# Thread: DIVISION BY 0

1. The one number we can't divide by is 0. Yet, the basis of division (in french anyway) is how many times the "dénominator" (in this case 0) can go in the "nominator" (for example : 1). We obtain: 1/0= ?????

Two answer occur to me: - either 0 can go infinite times in to 1
- either 0 can never go into one because it always gives 0 (more logical than the first ^^)

What do you think? Do you have another answer?

2.

3. Originally Posted by celtic_warrior
The one number we can't divide by is 0. Yet, the basis of division (in french anyway) is how many times the "dénominator" (in this case 0) can go in the "nominator" (for example : 1). We obtain: 1/0= ?????

Two answer occur to me: - either 0 can go infinite times in to 1
- either 0 can never go into one because it always gives 0 (more logical than the first ^^)

What do you think? Do you have another answer?
The first answer is correct. It is infinity. Dividing any number other than zero by zero results in infinity. Think about it like this

when m equals zero y also equals zero for all values of x. graph that expression and you find you have a ifinitily long line along the x axis

So lets look at this modification

when m equals 0, x will equal infinity, because when we graph this we get a value that equals 0 for all values of y.

the slope of this graph isn't zero this time it is infinitely large.

I know this isn't the best explanation, but It makes sense to me. Draw the graphs and think about it how I explained it.

4. the real interesting questions occour when you do this.

\dfrac{\infty}{0}

5. It's not infinity. It's 'undefined,' which is not the same.

6. Originally Posted by Chemboy
It's not infinity. It's 'undefined,' which is not the same.
No it's infinity.

7. Chemboy is right - it's undefined.

8. Infinity isn't even a number, so if you want division to be defined on numbers, then the answer isn't infinity.

9. Originally Posted by celtic_warrior
The one number we can't divide by is 0. Yet, the basis of division (in french anyway) is how many times the "dénominator" (in this case 0) can go in the "nominator" (for example : 1). We obtain: 1/0= ?????
Actually, if you were being rigorous, there is no 'division' in algebra, what you are doing is multiplication. The Reals are a field, which is defined via two binary operations, addition and multiplication. The reason why it seems there's a concept of division is that part of the definition of a field is that given any non-zero x there's a y such that x*y = 1. Typically we write y = 1/x. So when you 'divide by x' you're actually multiplying by 1/x . Now I specifically said that x is non-zero because for the field to be a coherent, consistent concept you cannot have a number y such that 0*y = 1. Therefore when you say "What is 1 divided by 0?" you are infact asking "What is the number I multiply zero by to get 1?" and the answer is that the Reals do not contain such a number any more than they contain a number which can be multiplied with itself to give -1.

10. Originally Posted by GenerationE
Originally Posted by celtic_warrior
The one number we can't divide by is 0. Yet, the basis of division (in french anyway) is how many times the "dénominator" (in this case 0) can go in the "nominator" (for example : 1). We obtain: 1/0= ?????

Two answer occur to me: - either 0 can go infinite times in to 1
- either 0 can never go into one because it always gives 0 (more logical than the first ^^)

What do you think? Do you have another answer?
The first answer is correct. It is infinity. Dividing any number other than zero by zero results in infinity. Think about it like this

when m equals zero y also equals zero for all values of x. graph that expression and you find you have a ifinitily long line along the x axis

So lets look at this modification

when m equals 0, x will equal infinity, because when we graph this we get a value that equals 0 for all values of y.

the slope of this graph isn't zero this time it is infinitely large.

I know this isn't the best explanation, but It makes sense to me. Draw the graphs and think about it how I explained it.
Neither answer is correct. Division by 0 is not defined. If it were there have to be a number which when multiplied by 0 would produce 1. Thereis no such real number. There is no such complex number. You can't do this in any field. You can't do it in any ring. You can't even do it with cardinal numbers.

Infinity is not a number. You can make sense of infinte numbers by working with cardinal and ordinal numbers. But you still can't divide by 0.

11. Just because infinity isn't a number doesn't mean that, infinity is a type of answer you would get. Is infinity defined at a specific number? No. Does that mean it can't be a possible way of expressing 1/0? So it follows that infinity should be considered a reasonable way of showing division by zero. The only expression I have ever heard of to be considered truly having no meaning is 0/0.

I just don't know how it can't be infinity. My professors with PhD's in Mathematics have told me it is infinity, and look even this page supports my explanation. Look at the graph in the middle of the page.

http://en.wikipedia.org/wiki/1/0

12. Well then, which infinity is it? Is it or ?

13. Here's an attempt to solve the problem from a few years ago .

And here's the mathematical framework underlying it.

http://www.bookofparagon.com/Mathema...achineVIII.pdf

IIRC the mathematical community responded lukewarm with something like: "the theory appears correct, but it doesn't bring us even the tiniest step further."

14. Originally Posted by GenerationE
Just because infinity isn't a number doesn't mean that, infinity is a type of answer you would get. Is infinity defined at a specific number? No. Does that mean it can't be a possible way of expressing 1/0? So it follows that infinity should be considered a reasonable way of showing division by zero. The only expression I have ever heard of to be considered truly having no meaning is 0/0.

I just don't know how it can't be infinity. My professors with PhD's in Mathematics have told me it is infinity, and look even this page supports my explanation. Look at the graph in the middle of the page.

http://en.wikipedia.org/wiki/1/0
1/0 is not infinity. 1/0 is not defined. If your professors with Ph.Ds in Mathematics have told you differently then they are wrong. I would be happy to debate it with them. I have one too. A real one.

If 1/0 were really iinfinity then 1/0 times infinity would be 1. But it is not.

15. Originally Posted by GenerationE
Just because infinity isn't a number doesn't mean that, infinity is a type of answer you would get. Is infinity defined at a specific number? No. Does that mean it can't be a possible way of expressing 1/0? So it follows that infinity should be considered a reasonable way of showing division by zero. The only expression I have ever heard of to be considered truly having no meaning is 0/0.

I just don't know how it can't be infinity. My professors with PhD's in Mathematics have told me it is infinity, and look even this page supports my explanation. Look at the graph in the middle of the page.

http://en.wikipedia.org/wiki/1/0
Look, you have to realise the difference between a value being a certain number, or a value approaching a certain number.
We define 1/0 as not defined. This is not debatable.
Mathematics is a defined system of describing things, and its axioms are not debatable. Sure, you could create other axioms but then you must construct another system, which probably wouldn't be accepted by most mathematicians.

16. Well then, which infinity is it? Is it or ?
I'd guess either side because 0 is neither positive or negative

17. Originally Posted by thyristor
Originally Posted by GenerationE
Just because infinity isn't a number doesn't mean that, infinity is a type of answer you would get. Is infinity defined at a specific number? No. Does that mean it can't be a possible way of expressing 1/0? So it follows that infinity should be considered a reasonable way of showing division by zero. The only expression I have ever heard of to be considered truly having no meaning is 0/0.

I just don't know how it can't be infinity. My professors with PhD's in Mathematics have told me it is infinity, and look even this page supports my explanation. Look at the graph in the middle of the page.

http://en.wikipedia.org/wiki/1/0
Look, you have to realise the difference between a value being a certain number, or a value approaching a certain number.
We define 1/0 as not defined. This is not debatable.
Mathematics is a defined system of describing things, and its axioms are not debatable. Sure, you could create other axioms but then you must construct another system, which probably wouldn't be accepted by most mathematicians.
While what you say is basically correct, but there is a bit of leeway in some cases.

In particular, the axiom of choice is a bit controversial. It is almost, but not quite, universally accepted. Paul Cohen has shown that it is independent of the other axioms of set theory. The majority of modern mathematics has been developed on the basis of an acceptance of the axiom of choice, which is a necessary ingredient in arguments that use transfinite induction.

But there is a school of mathematicians, constructivists led by Erret Bishop, who have managed to develop a great deal of mathematics without using the axiom of choice. This is serious mathematics and not some sort of wacko cult at all.

One reason for some reticence in accepting the axiom of choice, which seems intuitively obvious, is that one can prove theorems using it that are decidedly non-obvious, and a bit startling. The Banach-Tarski theorem is one such result (it is sometimes called a paradox, but it most certainly is not a paradox at all). http://en.wikipedia.org/wiki/Banach-Tarski_paradox

I think it is safe to say that the other fundamental axioms are universally accepted by working mathematicians.

18. Originally Posted by celtic_warrior
Well then, which infinity is it? Is it or ?
I'd guess either side because 0 is neither positive or negative
Well, it can't be both. If 1/0 is anything, it has to be one thing.

19. Originally Posted by MagiMaster
Well, it can't be both. If 1/0 is anything, it has to be one thing.
It's common to work in the set , the extended Complex numbers/plane, where all the infinities (since you can consider ) are equivalent since the set of equivalent to the Riemann sphere and the map between the sphere and the extended Complex plane is stereographic projection. If is that projection then is independent of , they all map to the 'North Pole' of the sphere.

If you're doing complex analysis such as contour integration then the phase of the infinity can play an important role though it's been too many years for me to remember a specific example of such a contour integral.

20. Well, alright. On the Riemann sphere, 1/0 is uniquely defined, but neither that nor the one point closure of the real line is the same as the real numbers that most people deal with. In the real numbers, neither nor are numbers.

21. Originally Posted by MagiMaster
Well, alright. On the Riemann sphere, 1/0 is uniquely defined, but neither that nor the one point closure of the real line is the same as the real numbers that most people deal with. In the real numbers, neither nor are numbers.
No, it isn't. x on the Riemann sphere or anywhere else. There seems to be some confusion between the notion of a pint at infinity in a topological sense and 1/0 within the algebraic construct of a ring or a field. They are decidedly not the same thing and the mere notation 1/0 clearly places one in an algebraic context.

You can deal with the extended reals, which is often done in abstract integration theory but while makes sense and is useful in that context you still cannot make sense of 1/0.

22. Ah. Alright. I haven't actually dealt with the Riemann sphere much myself.

23. Originally Posted by DrRocket
Originally Posted by MagiMaster
Well, alright. On the Riemann sphere, 1/0 is uniquely defined, but neither that nor the one point closure of the real line is the same as the real numbers that most people deal with. In the real numbers, neither nor are numbers.
No, it isn't. x on the Riemann sphere or anywhere else. There seems to be some confusion between the notion of a pint at infinity in a topological sense and 1/0 within the algebraic construct of a ring or a field. They are decidedly not the same thing and the mere notation 1/0 clearly places one in an algebraic context.

You can deal with the extended reals, which is often done in abstract integration theory but while makes sense and is useful in that context you still cannot make sense of 1/0.

Ok, I get that it is a very unusual thing. I'm just talking about the limit I guess. The limit of 1/n as n approaches 0 the limit approaches infinity correct?

I mean I know that some areas of math are fuzzy and can't be clearly defined. I'm just saying that as that limit goes to 0 the value trends toward infinity.

I was under the impression that 0/0 was the real culprit of confusion among the mathematicians.

I think that for 1/0 we should at least be able to say that infinity is in some maths acceptable.

24. Well, but , so you can't use that to say that 1/0 is .

Really though, mathematicians aren't confused about either 1/0 or 0/0. Just because we say that something is undefined doesn't mean we don't know what it is or what it should be. It is, and it should be, undefined.

25. Mathematics is always about definitions. When a mathematician talks about the real numbers he is in fact talking about a dedekind complete, totally ordered field. Nothing more, nothing less. So in the world of dedekind complete totally ordered fields, you cannot divide by zero. Trying to do anything else or adding some fancy point takes you outside the world of the real numbers.

Its like declaring that almost everybody owns a dragon, but your definition of a dragon is a small furry canine. It may be true, but it is not illuminating.

26. Originally Posted by GenerationE
...
I mean I know that some areas of math are fuzzy and can't be clearly defined. I'm just saying that as that limit goes to 0 the value trends toward infinity.

I was under the impression that 0/0 was the real culprit of confusion among the mathematicians.

I think that for 1/0 we should at least be able to say that infinity is in some maths acceptable.
No, areas of mathematics are not fuzzy. Mathematics is rather unique in the level of rigor, attention to detail, and clear unabiguous defninition employed. It something is fuzzy, it is NOT mathematics. It might be physics. It could easily be politics. It is most certainly not mathematics.

This is an important point. And it is one of the reasons that 1/0 is not defiined. One can quite literally define what is meant by division. It is done completely rigorously in the development of the real numbers. See Foundations of Analysis by Landau for a completely rigorous (and pretty boring) development of the real numbers using only the Peano Axioms (which basically define the natural numbers) and logical deductions from that basis, using the idea of Dedekind cuts.

Mathematics is exquisitely precise. And 1/0 is not. Neither is 0/0. We are not being stubborn on this point. Until you understand it, you cannot hope to learn more advanced rigorous mathematics.

27. then can zero be divided by zero?

28. Originally Posted by dancedancedance
then can zero be divided by zero?
Did you not read DrRocket's post, just above yours? In particular, pay attention to the second and third sentences of the penultimate line.

29. Originally Posted by dancedancedance
then can zero be divided by zero?
Asking that isn't bringing up anything new. It still involves division by zero, which in a practical context, is always undefined.

30. Division by 0 is infinity.

The method to prove that 1/0 is infinity is by using the concept of "limit" and graph. This is shown in Calculus textbook.

It is only undefined in computers...

31. Originally Posted by msafwan
Division by 0 is infinity. The method to prove that 1/0 is infinity is by using the concept of "limit" and graph. This is shown in Calculus textbook.It is only undefined in computers...
A reliable calculus text will never say that division by zero equals infinity. However, it will say that the limit of a certain rational function tends towards infinity from one side. It is not the same as saying division by zero equals or is defined to be infinity, which is not a number in the first place.

32. Originally Posted by msafwan
Division by 0 is infinity.

The method to prove that 1/0 is infinity is by using the concept of "limit" and graph. This is shown in Calculus textbook.
No it isn't (or you are using a bad textbook). Did you not read MagiMaster's post (#23)? If not, do so, and read epidecus' post above. They both explain well why you can't simply declare "Division by 0 is infinity." Because it ain't.

33. Because it ain't.
Because positive infinity not equal negative infinity?

34. I actually used to be a proponent for division by zero. But some people need to realize, like I did, that there is no real controversy in mathematical circles over these topics. It is the standard, definite approach. In the standard mathematics of any practical context, division by zero is undefined. Elementary algebra just can't handle zero in certain respects, and that's completely okay.

35. Originally Posted by tk421
you can't simply declare "Division by 0 is infinity." Because it ain't.
Quite. In fact the ONLY thing that "ain't" is infinity!

Look, here's something.....

Suppose we work in an area of mathematics where (the double arrow reads "implies that").

Suppose further that for ANY then . So , a comforting result.

So let , then , also very reassuring.

But running the argument backwards, we also have that , which is not at all reassuring - in fact it's a contradiction.

Obviously a similar contradiction arises for ANY number, real or complex, in the numerator, but does not arise for any NON-ZERO number in the denominator.

All that remains to be seen is that is not susceptible to normal arithmetic multiplication, and we're about done with this stupid thread

36. Well yea that means you put 0 into 1 infinitely many times. The denomenator should have atleast the slightest value more than zero. Your arguement is correct.

37. Well you it's not infinite but it goes into 1 infinitely many times.

38. Originally Posted by PhysicsApple
Well yea that means you put 0 into 1 infinitely many times.
In other words, you're saying ?

39. In real algebra, the null (0) can be reached from left and from right ... .

In complex algebra, the null (0) can be reached from all directions ... .

40. Originally Posted by epidecus
A silly little trick

41. is not 0 ... because we don't know when the series ends ... .

42. Originally Posted by river_rat

A silly little trick

1 - 1 + 1 - 1 + 1 - 1 + 1 ...
0 + 1 - 1 + 1 - 1 + 1 ...
1 - 1 + 1 - 1 + 1 ...
0 + 1 - 1 + 1 ...
ETC.

The problem is that each successive sum in the series indefinitely alternates between zero and one. It approaches no definite limit, so I believe the correct descriptive term is indeterminate? And this is not the case in 0+0+0+0... where each successive sum is always zero.

That is, unless you add some delimiters and change it to (1-1)+(1-1)+(1-1)+(1-1)+(1-1)+ ... I'm not completely sure though; there's always some anti-intuitive sublimity to these topics.

But one infinite series I do find confusing is , which apparently sums to !

43. [QUOTE=epidecus;374802]
Originally Posted by river_rat

But one infinite series I do find confusing is , which apparently sums to !

is negative infinity ... ,

so this series is divergent ... .

I'm sorry if I do mistake ... .

44. Not sure, trfrm.

I haven't studied infinite series yet, but I do find them interesting.

The 1/4 is a well-known derivation for this series, as seen by various summing methods.

The mistake might be here. The rearrangement might not be valid if we're assuming elementary order of operations. The non-commutativity / non-associativity of subtraction might also play a part. Still, I'm not very sure.

45. Originally Posted by epidecus
The mistake might be here. The rearrangement might not be valid if we're assuming elementary order of operations. The non-commutativity / non-associativity of subtraction might also play a part. Still, I'm not very sure.
Permission ... .

But, means for and any real numbers ... .

46. Originally Posted by trfrm

Permission ... .

But, means for and any real number ... .
Ah, you're right. Thought you were doing something else. My knowledge is standing on thin ice, but here's my own take...

INDETERMINATE

I can also factor out the negation in the third expression which results in the same thing. Neither of these come off as a definite divergence, but rather indeterminate limits. Again, I'm just playing around with numbers here, so don't take these seriously.

47. celtic_warrior
Posts 10 November 7th, 2008, 03:38 PM The one number we can't divide by is 0. Yet, the basis of division (in french anyway) is how many times the "dénominator" (in this case 0) can go in the "nominator" (for example : 1). We obtain: 1/0= ?????
Two answer occur to me: - either 0 can go infinite times in to 1
- either 0 can never go into one because it always gives 0 (more logical than the first ^^)
What do you think? Do you have another answer?
The symbol '0' isn't an integer, and should not be defined as such. It is the container for the elements of a system, aka the empty set, meaning there are no elements for consideration for any purpose. Using this concept, 1/0 is meaningless. Addition, where 0 is redundant, would only apply to N, the set of positive integers (excluding 0).
Since 'infinity' is not an integer, but a relation/condition where a value cannot be assigned, the expression 1=n0 is not true for any value of n.
The expression is a contradiction in terms, expecting something (1) from nothing (0)!

48. Originally Posted by phyti
The symbol '0' isn't an integer
What do you mean by this?

49. Originally Posted by celtic_warrior
The one number we can't divide by is 0. Yet, the basis of division (in french anyway) is how many times the "dénominator" (in this case 0) can go in the "nominator" (for example : 1). We obtain: 1/0= ?????

Two answer occur to me: - either 0 can go infinite times in to 1
- either 0 can never go into one because it always gives 0 (more logical than the first ^^)

What do you think? Do you have another answer?
I prefer your first answer. (How do you judge its consequenses?)
It seems it is the claim that you can infinitely many times not divide the number one.
Its done in a jiffy. We also pass an infinite number of points in any step.
I dont see the logic in your second step...can you clarify a little?

50. sigurd - fortunately for mathematics, your "preference" is totally irrelevant. The arguments against this garbage have been given in this thread and countless others like it.

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