But it's not undefined.

And sometimes we need to know the answer as WBahn asked in post# 9

The rest of your post is based upon not understanding number systems sufficiently.


Edit


Most people are quite relaxed with the idea that there is more than one solution to

√2 = ?

So why does there have to be only one solution to 0/0 = ?

In fact there can be infinitely many solutions or no solutions, depending upon the circumstances and number system. That is the difficulty that elementary theory seeks to avoid by discounting it.

In doing what most people call algebra, the rules fail if we allow division by zero and by consequence 0/0 is undefined. This is what I was attempting to demonstrate.

For the pedants amongst us, we are doing algebra on the field of real or complex numbers. It doesn't matter, in any field division by zero is undefined.

What seems to have happened in this thread is that what was a question of algebra morphed strangely to a question of analysis and limits of functions.

Of course for f(x) = x/x,

f(x) = 1 almost everywhere,

and for x=0 the limit is 1, but this is not the question.

f(x) as given is undefined at x=0.

Would you argue that for the function
g(x) = x/x for x ≠ 0
g(x) = 3 for x = 0

that g(0) is 1 or 3?

 

This is exactly the point I made in Post #2.

And since you defined g(x) to be 3 and x=0, the g(0)=3.

This extends directly to value of x for which x/x is perfectly well defined.

g(x) = x/x for x≠ 0, x≠ 13
g(x) = 3 for x = 0
g(x) = 42 for x=13

 

I agree with you WBahn.

I was replying to the strange post by studiot where he claimed that 0/0 is not undefined and did, in fact, have multiple values!

 

I don't take what he said as meaning that -- but perhaps I am reading something in that isn't there.

I took it that he is making a distinction between "absolutely undefined" and "indeterminate by itself, meaning we don't know what the value is or whether it has none, one, many, or an infinite number, but we may be able to determine which, if any, values it takes on in a specific situation based on other information relevant to the specifics of the situation."

I don't see that as an unreasonable view, though I tend to prefer the view that you and I offered up which is that it is indeterminate (or undefined, don't know that there's much of a distinction in this view) and that's all there is to it, but we can sometimes define a "companion" function that defines the value at those points in a reasonable way.

I think the latter is the more "pure" view, but the former is possibly more accessible for a wider, more practical, audience.

 

I was replying to the strange post by studiot where he claimed that 0/0 is not undefined and did, in fact, have multiple values!

and for x=0 the limit is 1, but this is not the question.

Is it? Does this limit exist? and is the function g(x)= (x/x) differentiable at x=0 ?

I also offered this once already in this thread. Here it is again.

Instead of mocking what you do not understand, I offered you the chance of explanation.

Zero (and infinity) does not exist in the most ancient and fundamental of number systems. These are called the natural or counting numbers, denoted by script capital N in mathematics.

The usual number system employed for algebra is called the real numbers, denoted capital script R, and this includes zero, but does not include infinity.

The next most complicated number system in maths is called the extended real number system and does include infinity. This system does not obey the normal rules of algebra. So all the clever manipulation diplayed in various posts has tried unseccessfuly to connect numbers drawn from different systems, that obey different rules of algebra and not suprisingly failed.

You further introduced the term 'field'. Do you understand what it means and why we wish to use it?

R does not constitute a field with respect to the binary operation of division.

 

Zero (and infinity) does not exist in the most ancient and fundamental of number systems. These are called the natural or counting numbers, denoted by script capital N in mathematics.

The usual number system employed for algebra is called the real numbers, denoted capital script R, and this includes zero, but does not include infinity.

The next most complicated number system in maths is called the extended real number system and does include infinity. This system does not obey the normal rules of algebra. So all the clever manipulation diplayed in various posts has tried unseccessfuly to connect numbers drawn from different systems, that obey different rules of algebra and not suprisingly failed.

You further introduced the term 'field'. Do you understand what it means and why we wish to use it?

R does not constitute a field with respect to the binary operation of division.

I wasn't talking about any ancient number system.

When we do algebra today in engineering we invariably do it on the field of real or complex numbers (http://mathworld.wolfram.com/Field.html) as you state:

The usual number system employed for algebra is called the real numbers, denoted capital script R, and this includes zero, but does not include infinity.

Every non-zero element in a field has a multiplicative inverse allowing division by any non-zero element to be defined.

Division by zero in a field is undefined: http://mathworld.wolfram.com/DivisionbyZero.html

0/0 is always undefined (http://mathworld.wolfram.com/Indeterminate.html) even if you are in an extended number system

 

I wasn't talking about any ancient number system.

I was offering you a small insight into the way number systems including the reals are built up from the natural numbers.

You should, of course, be aware that these are alive and well today as the first numbers we teach our children.

Since you know so much and keep bandying real numbers about, do you know what they are?

When we do algebra today in engineering we invariably do it on the field of real or complex numbers

Do we?

I can only suggest you read a modern text on the subject, such as Fraleigh or perhaps Lang.

Matrix algebra, Tensor algebra, come immediately to mind as not conforming to your strictures.

 

Thinking again I realise that the function g(x) = x/x does indeed have a limit of 1 at x=0, and is continuous and differentiable there, so Tesla you were correct.

However I would observe that my other comments stand.

Here is another way to look at it.

Take the natural numbers 1,2,3,.....

How many are there?

The idea of Infinity creeps in very quickly. Even the Aborigines had some idea,I understand from anthropologists that their counting runs 1,2,3, many

Now the Ancient Greeks proved that between any two rational numbers ( how many rational numbers are there?) there is another rational number.
They also realised that this implied that there must be an infinite number between any two rationals.

So is this one infinity ( the number of rationals) different from the infinity of rationals between any two of them?

Then moving on to your favourites, the reals and the complex numbers.

Are there more reals than complex numbers than reals, since through every real you can draw an infinite line of complex numbers to map out the complex plane?

 

(test post - please ignore)

 

Quote:
When we do algebra today in engineering we invariably do it on the field of real or complex numbers

Do we?

I can only suggest you read a modern text on the subject, such as Fraleigh or perhaps Lang.

Matrix algebra, Tensor algebra, come immediately to mind as not conforming to your strictures.

Can you really look back on this thread and conclude that those posts about 0/0 were about anything else other than algebra on the real or complex fields?

Even if we assume that folks were talking about something other than a field (I would have thought it have to be at least a ring to have a zero element, but I could be wrong), division by zero is always meaningless. I don't have your favourite text to hand, but I am always keen to learn, so please enlighten me as to where it is useful to assign a value (or values) to 0/0?

 

but I am always keen to learn, so please enlighten me as to where it is useful to assign a value (or values) to 0/0?

WBahn has already offered an example in his post#9.

I have always tried to answer the questions in your posts how about trying some of the ones in mine, particular my post#28 ?

 

I used to use Trig every day at work, and even in my own shop. Trig was useful in the real world.
I never had a practical use for Algebra, or more than a basic understanding of it.
Keep up the good work, guys. I say zero=zero.

This is an interesting thread.

 

WBahn has already offered an example in his post#9.

I have always tried to answer the questions in your posts how about trying some of the ones in mine, particular my post#28 ?

WBahn in post #9 gave an example of a function that evaluated to 0/0 at a point but had a limit at that point. This does not mean that 0/0 is meaningful. I prefer the simpler example,
f(x) = x/x

this equals 1 almost everywhere, but as given is undefined at x=0. If you don't agree with that, we should agree to disagree and hope that we both have done our last algebra and analysis exams where such pedantry matters.

As far as your question in post #28, I can't see the relevance, but there are the same number of reals as complex numbers. I had to think for a moment to construct a mapping, but it is a good exercise for you if you are wondering.

 

Tesla, you noted that this was a question to do with voltages.

A simple question perplexing us on another sub forum:

Can infinity times zero ever equal a constant?

I am an applied mthematician, not a clairvoyant. So perhaps you or ernie would enlighten us all as to the origin of this question.

I would observe that we have discussed at great lenght before at AAc various combinations of values that cause difficulty in Ohm's Law.

That is voltage without current, current without voltage (and zero ohms conductors.

 

Tesla, you noted that this was a question to do with voltages.

I am an applied mthematician, not a clairvoyant. So perhaps you or ernie would enlighten us all as to the origin of this question.

I've got no idea on the origin of the question or if it applies to voltages. WBahn answered it in the first reply. It is clearly indeterminate:
http://mathworld.wolfram.com/Indeterminate.html

On the question you asked, can you now see why there is the same number of real numbers as complex numbers?

 

On the question you asked

I actually posed several questions, not one.

Their result was meant to demonstrate (and your response did so quite ably) that we need infinity.

Taken together they can demonstrate that not only do we need infinity, but there is more than one infinity.

As to zero/zero well isn't obvious in an electrical forum?

What is the current when you apply zero voltage to an ideal conductor?

Here is another

What is the coefficient of friction when a massless object (eg a photon) slides along a surface?

 

I actually posed several questions, not one.

Their result was meant to demonstrate (and your response did so quite ably) that we need infinity.

So I got what, an 'A', 'B-'?

Sorry, someone that asks a question that they know the answer to, and then demands an answer, is trolling.

 

(test post - please ignore)

Congrats! t's good to see you got your posting problem fixed.

 

So I got what, an 'A', 'B-'?

Sorry, someone that asks a question that they know the answer to, and then demands an answer, is trolling.

Seems to me that your objective is to be as disruptive as possible.

You have repeatedly stated that 0/0 can never mean anything. Yet when asked how this plays with a simple engineering calculation of current in a 0/0 situation you offer the above abusive statement.

 

I used to use Trig every day at work, and even in my own shop. Trig was useful in the real world.
I never had a practical use for Algebra, or more than a basic understanding of it.
Keep up the good work, guys. I say zero=zero.

This is an interesting thread.

I imagine you used algebra all the time. You need to set a bunch of fence posts and have a budget that will only permit a maximum of so many bags of concrete. Each post is a 4x4 and will be set in a hole that is two feet deep. You know the total length of the fence and that there will be a post every eight feet. You need to determine what diameter the holes can be (and whether they would be big enough to provide a good footing, but that is a separate issue) while staying within budget.

The process that you use to answer that question is algebra, whether you recognize it as such or not.