# The value of 0/0

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OK I admit I thought I was losing my sanity when I thought of this, but then I realized people thought Galileo was crazy when he said that the Earth went around the Sun. I am a (fairly)good mathematician and I know better than to question integer division by zero is undefined, but what about 0/0. Basically, for the past one week I was thinking about 0/0. Not integer division by zero but only about zero divided by zero. To prevent "You must be out of your mind" replies, I would like to state the.... uh.... rules first.

Here's my theory. f(x) = x/x is a constant function giving the value 1. Therefore, 0/0 is also 1. IF you're going "OMG, Gasp.... what a sacrilege!!" wait... I have googled this item and have seen the replies. None of them are... uh... convincing...(I don't know the exact word for that feeling). So here's the deal

1. This is only a discussion/debate. Not a correctly-verified-and-proven theory.

2. First, I get to twist the facts to suit my theory. You mathematicians are going to state a theory without twisting the facts to REALLY disprove my theory.

3. If you can't do that you're going to get the chance to twist the facts to disprove my theory. (I have no doubt that's going to be as easy as "pi")

OK, before you start chewing my tail I'll admit a few things first. Granted, 0/0 IS undefined when taken in elementary maths. (Come on how can you divide "nothing" sweets among "nobody" persons, to quote the classic textbook example). But as a function. Like f(x) = x/x. Simplifying we get f(x) = 1; Which is a constant function; Meaning it IS 1 for all x belongs to R. Its like the other classic example for limits; f(x) = x2 - 1/ x-1 is not defined when x = 1. But we KNOW it IS 2 when x = 1; Simplifying f(x) = x2 - 1/ x - 1 gives f(x) = x + 1. I'm stating my case on this principle.

Ok, let the waves of opposition begin!

EDIT: Oops, I ought to have read the forum sticky before I posted this. Moderators can move this to the "Speculations" section if they want to.

Edited by Raptor115

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In computer floating point operations, dividing by zero is defined to result in either positive or negative infinity.

If you look at the algebraic equivalent of $\frac{0}{0} = x$ it's $x\times0 = 0$. In this case, x can literally be any number since anything times 0 is 0. A graph of the result set would be meaningless, since it would, quite literally include every number in existence.

More appropriately, we could write the formula for such a graph as $f(x) = 0 \times x$ which would result in a straight line from negative to positive infinity right across the y-axis of the graph. This is the reason the operation is undefined - it doesn't result in any useful answers.

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This is something I posted in an earlier thread:=

"f you calculate (8-x^3)/(2-x) when x=2 you get 0/0........................... I'll just say I make this example of 0/0 equal to 12........................"

Are you suggesting this particular case has two answers?

Edited by Joatmon
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@Joatmon

LOL you got the point wrong. Actually by the f(x) = x2-1/ x-1 example, I wasn't trying to equate 0/0 with something. I thought I said it was meaningless to equate 0/0 with anything. OK if i didn't I'll say it at the end of this reply. Actually I meant that, like we could simplify f(x) to give x+1, we could also simplify f(x) = x/x to f(x) = 1, which makes even f(0) to be 1.

@Greg

you said X x 0 =0. Basically, we know that if ax + c = bx + c, then a = b. then putting x= 0, a =1 , b =2, c= 3 we get 1x0 + 3= 2x0 + 3 => 3 =3 implying 1= 2. @Joatmon, like this we could equate 0 to all the numbers known. (Agreed, I AM twisting the facts now) All I'm saying is that, saying 0/0 = 1 doesn't change things that radically, cause you see we can prove 1 = 2 without proving 0/0 = 1. Besides I wasn't doing a graph of "Find x such that x x 0 = 0." I said it was f(x) = x/x.

OK, now for the divide-multiply headache.

0/0 = 1; 1 x 0 = 0; Finished. No violation of the law.

If you think I missed any of your points please point it out.

Also I do know 0/0 is meaningless in elementary arithmetic. As I said I probably was out of my mind last week and I couldnt rest till I started a debate.

BTW I didn't think there would be 2 responses in 15 mins (Responsive website this. Some websites you don't get replies for weeks.....)

Also, keep the arguments coming. I'm not going to stop till I find one that'll overload my brain. (Meaning it can convince my crazy half that this debate's a waste)

Edited by Raptor115
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@Greg

you said X x 0 =0. Basically, we know that if ax + c = bx + c, then a = b. then putting x= 0, a =1 , b =2, c= 3 we get 1x0 + 3= 2x0 + 3 => 3 =3 implying 1= 2. @Joatmon, like this we could equate 0 to all the numbers known. (Agreed, I AM twisting the facts now) All I'm saying is that, saying 0/0 = 1 doesn't change things that radically, cause you see we can prove 1 = 2 without proving 0/0 = 1. Besides I wasn't doing a graph of "Find x such that x x 0 = 0." I said it was f(x) = x/x.

The fact that it implies that 1 = 2 seems to only reinforce my point that it results in no useful answers. It's like saying that 1 + 2 = blue. Sure it's an answer. It may even be correct if blue = 3, but the answer, as given, doesn't really help us in any way.

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@Greg

I just re-read your first post and I realized I missed a big chunk of what you were trying to say. Also, I thought that you hadn't (I don't want to sound rude) read my post fully at first. I just told at the starting that 0/0 was and is meaningless (I don't know about "will be"). I was saying that f(x) = x/x could be simplified to f(x) = 1 and therefore f(0) is 1.

You just took it as f(0) = 0/0 and said it would make a point of (0, y-axis(more specifically -infinity to +infinity)). OK that's like saying f(x) = x2 - 1/ x- 1 gives (1, y-axis) when x = 1, when it gives (1,2) when simplified. I'm applying the same simplifying concept to f(x) = x/x to say f(0) = 1.

Umm... about your saying "it just doesn't help us in any way"..... I just realised its true. But it still doesn't convince my crazy half that this debate's a waste

Edited by Raptor115
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Additionally, let's consider fractions as proportions (since, that's really what they are).

If I have ten widgets and I give two away, it's easy to see that

a) I have 8 left, and

b) I have given away 20% of my initial widgets.

But if I start with 0 widgets, and I give 0 away, I still have what I started with, which is none. Can I even break up 0 into fractional parts? That's what you're really trying to do when you divide - break something up into smaller chunks.

Can I really break a pie up into 0 pieces? (Well I suppose if I eat the whole thing...) but it doesn't leave me with pieces of pie, it leaves me with no pie at all. Logically it makes no sense.

And how can I break up no pie into smaller pieces of no pie? (i.e. 0/0) If I have nothing to begin with, how do it make it smaller nothings? An answer of 1 implies that there is a whole "something" there. So what is the whole "something" that results from nothing divided by nothing?

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Can I really break a pie up into 0 pieces? (Well I suppose if I eat the whole thing...) but it doesn't leave me with pieces of pie, it leaves me with no pie at all.

LOL that's a nice one. OK consider --- The pie's in your stomach. How many pies have you in your hand --- 0. The stomach divides the pie into infinite (well, to be precise its as many atoms as were in the pie originally added with the atoms in the saliva and digestive juices that went with it) pieces using HCl. How many pies does it leave in your hand --- 0. Division happens...... OK that was a joke.

Basically that's exactly what I told in my first post. I mean, please read my post carefully once and you'll find that I SAID that 0/0 DOESN'T make sense in elementary arithmetic. I only said that in relation to the function f(x) = x/x. Its like a theoretical question. Like we know i2 = -1. We all know there ain't a real number like that. Yet we still don't disown it. Maybe my question comes along these lines.

i'm sorry for writing this para and I'll edit n' remove this tomorrow. But for Greg, sorry but i gtg sleep. Its 11.45 pm and I have to go school tomorrow and work with my friend on the trinomial theorem (a+b+c)n . We don't know if it's already been done, but we certainly want to do it ourselves without external help. We got our models right today and we need to come up with a gen. formula by this week. Keep the arguments coming --- and please include a bit of humour. That way, the debate won't be dry. Good luck at some other arguments Greg.

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0/0=

= π

∴ 0/0 = π

π = The Universe

0/0 = The Universe

lol.

Edit: Pi is exactly 3.

Re-Edit: But no, seriously. If you take 0 to be 'something', it is said to be able to fit into 'something else' indefinitely or infinitely; e.g. 8/0 = .

That's according to a section of an OCR mathematics book I read anyway.

It depends if you read it as dividing 8 into 0 sections... which would lead you to think it's disappeared. Or if you look at it as how many times 0 fits inside 8 (as many times as it pleases).

Re-Re-Edit: (Cannot sleep tonight.) If 1x0= 0 and 2x0=0, then 1x0=2x0. Therefore, if (1x0)/0= (2x0)/0, the zeros cancel to conclude 1 can equal 2. Maths is broken.

Edited by Iota
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Here's my theory.

The word theory means something special in the sciences, a well-substantiated, coherent explanation of some aspect of reality. In mathematics, the word means a consistent body of knowledge, such as knot theory, Galois theory, set theory, and so on. This is a scientific forum. The one thing "theory" does not mean is "a wild guess I came up with after finishing off a quart of tequila."

f(x) = x/x is a constant function giving the value 1. Therefore, 0/0 is also 1. IF you're going "OMG, Gasp.... what a sacrilege!!"

Yep. Sacrilege. And nonsense. First off, what made you pluck that function out of the clear blue sky? What about f(x)=2x/x? or f(x)=-3x/x?

Secondly, that's not how division is defined. a/b=c means that a-b*c=0. Now let's look at a=b=0. We're trying to find a number c such that 0-0*c=0. Any and every number satisfies this expression. 0/0 is indeterminate. It has to be. Arbitrarily assign it a value, any value, and you have just opened the mathematical door of death. It allows you to prove that 1=2. The first thing that a mathematical theory (remember the definition) must be is consistent, or contradiction-free. An axiomatic system that is known to allow consistencies isn't a theory. It's trash.

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0/0=

Not quite right on that point. I don't see 0/0 being anywhere. Its basically indeterminate

= π

∴ 0/0 = π

π = The Universe

0/0 = The Universe

lol.

LOL that's a good definition actually. Just kidding..... Don't take this seriously.

Edit: Pi is exactly 3.

Is this a joke?? OK lemme see Pi is exactly what you get when you cut the magic number 'e' off a "pie".

Re-Edit: But no, seriously. If you take 0 to be 'something', it is said to be able to fit into 'something else' indefinitely or infinitely; e.g. 8/0 = .

That's according to a section of an OCR mathematics book I read anyway.

It depends if you read it as dividing 8 into 0 sections... which would lead you to think it's disappeared. Or if you look at it as how many times 0 fits inside 8 (as many times as it pleases).

Bah, doesn't anyone ever read my first post fully? I've repeated it like 5 times ---- 0/0 doesn't make sense in elementary arithmetic. So don't use real life examples. Maybe if you can demonstrate the value of i in i2 = -1 with real-life examples, then I'll accept that 0/0 is NEVER 1 in any circumstance

Re-Re-Edit: (Cannot sleep tonight.) If 1x0= 0 and 2x0=0, then 1x0=2x0. Therefore, if (1x0)/0= (2x0)/0, the zeros cancel to conclude 1 can equal 2. Maths is broken.

I demonstrated that Maths is broken without even saying 0/0 = 1 in a previous reply. So I disagree. On the other hand you've taken (1x0)/0 = (2x0)/0. Well let's put it in a different way. We are going to take 0/0 = 1. Then, 1x0 = 0; 2x0 = 0. The whole thing becomes 0/0 = 0/0; 1=1. Maths is not broken by saying 0/0 =1. Its already broken if you consider my ax + c = bx +c example.

The word theory means something special in the sciences, a well-substantiated, coherent explanation of some aspect of reality. In mathematics, the word means a consistent body of knowledge, such as knot theory, Galois theory, set theory, and so on. This is a scientific forum. The one thing "theory" does not mean is "a wild guess I came up with after finishing off a quart of tequila."

OK. I'll change that to "hypothesis". Besides I didn't drink tequila, I drank too much coffee last week.... couldn't sleep much...... Just kidding.

Yep. Sacrilege. And nonsense. First off, what made you pluck that function out of the clear blue sky? What about f(x)=2x/x? or f(x)=-3x/x?

Secondly, that's not how division is defined. a/b=c means that a-b*c=0. Now let's look at a=b=0. We're trying to find a number c such that 0-0*c=0. Any and every number satisfies this expression. 0/0 is indeterminate. It has to be. Arbitrarily assign it a value, any value, and you have just opened the mathematical door of death. It allows you to prove that 1=2. The first thing that a mathematical theory (remember the definition) must be is consistent, or contradiction-free. An axiomatic system that is known to allow consistencies isn't a theory. It's trash.

First off, I didn't pick that function out of the clear blue sky, I picked it from my sky-blue Mathematics Textbook Volume - 2 (The first volume is orange-orange). Well there ain't a difficulty about f(x) = 2x/x which is f(x) = 2 and f(x) = -3x/x which is f(x) = -3. If 0/0 = 1, it would satisfy all the conditions whereas if 0/0 = something else it would not.

How? you ask? OK lets see; f(x) = 2x/x. Then, f(0) = 2 x 0/0 = 2 x1 = 2, which gives us the right answer. Same method to prove f(0) in -3x/x would give -3. If you'd started this hypothesis with 0/0 = 2 ( or 3 or 4 or -10) then you wouldn't be getting 2 x 0/0 = 2 x 1 = 2. you would get 2 x0/0 = 2x2 = 4 ( or 6 or 8 or -20).

Next, if 0/0 = 1, then 0 - (0x1) must be 0, right? Oh well, lets see; 0x1 = 0; Therefore 0 - 0; That's 0, right? So what's wrong? Well, you're wrong about assigning 0/0 opening the mathematical door of death. I already pointed out we could prove 1=2 without needing 0/0 to be 1. If you really believe assigning 0/0 opens the mathematical door of death, please explain more clearly.

OK, till now, except for D H, everybody posted why 0/0 isn't 1. I'm not asking why it isn't. I'm only asking why it shouldn't be 1.

If you can't understand the difference between those statements, the first one's like saying "why aren't we cloning humans?" The second one's like "what's gonna be the problem if we clone humans?"

Also, if you find my English difficult to understand, I'm sorry, I'm an Indian.

Edited by Raptor115
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Is this a joke?? OK lemme see Pi is exactly what you get when you cut the magic number 'e' off a "pie".

It is quite clearly a joke, yes.

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Bah, doesn't anyone ever read my first post fully? I've repeated it like 5 times ---- 0/0 doesn't make sense in elementary arithmetic.

It doesn't make sense, period.

I demonstrated that Maths is broken without even saying 0/0 = 1 in a previous reply.

You demonstrated nothing. Instead you made a bare assertion, a fallacy. Demonstrate this please, and do so without using an illegal operation such as dividing by zero or assuming that the only solution to x2=1 is one.

OK, till now, except for D H, everybody posted why 0/0 isn't 1. I'm not asking why it isn't. I'm only asking why it shouldn't be 1.

It's quite simple. Saying that 0/0 is any particular value opens the door to contradictions. Mathematics must be contradiction free.

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OK, till now, except for D H, everybody posted why 0/0 isn't 1. I'm not asking why it isn't. I'm only asking why it shouldn't be 1.

Also, if you find my English difficult to understand, I'm sorry, I'm an Indian.

First, your English is quite good. Better than some people for whom it's the native tongue, so don't worry about that.

Second, the reason it shouldn't be 1 are the same reasons it's not 1 - it leads to contradictions and problems in the most basic of maths, as DH pointed out. Whether you want to call it indeterminate or undefined, division by zero (which is really what you're talking about) renders a lot (if not all) of mathematics essentially unstable and meaningless.

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It doesn't make sense, period.

OK, maybe I'm ignorant, or maybe its just my mind. 0/0 doesn't make sense naturally. I concede. Then tell me how i = square root of -1 makes sense naturally.

You demonstrated nothing. Instead you made a bare assertion, a fallacy. Demonstrate this please, and do so without using an illegal operation such as dividing by zero or assuming that the only solution to x2=1 is one.

Probably you didn't read all my replies. I did demonstrate that Maths is broken if you consider equating with "multiply by zero" operations. Consider two equations ax + c and bx +c. We know that if ax + c = bx + c, then a = b. Lets try putting a few values. x = 3; c = 1; so we get 3a + 1 and 3b + 1. If we get 3a + 1 = 3b + 1, then a = b right? OK lets put different values. x = 0(GASP!!); a = 1; b =2; c =3. We get (1x0) + 3 and (2x0) + 3 => 0 + 3 and 0 + 3 => 3 and 3. Since 3 =3 we say a=b and we get 1 = 2. See, I neither divided by zero nor did I assume the solution of x2 = 1 is one (What does that have to do with 0/0??)

It's quite simple. Saying that 0/0 is any particular value opens the door to contradictions. Mathematics must be contradiction free.

I think I proved Maths has contradictions without assigning a particular value to 0/0.

First, your English is quite good. Better than some people for whom it's the native tongue, so don't worry about that.

Thanks. I just happen to be good at written English. When it comes to spoken English................

Second, the reason it shouldn't be 1 are the same reasons it's not 1 - it leads to contradictions and problems in the most basic of maths, as DH pointed out. Whether you want to call it indeterminate or undefined, division by zero (which is really what you're talking about) renders a lot (if not all) of mathematics essentially unstable and meaningless.

Again, I think I just proved that Maths is already unstable with "multiplication with zero" as it is with "division by zero". Maybe the solution is to say multiplication and division operations with 0 are indeterminate??

Edited by Raptor115
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Probably you didn't read all my replies. I did demonstrate that Maths is broken if you consider equating with "multiply by zero" operations. Consider two equations ax + c and bx +c. We know that if ax + c = bx + c, then a = b. Lets try putting a few values. x = 3; c = 1; so we get 3a + 1 and 3b + 1. If we get 3a + 1 = 3b + 1, then a = b right? OK lets put different values. x = 0(GASP!!); a = 1; b =2; c =3. We get (1x0) + 3 and (2x0) + 3 => 0 + 3 and 0 + 3 => 3 and 3. Since 3 =3 we say a=b and we get 1 = 2. See, I neither divided by zero ...

You did divide by zero. You did so by assuming ax=bx implies that a=b. This is not valid with x=0 because you are dividing by zero.

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You did divide by zero. You did so by assuming ax=bx implies that a=b. This is not valid with x=0 because you are dividing by zero.

OK I'll admit you stopped me in my tracks for a minute there. Then I got a curious kind of doubt. You know when you have two equations like ax2 + bx + c and fx2 + gx + h and when they're both equal, we have this "equating co-efficients of x2" and "equating co-efficients of x" stuff. How do you say they equate coefficients without dividing anything? Am I ignorant of any intermediate steps. I'm only 15 so I'm not that expert in Maths, but I can hold my own.

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Here is a different viewpoint to consider.

Let us say that I travel 0 miles in zero seconds.

What is my speed.

You assert it is 1 mph - which is true of course since at 1 mph the distance I travel in zero seconds = speed times time =1*0 = 0 miles.

However the same would also be true if I travelled at 2 mph or 20 mph or any other speed.

Therefore I can say that speed = distance /time for the case of 0/0 can be anything I want.

Which is another way of saying it is indeterminate.

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Here is a different viewpoint to consider.

Let us say that I travel 0 miles in zero seconds.

What is my speed.

You assert it is 1 mph - which is true of course since at 1 mph the distance I travel in zero seconds = speed times time =1*0 = 0 miles.

However the same would also be true if I travelled at 2 mph or 20 mph or any other speed.

Therefore I can say that speed = distance /time for the case of 0/0 can be anything I want.

Which is another way of saying it is indeterminate.

SIGH......... I tell it and tell it and tell it......... Please read the original post and all the replies. This is purely a hypothetical question which has no real answer. Its like asking to define i = square root of -1 with a real life example. I have seen this same viewpoint in many other websites.

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OK I'll admit you stopped me in my tracks for a minute there. Then I got a curious kind of doubt. You know when you have two equations like ax2 + bx + c and fx2 + gx + h and when they're both equal, we have this "equating co-efficients of x2" and "equating co-efficients of x" stuff. How do you say they equate coefficients without dividing anything?

Simply put, that is illegal at x=0.

Am I ignorant of any intermediate steps.

Let's look at ax=bx. Here's a couple of ways to look at this. One is to rewrite this as (a-b)*x = 0. This equation has two solutions, a=b or x=0. At x=0 you cannot make any inference about the relationship between a and b. Any values will satisfy (a-b)*0 = 0. Another way to look at is to multiply both sides of ax=bx by 1/x, yielding a=b. This multiplication by 1/x (division by x) is illegal at x=0. So once again the inference that ax=bx implies a=b is invalid at x=0.

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Simply put, that is illegal at x=0.

OK, but why? I actually meant how do you equate co-efficients of x2 of two equations when x is not zero. Am I ignorant of any intermediate steps in such a procedure. I know well enough the intermediate steps in equating ax = bx LOL.

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That's OK, I can take a hint, since you don't value anyone else's thoughts I will leave you to yours.

go well

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Ok, I'm only going to give this one shot.

Suppose $\frac{0}{0} = 1$. Because of the property of $0$ it follows that:

$0 = 0 \cdot 1 = 0 \cdot \frac{0}{0} = \frac{0 \cdot 0}{0} = \frac{0}{0} = 1$.

So by claiming that $\frac{0}{0} = 1$ you are actually claiming that $0 = 1$. This is a contradiction, because we know that they have different properties in the ring of the real numbers, and as such they are unique.

So if you want to keep the algebraic structure of the real numbers as a ring, then you can not claim that $\frac{0}{0} = 1$. And if you are thinking in the lines of "Fine, I can live with losing the ring structure of the real numbers", then you are not really defining $\frac{0}{0}$ as we know the number $0$, because, well, you just lost the meaning of the number $0$, so you have nothing to define.

Makes sense?

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Ok, let the waves of opposition begin!

...Just a story...

When I was your age or just a little older I developed a truly beautiful 58 step proof that anything divided by zero equals infinity. I showed it to all my teachers and anyone I could and they mostly just shrugged. While checking it for about the 100th time I found the 45th step was a very subtle assumption of the conclusion so the proof meant nothing.

People don't think about underpinnings and definitions of our ideas nearly enough so lose sight of what the results really mean and most of us get worse with age. Never lose your doubt. Try to keep as broad a perspective as possible.

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Ok, I'm only going to give this one shot.

Suppose $\frac{0}{0} = 1$. Because of the property of $0$ it follows that:

$0 = 0 \cdot 1 = 0 \cdot \frac{0}{0} = \frac{0 \cdot 0}{0} = \frac{0}{0} = 1$.

So by claiming that $\frac{0}{0} = 1$ you are actually claiming that $0 = 1$. This is a contradiction, because we know that they have different properties in the ring of the real numbers, and as such they are unique.

So if you want to keep the algebraic structure of the real numbers as a ring, then you can not claim that $\frac{0}{0} = 1$. And if you are thinking in the lines of "Fine, I can live with losing the ring structure of the real numbers", then you are not really defining $\frac{0}{0}$ as we know the number $0$, because, well, you just lost the meaning of the number $0$, so you have nothing to define.

Makes sense?

With all due respect to all others who replied to this thread, I have to say this is by far the best one. I was going along with the OP's replies for second, but this, this completely puts the issue to rest. Well played tmpst.

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