribute of the equality itself -

With the presence of an undefined term, it seems to me the equality must be false, not undefined.

If a term is given a name by specifying some property which it has, then it must first have been determined that there is a unique object satisfying that property. Otherwise a formula containing an invalid term is not a statement in the language of the theory and thus meaningless in the context of the mathematical theory under discussion.

e.g. There is no number in the reals such that x*0=1, thus 1/0 cannot be part of any statement in the language of the field of real numbers or have any well-defined truth value. Just like the English sentence "My 4tjf is happy", is meaningless, neither true or false.

 

[FactChecker]

Your expression, 1-1+1-1+1... needs to be arrived at in terms of some sequence of defined numbers. There are too many such sequences that can converge to any real number that you wish, or to +- infinity at any speed that you wish. If you are discussing a particular sequence of numbers, there may be a legitimate value that can be assigned to 0 times 1-1+1-1+1... But that would only be in the context of a limit of a sequence of well-defined real numbers.

 

[bahamagreen]

If a term is given a name by specifying some property which it has, then it must first have been determined that there is a unique object satisfying that property. Otherwise a formula containing an invalid term is not a statement in the language of the theory and thus meaningless in the context of the mathematical theory under discussion.

e.g. There is no number in the reals such that x*0=1, thus 1/0 cannot be part of any statement in the language of the field of real numbers or have any well-defined truth value. Just like the English sentence "My 4tjf is happy", is meaningless, neither true or false.

So how is you then use the equality containing the invalid term x in the language of your demonstration/explanation (x*0=1)?

When we write {formula1} = {formula2} we are asserting that the value of {formula1} equals the value of {formula2}. In order for that assertion to be either true or false both {formula1} and {formula2} must have a value. That is how we define equality.

It may seem to you that we could define equality differently: you might like to play with your definition and see what you can prove (clue: the hard task would be finding something you couldn't prove).

I define equality as the terms on each side of the equals sign representing the same value. If at least one of the sides is undefined, then the equality is false. What is the basis for insisting that both sides must have a value to deny that they are the same thing? It seems axiomatic that different things are not equal regardless of those things being different by value or different by one of them being undefined... comparing a value and an undefined term, it seems clear that they cannot be the same, therefore different, therefore the representation claiming equality is not true and must be false. In general, how can an equality claim be anything other than true or false - the equality can't be undefined as true or false; only the terms may be undefined.

{undefined} = 3 is clearly not true by definition of "undefined" and "3", so the equality is false. What would seem inconsistent and peculiar is if the convention required that the language was meaningless when an undefined term is present. The meaning of "undefined"=3 is perfectly clear and the evaluation that it is false is also clear. To say that the evaluation of its true/false is undefined makes no sense to me... its true/false is defined easily by noting that the terms don't represent the same object so the equality is not undefined, it is false.

My whole question was to verify the convention for dealing with undefined terms in equality claims with respect to the true/false evaluation of the equality. There must be some confusion about what I'm asking. I'm just not seeing how the attribute of a term being undefined has any power to jump over to the determination of the truth of an equality or how an equality can be undefined. If the misunderstanding is on my part, can someone point to the fundamental conventions about this? I have been searching for a few days.

 

[pbuk]

I define equality ...

Let's stick with what everyone else agrees on as the definition of equality.

What is the basis for insisting that both sides must have a value to deny

Because that is part of everyone else's definition.

It seems axiomatic that

The word "axiom" has a specific meaning in this context; again it is important to stick to what everyone agrees are axioms for the system under consideration.

{undefined} = 3 is clearly not true by definition of "undefined"

Here is the heart of your problem. You believe that {undefined} is a symbol of some generally used logical system: it is not. When we say that something is undefined we do not mean that is has a value equal to {undefined}, we mean that it doesn't have a value. {undefined} = 3 is not a well formed formula so it also does not have a value.

My whole question was to verify the convention for dealing with undefined terms in equality claims with respect to the true/false evaluation of the equality. There must be some confusion about what I'm asking. I'm just not seeing how the attribute of a term being undefined has any power to jump over to the determination of the truth of an equality or how an equality can be undefined. If the misunderstanding is on my part, can someone point to the fundamental conventions about this? I have been searching for a few days.

This question has been repeatedly answered (and so I am going to ask that the thread be locked): the convention is that terms which are not defined or resolve to values that are not defined cannot be used in a well formed formula in propositional logic - these links are a good place to start.

 

[Mark44]

The question has been asked and answered, so this seems like a good time to put this one to bed.