Why dividing zero by zero remains a problem

[mirelo]

Dividing any number by zero truly consists in two different mathematical operations:

1) Dividing zero by zero, which has any quotient, since the product of any number by zero is again zero -- so mathematicians call that quotient indeterminate.

2) Dividing any other (than zero) number by zero, which has no quotient, since, as the product of any number by zero is again zero, the product of no number by zero is different from zero -- so mathematicians call that quotient undefined.

The problem is that we define division as the inverse of multiplication. For instance, 2 * 3 = 6 only because 6 / 3 = 2 and 6 / 2 = 3. But if we concentrate on the division of zero by zero -- while leaving aside for a while the division of any other number by zero -- then we find something a bit shocking, namely that, according to the very definition of division, if 1 * 0 = 0, then it must be the case that 0 / 0 = 1. In fact, we may be fine with it until we discover that, for the same reason, it is also the case that 0 / 0 = 2, since 2 * 0 = 0.

And here is the problem: the very definition of division tells us to consider both 0 / 0 = 1 and 0 / 0 = 2 as valid in themselves, but the fact that 0 / 0 has any quotient, being thus indeterminate, tells us to repute it as invalid, since this indeterminacy would undermine mathematics as a whole -- although this is no mathematical reasoning. But no matter how much we love mathematics, our disregarding 0 / 0 to prevent mathematics form self-destructing will never turn into a mathematical argument itself. Historically, we all know what was mathematicians choice: even good mathematicians today tend to treat 0 / 0 as if it were just the same as 1 / 0, hence confusing being indeterminate with being undefined -- a mathematical error.

It is way too much easy for us to forget that mathematics itself is telling us that 0 / 0 = X is valid for each and every X, by the very definition of division, which demands of it only that X * 0 = 0. There is no mathematical reason for treating 0 / 0 as invalid, which does not mean there are no reasons to avoid dividing zero or any other number by zero in practice (for example, in computer programs). But no matter how disastrous that division may be -- as indeed it is -- it is mathematically valid. Even worse, it makes numbers false, because a true number 1 must never be identical to the number 2, which it is once it becomes the quotient of 0 / 0.

One thing is to forbid such a mathematical operation, like the Catholic Church has forbidden so many things in medieval times. Another thing is to have a rational motivation for that, not to mention a mathematical reason.

 

[HallsofIvy]

I'm not clear on what you mean by "valid" here. But by any reasonable definition of "valid" your statement, that "mathematics itself is telling us that 0/0= X is valid for each and every X, by the very definiton of division" is simply wrong.

The "definition of division" requires that the division of two numbers, a/b, be a specific number. Since 0= 0*X for any number, X, there is no specific number that can be set equal to 0/0. 0/0 is NOT "mathematically valid.

 

[mirelo]

I'm not clear on what you mean by "valid" here. But by any reasonable definition of "valid" your statement, that "mathematics itself is telling us that 0/0= X is valid for each and every X, by the very definiton of division" is simply wrong.

The "definition of division" requires that the division of two numbers, a/b, be a specific number. Since 0= 0*X for any number, X, there is no specific number that can be set equal to 0/0. 0/0 is NOT "mathematically valid.

I think you are a bit confused here: any "specific" number can be set equal to 0 / 0 -- since, as you put it yourself, 0 = 0 * X for any X -- which is precisely why 0 / 0 is indeterminate -- rather than undefined. What you are saying would be true if we were talking about, say, 1 / 0. But we are not: we are talking about 0 / 0.

 

[Mark44]

mirelo,
Why is it so hard for you to accept that division, just like the other three arithmetic operations, requires a unique number to be produced by the operation? You're beating a dead horse here.

 

[mirelo]

mirelo,
Why is it so hard for you to accept that division, just like the other three arithmetic operations, requires a unique number to be produced by the operation? You're beating a dead horse here.

I find it hard to accept anything that is not true. Division's only requirements are:

1) Its quotient must be a number.
2) The product between that number and the divisor must result in the dividend.

That's all. The requirement of any quotient being unique follows not from division itself, but from our trying to prevent numbers from becoming all the same, hence false, by means, precisely, of the division of zero by zero. Which is why if you eventually try to come up with a reason for your "requirement," then the goal of avoiding the indeterminacy of dividing zero by zero will be, curiously enough, your only argument.

 

[Hurkyl]

1) Dividing zero by zero, which has any quotient,

No, "dividing zero by zero" is simply not meaningful -- it's undefined.

mathematicians call that quotient indeterminate.

You appear to be thinking of an indeterminate form or similar ideas. There are often theorems in mathematics that are something like

If "b/a" is meaningful expression, then the result of the expression is the answer you're looking for​

and then the expression "0/0" might be called an indeterminate form, because it means we cannot use this theorem to tell us about the problem.

Of course, in certain cases a theorem might tell us different information. When solving ax=b for x, we consider the form b/a. In the case of the form "0/0", the theorem tells us that every number is a solution for x. (but, the theorem by no claims that x is a quotient of zero by zero)

The problem is that we define division as the inverse of multiplication.
...
according to the very definition of division, if 1 * 0 = 0, then it must be the case that 0 / 0 = 1.

You have the definition of division wrong. While there are a variety of ways to define division, all of them require (explicitly or implicitly) the denominator be nonzero.

Another thing is to have a rational motivation for that, not to mention a mathematical reason.

We do have a rational and mathematical reason for leaving 0/0 undefined -- arithmetic is more useful that way.


There are, of course, other ways to do arithmetic. A wheel (see http://en.wikipedia.org/wiki/Wheel_theory) allows division by zero. You can see just from the wiki page it's rather awkward to use. And, of course, it's not useful at all unless you have an application for wheel theory...

 

[Mark44]

Division's only requirements are:

1) Its quotient must be a number.
2) The product between that number and the divisor must result in the dividend.

Right. And by "a number" it means one specific number, not two of them, not an infinite number of them.

 

[Char. Limit]

I find it hard to accept anything that is not true. Division's only requirements are:

1) Its quotient must be a number.
2) The product between that number and the divisor must result in the dividend.

Actually, Condition 1 should read "Its quotient must be a unique number." Uniqueness is a requirement for division.

 

[micromass]

I don't understand what the problem is here. An operation is defined as a map $$O:S\times S\rightarrow S$$

such that three conditions are satisfied:
- For every elements x and y of S, it must hold that S(x,y) is in S,
- O(x,y) is defined for every x and y,
- O(x,y) must be unique.

Your definition violates the third rule. Thus, by definition, it is not an operation.

Of course, you could say that you could just eliminate rule 3 and just use 1 and 2. Alright, that will give you a multi-valued operation. So your division is a multi-valued operation. However, mathematicians don't usually work with them because they are very awkward to use and don't give anything special. That is, everything works fine with usual operations, so they're is no need for multi-valued operations.

There's nothing stopping you from using your division. But you need to be aware that it is not what mathematicians usually tend to do.

 

[jhae2.718]

And what would the point be in using such a multivalued division? In essence this simply results in a pointless and superfluous arithmetic where indeterminate forms have meaning, but this meaning is of no use.

 

[mirelo]

Right. And by "a number" it means one specific number, not two of them, not an infinite number of them.

It is "a number" just because I was talking about a single quotient, and not because a division must have a single quotient. This is just a matter of rephrasing:

Division's only requirements are:

1) Any quotient must be a number.
2) The product between that number and the divisor must result in the dividend.

 

[Mark44]

It is "a number" just because I was talking about a single quotient, and not because a division must have a single quotient. This is just a matter of rephrasing:

Division's only requirements are:
1) Any quotient must be a number.
2) The product between that number and the divisor must result in the dividend.

Again, if by "a number" you mean "a unique number" then fine. Otherwise what you are talking about is your own definition for division, which is at odds with how division is defined and understood by anyone capable of doing arithmetic.

 

[mirelo]

No, "dividing zero by zero" is simply not meaningful -- it's undefined.

Despite my distaste in discussing with someone who blocked one thread I initiated and deleted another, I must remind you that 0 / 0 is indeterminate rather than undefined. As I said before, indeterminate means to have any possible value, while undefined means to have no possible value. Perhaps you'll pay more attention to this than to me:

http://en.wikipedia.org/wiki/Indeterminate_form"

An excerpt in case you don't go there:

The most common example of an indeterminate form is 0/0. As x approaches 0, the ratios x/x3, x/x, and x2/x go to , 1, and 0 respectively. In each case, however, if the limits of the numerator and denominator are evaluated and plugged into the division operation, the resulting expression is 0/0. So (roughly speaking) 0/0 can be 0, or , or it can be 1 and, in fact, it is possible to construct similar examples converging to any particular value. That is why the expression 0/0 is indeterminate.

You appear to be thinking of an indeterminate form or similar ideas. There are often theorems in mathematics that are something like

If "b/a" is meaningful expression, then the result of the expression is the answer you're looking for​

and then the expression "0/0" might be called an indeterminate form, because it means we cannot use this theorem to tell us about the problem.

Of course, in certain cases a theorem might tell us different information. When solving ax=b for x, we consider the form b/a. In the case of the form "0/0", the theorem tells us that every number is a solution for x. (but, the theorem by no claims that x is a quotient of zero by zero)

How can a number be a solution for X without being X, only God knows (or perhaps you are him).

You have the definition of division wrong. While there are a variety of ways to define division, all of them require (explicitly or implicitly) the denominator be nonzero.

The denominator being nonzero is a practical limitation of use, not a part of the definition.

We do have a rational and mathematical reason for leaving 0/0 undefined -- arithmetic is more useful that way.

Arithmetics is not only "more useful" that way, it is only possible that way. Are you capable of doing any arithmetics by admitting 0 / 0? I am not. I am not discussing the usefulness of 0 / 0, but its mathematical validity.

There are, of course, other ways to do arithmetic. A wheel (see http://en.wikipedia.org/wiki/Wheel_theory) allows division by zero. You can see just from the wiki page it's rather awkward to use. And, of course, it's not useful at all unless you have an application for wheel theory...

I am not interested in alternate arithmetics, I am not trying to create another kind of mathematics, my interest for now is restricted to the mathematical validity of 0 / 0, if you don't mind.

 

[Dickfore]

Actually, division is defined through multiplication and its inverse:

$$a/b \equiv a \cdot b^{-1}$$

The problem with division by zero comes from the fact that 0 has no multiplicative inverse in ordinary numbers, because, by using the distribution law:

$$x*(0 + y) = x*0 + x*y$$

Using the fact that 0 is the neutral element of addition:

$$0 + y = y$$

so we may write:

$$x*y = (x*0) + x*y$$

which is valid for any x and y. Thus, we must have:

$$x*0 = 0*x = 0$$

for all $x$. But, this means that there is no $x$ such that:

$$0*x = x*0 = 1, \, \mathrm{FALSE}!$$

So, in order to define division by zero, you must renounce the distributive law.

 

[micromass]

Despite my distaste in discussing with someone who blocked one thread I initiated and deleted another, I must remind you that 0 / 0 is indeterminate rather than undefined. As I said before, indeterminate means to have any possible value, while undefined means to have no possible value. Perhaps you'll pay more attention to this than to me:

This is NOT true. 0/0 is an undefined form, it is not undetermined.
Wiki is not wrong however, but you understand them wrong. 0/0 is undetermined when you are working with limits. Thus if your limit evaluated to 0/0, then you'll have to try something else that works. So the limit is undetermined.
However, when you do some arithmetic, then 0/0 is an undefined form. And this is the thing you're doing here...

 

[micromass]

On another note, I really fail to see why the mentors don't lock this post. The OP has no intention of changing his mind on this, this is clear from his way of discussing.
Thus this is just a statement of facts that the OP beliefs, and not a genuine question. I really don't see why this needs to be discussed...

 

[mirelo]

Again, if by "a number" you mean "a unique number" then fine. Otherwise what you are talking about is your own definition for division, which is at odds with how division is defined and understood by anyone capable of doing arithmetic.

The definition of division does not require it to have just one quotient, although you could not do any arithmetics if it didn't, since it makes numbers false, hence useless. I am not telling you that by accepting 0 / 0 you will make any kind of useful arithmetics, which you will not. What I am telling you is that mathematics provides nothing to invalidate 0 / 0. We are the ones who invalidate it, so as to keep doing useful arithmetics, without waiting at all for mathematics to justify that.

 

[mirelo]

Again, if by "a number" you mean "a unique number" then fine. Otherwise what you are talking about is your own definition for division, which is at odds with how division is defined and understood by anyone capable of doing arithmetic.

Do you know how "a number" can mean other thing than "a unique (or single) number"? I don't. That is not the point. The point is whether division must have only one quotient by definition, which it doesn't. And being capable of doing arithmetic has little to do with that, so it won't help you to understand what I am saying.

 

[mirelo]

Actually, division is defined through multiplication and its inverse:

$$a/b \equiv a \cdot b^{-1}$$

The problem with division by zero comes from the fact that 0 has no multiplicative inverse in ordinary numbers, because, by using the distribution law:

$$x*(0 + y) = x*0 + x*y$$

Using the fact that 0 is the neutral element of addition:

$$0 + y = y$$

so we may write:

$$x*y = (x*0) + x*y$$

which is valid for any x and y. Thus, we must have:

$$x*0 = 0*x = 0$$

for all $x$. But, this means that there is no $x$ such that:

$$0*x = x*0 = 1, \, \mathrm{FALSE}!$$

So, in order to define division by zero, you must renounce the distributive law.

I have already made it clear since the beginning the difference between 0 / 0 and 1 / 0. The first is indeterminate, since, as you noted, any number multiplied by zero results in zero, while the second, for the same reason, is undefined.

X * 0 = 0 for any X, hence 0 / 0 = X for any X -- which makes 0 / 0 indeterminate -- while any X <> 0 will make X / 0 undefined.

 

[jhae2.718]

In arithmetic, a/0 is undefined for all possible a. Quantities such as 0/0 are considered indeterminate in cases of limits, where alternative methods such as L'Hôpital's rule can yield finite solutions.

 

[Mark44]

I have already made it clear since the beginning the difference between 0 / 0 and 1 / 0. The first is indeterminate, since, as you noted, any number multiplied by zero results in zero, while the second, for the same reason, is undefined.

You have completely misunderstood the concept of an indeterminate form, as described in the wiki page you posted. Indeterminate forms such as [0/0] and the others are often presented in brackets to reinforce the idea that they are not numbers. Whether you have 0/0 or 1/0, the division is not defined.

 

[mirelo]

This is NOT true. 0/0 is an undefined form, it is not undetermined.
Wiki is not wrong however, but you understand them wrong. 0/0 is undetermined when you are working with limits. Thus if your limit evaluated to 0/0, then you'll have to try something else that works. So the limit is undetermined.
However, when you do some arithmetic, then 0/0 is an undefined form. And this is the thing you're doing here...

Undefined means there is no possible quotient, while indeterminate means there are possible quotients, you just have no means of choosing one.

Undefined:

1 / 0 = ? (there is no possible ?)

Indeterminate:

0 / 0 = 1, 2, 3... (there is no way of "choosing" a quotient)

 

[micromass]

Undefined means there is no possible quotient, while indeterminate means there are possible quotients, you just have no means of choosing one.

Undefined:

1 / 0 = ? (there is no possible ?)

Indeterminate:

0 / 0 = 1, 2, 3... (there is no way of "choosing" a quotient)

This is not what indeterminate means. Indeterminate applies to limits. And it means that there is a unique answer to the limit, but we just haven't found it yet.

Indeterminate forms do NOT apply to quotients, and they do NOT allow you to choose a quotient in any way...

 

[Dickfore]

I have already made it clear since the beginning the difference between 0 / 0 and 1 / 0. The first is indeterminate, since, as you noted, any number multiplied by zero results in zero, while the second, for the same reason, is undefined.

Suppose $0/0 = a$. Then:

$$0 * x = 0$$

$$a = \frac{0}{0} = \frac{0 * x}{0} = \frac{0}{0} * x = a * x$$

which is true for all x. This means that $a = 0$. But, then:

$$x = x + 0 = x + \frac{0}{0} = \frac{0*x + 0}{0} = \frac{0 + 0}{0} = \frac{0}{0} = 0$$

which is an absurd result.

 

[jhae2.718]

You are incorrect in your definitions. An expression E is undefined for some value x if there is no clear mapping by E of x to another defined value.

Indeterminate forms occur in the case of limits, where an indeterminate form is a form that can by converted into a determinate form through various methods.

 

[mirelo]

On another note, I really fail to see why the mentors don't lock this post. The OP has no intention of changing his mind on this, this is clear from his way of discussing.
Thus this is just a statement of facts that the OP beliefs, and not a genuine question. I really don't see why this needs to be discussed...

Perhaps if you provide me with arguments you will get me willing to change my mind.

As for the validity of 0 / 0 being a genuine question, the millenarian discussion surrounding the topic testifies it is.

 

[jhae2.718]

The problem is that your logic is flawed, and you seem unwilling to admit this.

 

[micromass]

Nobody has the "intention of changing his mind": everybody (which includes you) has a good reason to think the way he does, and will only change his mind if forced by arguments (hopefully).

Well, there are two possible situations which could occur:
- you want to ask us a question because you don't quite understand why mathematicians have not defined 0/0. That's a discussion that I would appreciate.
- you come here claming mathematics is wrong and you are not willing to listen to arguments. In fact, you just admitted not wanting to change your mind. This is when a topic deserves to be locked in my opinion.

I see this as a serious forum. And people claiming that mathematics is all wrong without understanding the mathematics itself, should go have their discussion elsewhere in my opinion...

As for the validity of 0 / 0 being a genuine question, the millenarian discussion surrounding the topic testifies it is.

There is no discussion on the value of 0/0. All mathematicians agree that it is undefined.

 

[berkeman]

Thread closed pending Moderation.


EDIT -- This thread is now closed permanently, for obvious reasons.