Is Dividing Zero by Zero Valid in Mathematics?

[mirelo]

If we ask whether dividing zero by zero equals one, then we must answer yes, because if we multiply one by zero then we recover zero. However, if we ask whether it equals two rather than one, then we must still answer yes, because if we multiply two by zero then we also recover zero. Likewise, the result of dividing six by two is three only because multiplying three by two results in six. Hence, the result of dividing zero by zero is any number, as multiplying any number by zero results in zero. So dividing zero by zero makes any number the same as any other number. And if all numbers are the same, then all operations between them are also the same. Yet still, in taking a single division of zero by zero with any single quotient, we must recognize it as a perfectly valid operation, since this quotient multiplied by zero results in zero. Nothing at each single division of zero by zero alone makes it invalid: each one is a place where all numbers falsify themselves. Conversely, these operations are only possible if any number is different from any other number, hence true, so every number must be true to be false -- in the division of zero by zero -- and false to be true -- since the division of zero by zero remains valid.

Likewise, whenever I say, "right now, I am lying," if what I am saying is true, then it must be false, and if it is false, then it must be true.

 

[Mark44]

Mathematicians want arithmetic operations to have a single answer. 0/0 doesn't meet that criterion, since, as you point out, we could define 0/0 to be 1, 2, -5, $\pi$, whatever.

Dividing a nonzero number by zero raises a different problem - there is no number such that zero times that number equals the nonzero number in the numerator. Either way, division by zero is not defined.

Some of the other things you said in the last part of the first paragraph don't make much sense. For example, "a place where all numbers falsify themselves." Numbers are not Boolean expressions whose values can be only true or false. A real number identifies a point on the number line. False and true are nowhere to be found on the real number line (assuming that you are not a C programmer). However, a particular statement can be always true, always false, or true some of the time. Some examples are
2(x + 3) = 2x + 6 (true for all x)
3x + 2 = 3x + 1 (never true)
x² = 4 (true only for x = 2 or x = -2)

This statement of yours is incomprehensible:
"Conversely, these operations are only possible if any number is different from any other number, hence true, so every number must be true to be false-- in the division of zero by zero -- and false to be true -- since the division of zero by zero remains valid."

Again, you are applying the attributes of true and false to things (numbers) to which these concepts don't apply.




Another

 

[atomthick]

If you ask me what is the result of 0/0 my answer will be "it can be any value.". But if I had more information on how that 0/0 was obtained I might be able to give you only one answer.

Example: 0/0 is like trying to find the length of a pipe by looking through it's hole to the other side. As far as you can tell the pipe can have any length (actualy it only has one fixed length) but if you change the angle and view the pipe from the outside you can see a right angle triangle in which the length of the pipe is the hypotenuse. See, when you first looked through the hole you were trying to compute 0/cos(pi/2) which is 0/0.

My advice is that whenever you get to 0/0 it only means you need to look fore more information, look at the problem in a different way

 

[mirelo]

Mathematicians want arithmetic operations to have a single answer. 0/0 doesn't meet that criterion, since, as you point out, we could define 0/0 to be 1, 2, -5, $\pi$, whatever.

Dividing a nonzero number by zero raises a different problem - there is no number such that zero times that number equals the nonzero number in the numerator. Either way, division by zero is not defined.

Some of the other things you said in the last part of the first paragraph don't make much sense. For example, "a place where all numbers falsify themselves." Numbers are not Boolean expressions whose values can be only true or false. A real number identifies a point on the number line. False and true are nowhere to be found on the real number line (assuming that you are not a C programmer). However, a particular statement can be always true, always false, or true some of the time. Some examples are
2(x + 3) = 2x + 6 (true for all x)
3x + 2 = 3x + 1 (never true)
x² = 4 (true only for x = 2 or x = -2)

This statement of yours is incomprehensible:
"Conversely, these operations are only possible if any number is different from any other number, hence true, so every number must be true to be false-- in the division of zero by zero -- and false to be true -- since the division of zero by zero remains valid."

Again, you are applying the attributes of true and false to things (numbers) to which these concepts don't apply.




Another

Would you say a number two that is identical to the number one is a true number?

 

[mirelo]

If you ask me what is the result of 0/0 my answer will be "it can be any value.". But if I had more information on how that 0/0 was obtained I might be able to give you only one answer.

Example: 0/0 is like trying to find the length of a pipe by looking through it's hole to the other side. As far as you can tell the pipe can have any length (actualy it only has one fixed length) but if you change the angle and view the pipe from the outside you can see a right angle triangle in which the length of the pipe is the hypotenuse. See, when you first looked through the hole you were trying to compute 0/cos(pi/2) which is 0/0.

My advice is that whenever you get to 0/0 it only means you need to look fore more information, look at the problem in a different way

So you are telling me that your pipe can have any length I wish?

 

[Jack Fawkes]

So you are telling me that your pipe can have any length I wish?

No, the length of the pipe is determined by the context (pattern) within the system.

Example...

Say that this pipe described earlier is not just alone, but it is in fact flanked on both sides by an infinite amount of pipes. Let's say that you measure each pipe starting at the end of both sides, eventually approaching the middle one. We'll say that you measure each and they all have a length of 1 meter. When you reach the middle pipe, you don't measure it like the other pipes but instead perform the same experiment described earlier. It could still have any value, but logic would dictate that this pipe would carry the same pattern as those pipes to it's left and to it's right. Mathematics is nothing more than an deductive analysis of patterns within a system.

x/0 must have only one value under all mathematical axioms, but what if there existed a number whose value could change based upon context? A variable number. This is part of an idea I have been working on for nearly 2 years. I will explain later.

 

[mirelo]

Mathematicians want arithmetic operations to have a single answer. 0/0 doesn't meet that criterion, since, as you point out, we could define 0/0 to be 1, 2, -5, $\pi$, whatever.

Dividing a nonzero number by zero raises a different problem - there is no number such that zero times that number equals the nonzero number in the numerator. Either way, division by zero is not defined.

Some of the other things you said in the last part of the first paragraph don't make much sense. For example, "a place where all numbers falsify themselves." Numbers are not Boolean expressions whose values can be only true or false.

One thing is to say numbers are not capable of expressing their own falsity, and an entirely other thing is to say they are not capable of being false. In a sense, even boolean expressions cannot express their own falsity: whenever you evaluate a boolean expression incorrectly, the boolean result you get is false, independently of its (false) falsity or truth, which it cannot at once express and embody.

A real number identifies a point on the number line. False and true are nowhere to be found on the real number line (assuming that you are not a C programmer).

By the above, either you are saying that numbers are not boolean values or that a false number is not a true number, none of which is much of a novelty.

However, a particular statement can be always true, always false, or true some of the time. Some examples are
2(x + 3) = 2x + 6 (true for all x)
3x + 2 = 3x + 1 (never true)
x² = 4 (true only for x = 2 or x = -2)

If "zero divided by zero equals nine" is a valid mathematical operation -- as it is, because nine multiplied by zero indeed results in zero -- then each mathematical operation you repute as true can be false, and vice-versa.

This statement of yours is incomprehensible:
"Conversely, these operations are only possible if any number is different from any other number, hence true, so every number must be true to be false-- in the division of zero by zero -- and false to be true -- since the division of zero by zero remains valid."

This means you can only divide zero by zero if zero remains different from, say, one: it must remain a true number so you can still mathematically operate. On the other hand, its division by zero makes it false, along with all other numbers, and since you cannot deny validity to any single division of zero by zero giving you a particular quotient -- as the product of that quotient by zero still gives you zero -- zero and all other numbers being true must make them false.

Again, you are applying the attributes of true and false to things (numbers) to which these concepts don't apply.

Another

Again, although numbers cannot express their own truth or falsity this has no chance of meaning they cannot be themselves true or false: would you say a number two that equals the number one is a true number?

 

[mirelo]

No, the length of the pipe is determined by the context (pattern) within the system.

Example...

Say that this pipe described earlier is not just alone, but it is in fact flanked on both sides by an infinite amount of pipes. Let's say that you measure each pipe starting at the end of both sides, eventually approaching the middle one. We'll say that you measure each and they all have a length of 1 meter. When you reach the middle pipe, you don't measure it like the other pipes but instead perform the same experiment described earlier. It could still have any value, but logic would dictate that this pipe would carry the same pattern as those pipes to it's left and to it's right. Mathematics is nothing more than an deductive analysis of patterns within a system.

x/0 must have only one value under all mathematical axioms, but what if there existed a number whose value could change based upon context? A variable number. This is part of an idea I have been working on for nearly 2 years. I will explain later.

There is a difference between saying the length of a pipe is relative and saying it can have any value, including zero. What I am saying is that numbers must be false to be true -- and vice-versa -- just like the liar statement must be false to be true. Which means your pipe may have no length at all.

 

[atomthick]

So you are telling me that your pipe can have any length I wish?

If you are a philosopher yes it means the pipe can have any length, but if you are a physicist or a mathematician it actualy means that you haven't solved the problem. Listen, you could say the same thing about the pipe even if you haven't looked through the hole, you could do it even if someone asks you on the street and you know nothing about that pipe, this is what actualy 0/0 means, you know nothing worthy about the problem you are trying to solve, it means you don't have the right information to deduce the answer. It can be any value.

 

[mirelo]

If you are a philosopher yes it means the pipe can have any length, but if you are a physicist or a mathematician it actualy means that you haven't solved the problem. Listen, you could say the same thing about the pipe even if you haven't looked through the hole, you could do it even if someone asks you on the street and you know nothing about that pipe, this is what actualy 0/0 means, you know nothing worthy about the problem you are trying to solve, it means you don't have the right information to deduce the answer. It can be any value.

You are confusing possibility with actuality: the pipe will eventually have a well-defined length -- when you have "solved the problem" -- but zero divided by zero will never have a well-defined value -- it will be forever indeterminate. According to you, it is like once you had finally figured out how long is your pipe (I should rephrase), then zero divided by zero would suddenly become well-defined, which will never be the case.

 

[dkotschessaa]

Off topic, but you've got me wondering... Can you determine the length of a pipe, by measuring the circumference, then finding the ratio between that circumference and the apparent circumference of the other end as viewed through it? In other words if you held a ruler up to the end that was towards you and measured 1 inch, and if with that same ruler in the same place the other end appeared to be half an inch... Well I'd have to experiment.

I realize that you just meant it as an analogy.

-M

 

[atomthick]

Off topic, but you've got me wondering... Can you determine the length of a pipe, by measuring the circumference, then finding the ratio between that circumference and the apparent circumference of the other end as viewed through it? In other words if you held a ruler up to the end that was towards you and measured 1 inch, and if with that same ruler in the same place the other end appeared to be half an inch... Well I'd have to experiment.

I realize that you just meant it as an analogy.

-M

I believe you can ;)

 

[mirelo]

Off topic, but you've got me wondering... Can you determine the length of a pipe, by measuring the circumference, then finding the ratio between that circumference and the apparent circumference of the other end as viewed through it? In other words if you held a ruler up to the end that was towards you and measured 1 inch, and if with that same ruler in the same place the other end appeared to be half an inch... Well I'd have to experiment.

I realize that you just meant it as an analogy.

-M

Back to the topic, do you agree that numbers must be true to be false and false to be true? And if not, then why?

 

[dkotschessaa]

Back to the topic, do you agree that numbers must be true to be false and false to be true? And if not, why?

I believe, as was explained to you more than once, that numbers themselves cannot be assigned values of "true" or "false," and that dividing anything by zero is just bad algebra.

Sorry for the earlier off-topic post.

-DaveKA

 

[Hurkyl]

This is a forum for mathematics, not a forum for whatever random musings you may have.