How can the illicit operation 0/0 be remedied?

[roger]

How can the illicit operation 0/0 be remedied ?

so that it becomes meaningful ?

 

[matt grime]

it can't: algebraically it is meaningless, leave it alone, please. and can we have that FAQ now? how many times must these bloody questions have to be answered?

 

[TD]

How can the illicit operation 0/0 be remedied ?
so that it becomes meaningful ?

*IF* you are referring to a function which apparently yields the indeterminate form 0/0 when the function itself can be written as f(x)/g(x), you can apply L'Hopitals rule.

 

[roger]

it can't: algebraically it is meaningless, leave it alone, please. and can we have that FAQ now? how many times must these bloody questions have to be answered?

I already looked this up on this forum , but as I far I am aware, no one has actually attempted to remedy it. The debate was surrounding whether or not it was defined and why.

I understand, that it is meaningless, I do not dispute this. But please at least give a reason for your assertion that it can't be remedied.



Roger

 

[roger]

I wasn't refferring to any functions though.

 

[matt grime]

Remedt it? There are many explanations of why it cannot be meaningully remedied as an algbraic quantity. Look through them again with particular reference to what a field is, or look here

http://www.maths.bris.ac.uk/~maxmg/maths/philosophy/divide.html

it is about 1/0 but algebraically the same argument is valid for 0/0. If one were algebraically valid in a field then the other would be to

 

[russ_watters]

I already looked this up on this forum , but as I far I am aware, no one has actually attempted to remedy it. The debate was surrounding whether or not it was defined and why.
I understand, that it is meaningless, I do not dispute this. But please at least give a reason for your assertion that it can't be remedied.
Roger

Um - if you understand that it is meaningless, then you should understand that that means that it can't be "remedied".

 

[roger]

''However, this doesn't stop us defining the symbol 1/0 and adding it into the real numbers, say, however this then makes the reals cease to be a field...''

What would the symbol 1/0 be defined to be outside the reals ?
I didn't originally ask why 0/0 can't be remedied within a field.


''We also require that addition and multiplication behave just as you expect: x*(u+v)=x*u+x*v ''

In what sense is it to be expected ? isn't that just a rule ?


When you said 0*x =0 can be deduced, is that because 0+x=0 for all x and x*y=y*x has already been defined ?



''..and 1/y is defined to be the multiplicative inverse of y, that is the number such that y*(1/y)=1.''

Wouldn't that be circular reasoning, since you're using 1/y itself, as part of the definition of 1/y ?



Roger

 

[matt grime]

''However, this doesn't stop us defining the symbol 1/0 and adding it into the real numbers, say, however this then makes the reals cease to be a field...''
What would the symbol 1/0 be defined to be outside the reals ?

if you read any of the many posts on this you'd know all about the extended reals or complex numbers

I didn't originally ask why 0/0 can't be remedied within a field.

then this makes it a poorly stated question

''We also require that addition and multiplication behave just as you expect: x*(u+v)=x*u+x*v ''
In what sense is it to be expected ?

look up the axioms of a field

isn't that just a rule ?

axiom

When you said 0*x =0 can be deduced, is that because 0+x=0 for all x and x*y=y*x has already been defined ?

get a pencil and paper and work out why 0*x=0 is a consistent statement within a ring or field

''..and 1/y is defined to be the multiplicative inverse of y, that is the number such that y*(1/y)=1.''
Wouldn't that be circular reasoning, since you're using 1/y itself, as part of the definition of 1/y ?
Roger

Eh? Of course 1/y appears in the definition of 1/y otherwise it wouldn't define 1/y, would it? No it is not circular, and it is the definition of 1/y: the multiplicative inverse of y.

 

[Gokul43201]

roger, please search for 0/0 in this forum and you'll find several threads that answer your question. Here's one : https://www.physicsforums.com/showthread.php?t=78322&page=3&highlight=0/0

With that, this may be closed.

Edit : So far, this thread doesn't seem to be going the way such threads usually do, so as long as there is something new being discussed, I think it can stay open. If it's all redundant, then a link will serve. My above link was not the best one for the purpose.

 

[roger]

Gokul, I appreciate what you're saying, I actually looked up your link prior to posting this anyway.

Matt,

Is it because in a field 0*x= (0-1)x+x=-x+x=0 ?
using induction ?

when you said ''We also require that addition and multiplication behave just as you expect: x*(u+v)=x*u+x*v ''

I simply wanted to know what you meant by ''expect'' ?

because at the end of your note, you conclude saying that ''...it is all a matter of definition, and not actually a debate about the inherent meaning of division as some sort of real life operation.''

 

[matt grime]

Induction? On what? We don't have the natural numbers anywhere.

 

[roger]

I was doing, or attempting to do induction on 0+x to show that 0*x=0

could you please answer my question in my previous post ?


Roger

 

[matt grime]

Which question? I answered those I thought needed answers and left as a simple exercise those things you ought to be able to do for yourself (or at least be able to look up somewhere if you can't figure it out). Remember that induction only applies to statements indexed by the natural numbers. The proof that 0*x=0 even appears on my web pages, one page above the link I posted)

 

[roger]

when you said ''We also require that addition and multiplication behave just as you expect: x*(u+v)=x*u+x*v ''

I simply wanted to know what you meant by ''expect'' ?

That question.

and what do you mean by indexed by natural numbers, I looked it up , on mathworld but there's no mention of this..

 

[matt grime]

I meant what i said in my first reply. GO LOOK UP THE AXIOMS OF A FIELD.

Do you know what induction is?

 

[roger]

yes I Looked it up but I couldn't find it.

and induction is where for an infinite set of propositions, you prove the first case which then implies the truth of the rest.

 

[DeadWolfe]

yes I Looked it up but I couldn't find it.

http://www.google.ca/search?hl=en&q=field+axioms&btnG=Google+Search&meta=

Very first page.

 

[matt grime]

yes I Looked it up but I couldn't find it.
and induction is where for an infinite set of propositions, you prove the first case which then implies the truth of the rest.

I think you'd better go and search for mathematical induction again and see what kind of set of propositions are allowed.

 

[roger]

matt, are you aware of any typing errors in Tims book ''a very short introduction'' In the section on limits it says to find the instantaneous speed to multiply the infinitesimal distance by infinitesimal time ?

And I wonder if there are any other similar books but more formal and less basic ?

 

[houserichichi]

when you said ''We also require that addition and multiplication behave just as you expect: x*(u+v)=x*u+x*v ''
I simply wanted to know what you meant by ''expect'' ?

Perhaps this is a dead issue but...we expect the real numbers to follow the rule in the quote above. Why? Because that's what we're taught when first dealing with any form of algebraic manipulation. It isn't until one studies abstract mathematical gadgets (like fields) that they discover that such rules are satisfied (hence expected) because they are requirements of the gadgets we didn't know we were working with.

More succinctly, the majority of us grow up not knowing what a field is or that they exist. Rules like x*(u+v)=x*u+x*v and x+y=y+x are things we take for granted. Turns out that the reals actually form this mathematical concept of a field (or satisfy its axioms...you choose your semantics) and thus have such things imposed on them. All fields share these exact same rules. The reals are not unique with respect to the field axioms.

 

[matt grime]

matt, are you aware of any typing errors in Tims book ''a very short introduction'' In the section on limits it says to find the instantaneous speed to multiply the infinitesimal distance by infinitesimal time ?
And I wonder if there are any other similar books but more formal and less basic ?

The book has a chapter called "Further Reading", or at least there is certainly a section called that. Don't you think that might be a good place to look for, um, advice on further reading?

 

[Martin Rattigan]

IBM's Assembler language deals with integer values (within defined bounds and allows division with the result defined as the integral part of the division. The value n/0 is not defined if n is nonzero but 0/0 is rectified by defining it as 0, which turns out to be very useful for testing for specific values. The resulting system isn't a field (with or without the rectification) but then the original question didn't ask for that, it was just assumed by matt.

 

[JasonRox]

This is hilarious.

and induction is where for an infinite set of propositions, you prove the first case which then implies the truth of the rest.

I wish it were that easy.

 

[matt grime]

IBM's Assembler language deals with integer values (within defined bounds and allows division with the result defined as the integral part of the division. The value n/0 is not defined if n is nonzero but 0/0 is rectified by defining it as 0, which turns out to be very useful for testing for specific values. The resulting system isn't a field (with or without the rectification) but then the original question didn't ask for that, it was just assumed by matt.

If we aren't required to make the definition of 0/0 fit in with anything then the question is vacuous. My exact feelings about how this question should be answered are unprintable.

 

[D H]

IBM's Assembler language deals with integer values (within defined bounds and allows division with the result defined as the integral part of the division. The value n/0 is not defined if n is nonzero but 0/0 is rectified by defining it as 0.

Argghh! That is 1/0 times as bad as the IEEE floating point standard, with its 0/0 = NaN (not-a-number), 1/0 = Inf (infinity). The standard's default behavior is to let these attrocities pollute all of my calculations. I have yet to see a situation where the existence NaNs and Infs provide any utility. Switching the default behavior to something reasonable (i.e., drop core RIGHT NOW) still requires assembly language on many computers/many compilers. One positive regarding the standard: at least I know I screwed up somewhere when I see NaNs and Infs all over the place. If 0/0 results in 0 (with no warning), how could I ever trust the results?

 

[Martin Rattigan]

The results from this type of arithmetic are only ever used as parameters in compiling the program, not in its functioning, so any problems would usually be apparent fairly quickly. I have often found it useful to give a Kronecker delta facility when I can't explicitly test values. So if u and v are locations in the program, say, then

1-(u-v)/(u-v)

gives 0 if they are different locations and 1 if they're the same, and can be used in places where there is no mechanism for an explicit test.

Of course this is totally off the subject.

 

[Hurkyl]

I have yet to see a situation where the existence NaNs and Infs provide any utility. Switching the default behavior to something reasonable (i.e., drop core RIGHT NOW) still requires assembly language on many computers/many compilers.

You could always test the results of your calculations, and print useful error messages in the case of an Inf or Nan.

 

[Jippo]

This is hilarious.
I wish it were that easy.

 

[Neohaven]

How can the illicit operation 0/0 be remedied ?
so that it becomes meaningful ?

0/0 is irrelevent. it is an oxymoron to say that anything is EQUAL to the whole set of the Real Numbers... as an example :

Let's suppose that 0/0=1, by saying that x/x=1 with x=any real number, which is how most people prove that 0/0=1.

Then, we have 0/0=1, and by multiplicating by x, we now have 0/0=x. let`s now mention that x is ANY real number... thus, 0/0=R, which is an oxymoron.

 

[Hurkyl]

we now have 0/0=x. let`s now mention that x is ANY real number... thus, 0/0=R, which is an oxymoron.

No, it means you've proven that all real numbers are equal.

I'm going to close this, not because of any specific post, but just to combat necromancy of 0/0 threads.