# How can the illicit operation 0/0 be remedied?

1. Nov 23, 2005

### roger

How can the illicit operation 0/0 be remedied ?

so that it becomes meaningful ?

2. Nov 23, 2005

### 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?

3. Nov 23, 2005

### TD

*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.

4. Nov 23, 2005

### roger

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

5. Nov 23, 2005

### roger

I wasn't refferring to any functions though.

6. Nov 23, 2005

### 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 [Broken]

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

Last edited by a moderator: May 2, 2017
7. Nov 23, 2005

### Staff: Mentor

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

8. Nov 23, 2005

### roger

''However, this doesn't stop us defining the symbol 1/0 and adding it in to 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 cant 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

9. Nov 23, 2005

### matt grime

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

then this makes it a poorly stated question

look up the axioms of a field

axiom

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

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.

10. Nov 23, 2005

### Gokul43201

Staff Emeritus

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.

Last edited: Nov 23, 2005
11. Nov 23, 2005

### 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.''

12. Nov 24, 2005

### matt grime

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

13. Nov 24, 2005

### roger

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

Roger

14. Nov 24, 2005

### 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)

Last edited: Nov 24, 2005
15. Nov 24, 2005

### 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..

16. Nov 24, 2005

### 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?

17. Nov 24, 2005

### roger

yes I Looked it up but I couldnt 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.

18. Nov 24, 2005

19. Nov 25, 2005

### matt grime

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

20. Nov 26, 2005

### 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 ?