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The value of 0/0


Raptor115
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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

Sorry for that.... Really. I was a bit sleepy at that time (Well I can only switch on my comp after 8 pm, what with school and after-school tuition). The thing is..... I just thought I said that you can't define 0/0 with real-life examples in the original post and so I thought you hadn't read it fully. Like I said, if someone can define i = the square root of -1 with a real-life example, I'll say, that guy has real genius.

 

 

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

 

Suppose [math]\frac{0}{0} = 1[/math]. Because of the property of [math]0[/math] it follows that:

 

[math]0 = 0 \cdot 1 = 0 \cdot \frac{0}{0} = \frac{0 \cdot 0}{0} = \frac{0}{0} = 1 [/math].

 

So by claiming that [math]\frac{0}{0} = 1[/math] you are actually claiming that [math]0 = 1[/math]. 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 [math]\frac{0}{0} = 1[/math]. 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 [math]\frac{0}{0}[/math] as we know the number [math]0[/math], because, well, you just lost the meaning of the number [math]0[/math], 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.

OK..... lets see if me can twist this one...... OK Here goes, its like saying x = x.1 = x. (x/x) = x/x = 1? Where did the extra x go? Oh well..... if you take x as zero that might work out... but actually I thought of this for a long time and I saw in another website that they had an argument like if f(x) = x/x = 1 for x = 0 then f(x) = x2 /x = x = 0 for x = 0. Well lets see what you did. 0 = 0.1 (OK, its the same value) = 0 x 0/0 (Putting what I just said) = (0 x 0) / 0 (Well there ain't no bracket in the first place, so I'm gonna rewrite this as 0 x (0/0)). So we have 0 x (0/0) which is 0 x 1 (If 0/0 = 1) which is equal to 0. So 0 = 0. Nothing's changed right?? OK.... if you still don't understand we'll put it like x2 / x simplifies to x. Therefore, 0 x 0/0 is 0.

 

I must say that it seems like I'm twisting the facts to prove my "hypothesis", but hey, I'm not like "I KNOW 0/0 is 1 and everybody else is a fool", I'm like "OK, if we say 0/0 = 1, then sure there are problems, but can't we solve those problems. I mean like, before the Indians came up with 0, the mathematicians must have struggled a lot with this concept of nothingness. Now I'm just trying to come up with something for 0/0, which doesn't cause contradictions. Now don't say there's a contradiction as soon as you say 0/0 = 1. I think we can solve the contradictions. Now, if someone presented me that 0/0 = 1 has a contradiction which would be unable to solve, I'll give it up.

 

 

...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.

LOL, you must have spent a month dreaming about it for sure. Well.... if I was to say this at school, the Maths teachers would come at me with a slipper in their hand.... Its hard not to lose a doubt. I had an idea about producing unlimited electricity by placing motors and generators on the same axle, but everybody said it wouldn't work cause there'd always be some kind of loss... and I gave up on it.... Well that's a topic for the physics forums and not for maths so don't ask about it in this thread.

 

Oh and please remember.... 0 is not equal to 1. Neither did I say x/0 = 1. I was only thinking about 0/0 and NOT about 0 or x/0.

Edited by Raptor115
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Read what I said again, and don't look for a contradiction or a definition, real life or otherwise, because I didn't provide either.

 

The whole point of my example was that 0/0 is undefined, not because it can't be or because it has only one definition, but because it has to many (an infinite number in fact).

 

If you want a more mathematical example it's like asking 'what is the solution to the equation y = x ?'.

 

go well

Edited by studiot
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Read what I said again, and don't look for a contradiction or a definition, real life or otherwise, because I didn't provide either.

 

The whole point of my example was that 0/0 is undefined, not because it can't be or because it has only one definition, but because it has to many (an infinite number in fact).

 

If you want a more mathematical example it's like asking 'what is the solution to the equation y = x ?'.

 

go well

Exactly my point. If you take it as y =x, you'll have infinite solutions. Rewriting it, x - y = 0. Now add another equation, like x + y = 2. Then solving both you get x = y= 1. Now when I said 0/0 doesn't make sense in elementary arithmetic, I meant exactly what you mean --- It has infinite solutions. Now take some other functions like f(x) = x/x or 2x/x or -3x/x or 4- x2 / 2 - x. In all these equations, we can simplify to a form where we KNOW the answer like f(x) = 1 or 2 or -3 or 2+x. Now comparing all these.... if we take 0/0 = 1, we get the answer we're supposed to get when we simplify them. That's why I thought if we define 0/0 = 1, we could maybe simplify a few things, rather than contradict them. And that's what I've been saying all along in this article. Give me one contradiction with 0/0 = 1 which I can't solve and I'll give it up.

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Now, if someone presented me that 0/0 = 1 has a contradiction which would be unable to solve, I'll give it up.

I just did, because [math]1=0[/math] is a contradiction you can't solve. The number 1 has the property [math]x \cdot 1 = x[/math], and the number 0 has the property [math]x \cdot 0 = 0[/math] for all [math]x \in \mathbb{R}[/math]. If [math]0=1[/math] then [math]x = x \cdot 1 = x \cdot 0 = 0[/math] for all [math]x \in \mathbb{R}[/math]. From defining [math]\frac{0}{0} = 1[/math] it follows that there is only one real numbe, e.g. [math]\mathbb{R} = \lbrace 0 \rbrace[/math]. You can't make this work.

 

(And now I gave it two shots...)

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I just did, because [math]1=0[/math] is a contradiction you can't solve. The number 1 has the property [math]x \cdot 1 = x[/math], and the number 0 has the property [math]x \cdot 0 = 0[/math] for all [math]x \in \mathbb{R}[/math]. If [math]0=1[/math] then [math]x = x \cdot 1 = x \cdot 0 = 0[/math] for all [math]x \in \mathbb{R}[/math]. From defining [math]\frac{0}{0} = 1[/math] it follows that there is only one real numbe, e.g. [math]\mathbb{R} = \lbrace 0 \rbrace[/math]. You can't make this work.

 

(And now I gave it two shots...)

LOL.... I thought I said I was giving the value of 0/0 and not 0. You yourself say 0/0 is not 0. Then how do you say 0 = 0/0 =1?

 

To The Moderators:

I'll be going away for a week, and to a place where there is no internet connection. So please don't close this thread. The debate's going nicely.

Edited by Raptor115
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LOL.... I thought I said I was giving the value of 0/0 and not 0. You yourself say 0/0 is not 0. Then how do you say 0 = 0/0 =1?

 

Please read my first answer. By giving 0/0 the value 1, you are in face giving 0 the value 1. If you combine my two posts, you can see that I have shown that [math](\frac{0}{0}=1) \Rightarrow (0=1) \Rightarrow (\mathbb{R} = \lbrace 0 \rbrace)[/math]. If you can find a flaw in my deduction, please feel free to point it out. If not, then I am done.

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.............I said 0/0 doesn't make sense ..............................I meant exactly what you mean

 

 

 

I didn't say that and I didn't mean that!

 

 

 

It makes perfect sense, it is undefined.

 

 

 

What doesn't make sense is applying rules defined for certain objects to objects for which they were not defined. This is what you should be avoiding, - not trying to do.

 

 

 

When you are prepared to look properly at what others are saying ( and all here have so far only told part of the story) you will make progress. Untill then you will be at a brick wall like Zeno was two and a half thousand years ago.

 

 

 

Consider the equation ab = 0

 

 

 

Now the normal rules we use state that either a = 0 or b = 0 or both.

 

 

 

However there are systems of arithemtic where this is not true. You need to to study these to move on.

Edited by studiot
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LOL.... I thought I said I was giving the value of 0/0 and not 0. You yourself say 0/0 is not 0. Then how do you say 0 = 0/0 =1?

 

To The Moderators:

I'll be going away for a week, and to a place where there is no internet connection. So please don't close this thread. The debate's going nicely.

 

Actually, one more thing. By twisting tmpst's first post on this thread, you made an unforgivable fundamental mistake. tmpst's proof that 0/0 = 1 => 0=1

is perfectly correct (based on your assumption of 0/0 = 1) and in your twisting of it you denied a key mathematical concept, which is the fact that by saying that 0*0/0 equals 0* (0/0) = 0 (as you showed) but does not equal (0*0)/0 = 1 (as tmpst showed), you are basically denying the associative property of multiplication which says: When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors. For example, (2*3)*4 = 2*(3*4).

 

I.e. 0 = 0*0/0 = 0*0*(1/0) = 0*(0*(1/0)) = 0*(0/0) (Which you showed = 0) = (0*0)*(1/0) = (0*0)/0 = 1 (which tmpst showed)

 

Therefore, what you call your "hypothesis" not only brings about the contradiction 1 = 0 as tmpst showed, it also violates one of the essential properties of multiplication, which is the associative property. So, there you go. Go to sleep kid, and when you wake up, put your mind to better use...

 

If you do say anything after this, other than "I agree", then you are arguing for the sake of argument, and your goal is not to arrive at a logical conclusion, but to raise anarchy and mayhem...

Edited by somecallmegenius
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  • 2 weeks later...

Please read my first answer. By giving 0/0 the value 1, you are in face giving 0 the value 1. If you combine my two posts, you can see that I have shown that [math](\frac{0}{0}=1) \Rightarrow (0=1) \Rightarrow (\mathbb{R} = \lbrace 0 \rbrace)[/math]. If you can find a flaw in my deduction, please feel free to point it out. If not, then I am done.

OK.... You've said it. I guess it all comes down to the value of 0*0.

 

Actually, one more thing. By twisting tmpst's first post on this thread, you made an unforgivable fundamental mistake. tmpst's proof that 0/0 = 1 => 0=1

is perfectly correct (based on your assumption of 0/0 = 1) and in your twisting of it you denied a key mathematical concept, which is the fact that by saying that 0*0/0 equals 0* (0/0) = 0 (as you showed) but does not equal (0*0)/0 = 1 (as tmpst showed), you are basically denying the associative property of multiplication which says: When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors. For example, (2*3)*4 = 2*(3*4).

 

I.e. 0 = 0*0/0 = 0*0*(1/0) = 0*(0*(1/0)) = 0*(0/0) (Which you showed = 0) = (0*0)*(1/0) = (0*0)/0 = 1 (which tmpst showed)

 

Therefore, what you call your "hypothesis" not only brings about the contradiction 1 = 0 as tmpst showed, it also violates one of the essential properties of multiplication, which is the associative property. So, there you go. Go to sleep kid, and when you wake up, put your mind to better use...

 

If you do say anything after this, other than "I agree", then you are arguing for the sake of argument, and your goal is not to arrive at a logical conclusion, but to raise anarchy and mayhem...

OK... I agree that you and tmpst have the best argument so far(BTW I WAS arguing for the sake of argument. Otherwise, why would I consider myself to be out of my own mind). In any case, I think it all comes down to this --- How do you say 0*0 = 0. OK this might seem like a stupid question, but really, I'm sincere in asking it, because ---- Let's consider: 2*3. In Maths, multiplication is adding the first number for the number of times given by the second number, right? So this is 2+2+2 = 6. We could write this as 3*2 --- 3+3 = 6. So we get the Commutative property and by further proof, the Associative property. Now for multiplication with 0. 6*2 means 6+6. 6 * 1 means 6. 6*0 means ___?(Nothing?. Oh, that's what zero is..., right? Wait.., we write 0 one time so its 0*1) Using commutative property we write 6*0 as 0*6 --- 0+0+0+0+0+0 = 0. Now, how do we define 0*0?? Please, I'm really serious. I'm not joking or making fun. The point is if we write a number once, it's that_number * 1. If we multiply 0 with 0. That is, 'Nothing' added 'nothing' times. What does that mean?

 

Consider the equation ab = 0

 

Now the normal rules we use state that either a = 0 or b = 0 or both.

 

However there are systems of arithemtic where this is not true. You need to to study these to move on.

There are?? Can you please point out and explain some of these systems.

Edited by Raptor115
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There are?? Can you please point out and explain some of these systems.

 

Please is a very good word +1

 

If a and b are both vectors then both a . b and a x b can equal zero (though not simultaneously) without either a or b (or both) being equal to zero.

 

Do you know what these imply?

Edited by studiot
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Hi.

 

I didn't have time to read all the topic. So excuse me if what I'm saying has already been told.

Please take time to read the following paragraph taken from Wikipedia:

 

In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as a /0 where a is the dividend (numerator). Whether this expression can be assigned a well-defined value depends upon the mathematical setting. In ordinary (real number) arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a (a≠0), and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 has no defined value and is called an indeterminate form.

 

Please pay attention to the difference between "undefined" and "indeterminate".

 

And about the following:

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.

 

I remember someone answering that, equating coefficients is not possible for X=0 . Well that is not true at all.

The thing is that, no division happens at all.

Let me first give you the correct form of the question:

 

The following equation is true for all values of X . Find d,e and f . (values of a ,b and c are given on the question)

ax2 + bx + c = fx2 + gx + h

 

So this is what happens:

(a-f)x2 + (b-g)x + (c-h) = 0

Since the above equation is true for ALL values of X, the only logical answer would be: a=f b=g c=h

You could solve the equation for three values of X. let's say 1 and 2 and 3

1: (a-f) + (b-g) + (c-h) = 0

2: (a-f)*4 + (b-g)*2 + (c-h) = 0

3: (a-f)*9 + (b-g)*3 + (c-h) = 0

It is easy to solve afterwards.

 

You could also use Proof by contradiction.

 

Link to full wikipedia article: Division By Zero

Edited by Hooman
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[math]\lim_{x \rightarrow 0^{\pm}} \frac{n}{x} = \pm \infty \; \; \; \; \; \; n \neq 0[/math]

 

[math]\lim_{x \rightarrow 0^{\pm}} \frac{n}{x} = 1 \; \; \; \; \; \; n = x[/math]

 

[math]\lim_{x \rightarrow 0^{\pm}} \frac{x}{n} = 0 \; \; \; \; \; \; n \neq 0[/math]

 

Reference:

Division by zero - Wikipedia

Edited by Orion1
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Figure representation

 

1

------------------------------------------

1/2

------------------------!

1/3

--------------!

.

.

1/0.5

---------------------------------------------------!--------------------------------------------------

1/0.25

---------------------------------------------------!--------------------------------------------------!---------------------------------------------!----------------------------------------

.

.

1/0??? Imaginary

.

.

1/2

-------------------------!

0.5/2

------------!

0.25/2

------!

.

0.00000000000001/2

!

 

0/2

?

 

0/0

?

 

How to draw the Figure? 1/0, 0/2, 0/0

Edited by alpha2cen
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1/0 = infinite means 1/0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000...001?

 

Is there no difference between 0 and 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000...001 ?

 

Is 1/0 infinite or calculation impossible?

Edited by alpha2cen
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  • 2 weeks later...

OK I admit I thought I was losing my sanity when I thought of this. . .

 

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. . .

 

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")

 

 

No, no need for twisting the facts; by doing that we will not arrive at the Truth.

 

I am no mathematician.

 

But let me give you my humble opinion. See if it convinces you.

 

When you say;

The Axiom:

f(x) = x/x is a constant function giving the value 1.

And then reach the conclusion;

Therefore, 0/0 is also 1 . . .

There is a logical fallacy in this theory [OK, if you are happy to call it a theory, let us call it a theory, till the time we both are convinced it is otherwise].

Explanation:

In,

 

f(x) = x/x

 

The operation involved is the 'Arithmetic Division'.

What is Arithmetic division?

'Arithmetic Division' is partitioning of a Quantity into stipulated/required number of parts of equal Quantity/size.

I.E,

 

If 'A' is a quantity, and you require it to be divided into 'B' number of parts of equal quantity, then one has to carry out the 'Arithmetic operation of Division' and divide 'A' by 'B' which will give us 'B' number of parts of 'A' all of equal quantity/size 'C'.

 

Mathematically it can be written as,

 

A/B = C.

 

So, all through our exercise, we see that the 'Arithmetic Division' is a mathematical operation carried out to divide a Quantity into a required number of equal Quantities. [That required number also ultimately becomes a Quantity].

 

So here your theory needs to consider the following:-

 

  • '0' is a number that does not represent a quantity. Instead, it represents the 'Lack of a quantity' i.e. '0' is not a Quantity.
  • When you say you want to divide some quantity by '0', you are saying you don't want to divide at all. I.E. When you are dividing some quantity, you are dividing it into a 'CERTAIN' number of equal parts. But if you say you want to divide that quantity by/into '0' number of equal parts, you are in fact saying, you don't want to divide it at all. You cannot divide something into nil number of equal parts.
  • When you are dividing '0' into a 'CERTAIN' number of equal parts, you are in fact not dividing anything at all. So no question of getting any number of equal parts. The result is '0'.
  • When you say 0/0, you are saying; you are dividing Nothing into Nil number of parts. In fact you are not doing anything at all, leave aside Division. How can the results of not doing anything, be equal to 1 or any quantitative number?

So 0/0 is not a valid Arithmetic Division.

 

But then, for your theory to hold well and to become complete in itself, you have to make a small correction.

 

You have to say it thus;

 

f(x) = x/x is a constant function giving the value 1, for all quantitative numbers/values of x.

 

Then the question of saying,

 

0/0

 

does not arise at all.

 

Zero is not a quantity in the sense that it represents a lack of quantity.

 

I respect your freedom to find new theories and welcome your endeavor to bring us new knowledge. I suggest you equip yourself further, to bring us best theories; my good wishes are with you.

 

. . . IF you're going "OMG, Gasp.... what a sacrilege!!" . . .

 

No it's not a sacrilege. All is fair while seeking knowledge. [Like we say 'All is fair in love and war'. Not war, because war itself is not fair.]. The more you meddle with knowledge, as long as you are willing to discuss before deciding, the more it becomes sacred, because in the end, it convinces us what is right and what is not.

 

Nothing is more sacred than the quest of knowledge.

 

Thank you

Edited by Anilkumar
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[math]\lim_{x \rightarrow 0^{\pm}} \frac{n}{x} = \pm \infty \; \; \; \; \; \; n > 0[/math]

 

[math] \lim_{x \rightarrow 0^{\pm}} \frac{n}{x} = 0 \; \; \; \; \; \; n = 0[/math]

 

Consider a function with zero in the numerator, it seems that a zero in the numerator becomes the dominant function for all values of x, including zero. Stepping off zero in any x axis direction with any infinitesimal quantity of 'n' and the result is infinity, therefore:

 

[math]\frac{0}{0} = 0[/math]

 

[math]0 \times 0 = 0[/math]

Edited by Orion1
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  • 1 month later...

Sorry to revive a month-old thread, but I was just reading along and found this topic really interesting.

 

EDIT: Okay, nevermind! Had a question about a certain proof, but I missed the obvious...

Edited by Amaton
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  • 4 weeks later...

Right. Experimental post coming up.

 

Isn't the problem here that there are two very different ways of defining/conceptualising zero? If it is literally 'nothing', the absence of any positive quantity, then there is nothing to divide or multiply. If it the empty set, however, then it is not nothing since it is a set. A set divided by a set gives 1. So the whole thing would come down to how we define zero in our system. It seems to me that if we define zero as the empty set then we have created a contradiction right there, before doing any operations on it, and that this may be what the OP is suggesting.

 

If this as nonsense I apologise. Just trying it out.

Edited by PeterJ
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I am not sure if this is helpful. Even though dividing by 0 is undefined in mathematics, dividing by 0 in reality is a useful concept. I believe math is here mainly to allow us to describe the physical world. The whole concept came from the need to interact with the world. Cave men knew nothing about math, but they knew it was harder to climb up than down. And if he brought home an antelope, he didn't have to hunt as often as when he brought home a rabbit.

Throughout the years humans just invented more ingenious ways to describe our world. Euler made e so powers were easier to work with. When mathematics can't give us the answers we want or know exist, sometimes we have to ask the question in different ways. Take velocity. It is the instantaneous speed at a single point in space and time. Which in essence is division by 0. Newton and Leibniz and some others got around this by introducing the concept of a limit. This concept allows us to divide by numbers closer and closer to 0. By doing this we approach some number which would be the answer if math allowed us to actually divide by 0. So even though math may not always be able to give an exact answer, it can get arbitrarily close enough to not matter.

Since pi doesn't have an exact solution, when you divide by pi you get a fraction that doesn't really have an exact value, so is division by pi or e undefined too? I don't know, but can we use it to describe the physical world? Yes(pi r^2)(e^i theda). And regardless of whether math allows us to divide by 0, pi, e or not, it gets us close enough to the exact answer as we need.

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dividing by 0 in reality is a useful concept.

 

Theoretically, any number divided by itself should equal 1. 0 represents the absence of value, the absence of units of matter, so why am I not seeing random 1s pop up everywhere if nothingness can just divide by itself and get something? Dividing by 0 isn't useful at all precisely because it has little physical meaning.

When you have limits anyway, the limit doesn't mean that the dependent variable every equals that value, it strictly means it y values only approaches it. In fact, the limit of the sequence in the topic post doesn't exist when 0 is approached from both sides. If it's approached from one side, then the limit is plus or minus infinity, but that still doesn't mean you can do direct substitution and logically say that x=infinity.

Edited by EquisDeXD
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Theoretically, any number divided by itself should equal 1. 0 represents the absence of value...[/Quote]

 

No, theoretically, any infinitesimal or maximal quantity divided by itself equals one. Zero is neither an infinitesimal nor is it a maximal.

 

If it's approached from one side, then the limit is plus or minus infinity, but that still doesn't mean you can do direct substitution and logically say that x=infinity.[/Quote]

 

In mathematics, specifically calculus, a limit is the value that a function or sequence 'approaches' as the input or index approaches some value. It is not the purpose to determine a limit by direct numerical integration of any function.

 

My ad argumentum stated in post #41 remains valid.

Edited by Orion1
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No, theoretically, any infinitesimal or maximal quantity divided by itself equals one. Zero is neither an infinitesimal nor is it a maximal.

Which is why it doesn't work.

 

 

In mathematics, specifically calculus, a limit is the value that a function or sequence 'approaches' as the input or index approaches some value. It is not the purpose to determine a limit by direct numerical integration of any function.

 

My ad argumentum stated in post #41 remains valid.

 

No it doesn't, if the limit goes to 0, and if there's more than one possible answer, then the limit doesn't exist.

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Theoretically, any number divided by itself should equal 1. 0 represents the absence of value, the absence of units of matter, so why am I not seeing random 1s pop up everywhere if nothingness can just divide by itself and get something?

 

I don't think that question makes sense, especially if we're discussing the practicality of zero/zero in the real world. What sense does it make to say something just arbitrarily divides by itself? Rather, it makes sense to say how many times the quotient fits into the dividend, which would happen to be our divisor.

 

If I have zero apples, what number can I multiply this amount by so that I end up with zero apples? Any real number satisfies this.

 

Of course, this isn't very good since defining it as such would lead to disastrous contradictions, due to the implications of every real number being equal to every other real number.

 

However, it does provide some intuitive structure. Consider the relation [math]f(x,y)=\frac{x}{y}[/math]. Graphically, as [math]x[/math] and [math]y[/math] simultaneously approach zero, the function starts to look like the Cartesian axes... Meaning that the relation virtually approaches this: a vertical line at [math]x=0[/math] and a horizontal at [math]y=0[/math]. Which would agree with "zero/non-zero = zero" and "zero/zero = all reals".

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