Why Does This Happen? Investigating Common Causes

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In summary: I don’t know, I’m a newb. I can tell you one thing, I’ll only be doing long division with a pencil and paper.In summary, the conversation discusses the meaning and validity of dividing by zero. While some argue that it is a proof of the number of zeros when dividing by zero, others argue that it is a meaningless operation with no defined answer. The conversation also delves into the concept of extending the multiplicative group by the zero of the additive group, which ultimately leads to contradictions. Overall, it is agreed that dividing by zero is not allowed and any attempt to attach meaning to it is futile. Some posters share their frustration with the question, while others provide explanations and proofs to support their arguments.
  • #1
Atlas3
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Why does this?

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  • #2
I have no idea what that image is trying to convey. I can give you one message: dividing by 0 is always forbidden and makes absolutely zero sense.
 
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  • #3
Of course, it is forbidden. But those two examples are a proof. If you have a zero, and zero goes into zero, you have exactly one zero, right? It is not "fringe" I didn't particularly care to be called fringe for asking about this. I could not find a proper group for number theory in this forum. I think from what I know of Discrete Mathematics, it is a truth. That's all that image is trying to convey.
 
  • #4
Atlas3 said:
Of course, it is forbidden. But those two examples are a proof. If you have a zero, and zero goes into zero, you have exactly one zero, right? It is not "fringe" I didn't particularly care to be called fringe for asking about this. I could not find a proper group for number theory in this forum. I think from what I know of Discrete Mathematics, it is a truth. That's all that image is trying to convey.

What does "if you have a zero, and zero goes into zero, you have exactly one zero" mean?
 
  • #5
Atlas3 said:
... those two examples are a proof...
A proof of what? That if you perform a meaningless operation that you can pretend that the answer is valid?
 
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  • #6
If you do the multiplication to check like you would after a long division and ask yourself HOW MANY zeroes you end up with you have one. Same with zero into zero like I was taught to think, how many times does two go into four, well twice.
 
  • #7
Atlas3 said:
If you do the multiplication to check like you would after a long division and ask yourself HOW MANY zeroes you end up with you have one. Same with zero into zero like I was taught to think, how many times does two go into four, well twice.
@Atlas3, you are, in military parlance, pissing up a rope.
 
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  • #8
A correct answer to this question is: junk the book.

Zero isn't part of the multiplicative group of our ring. Hence the question about division by zero is as meaningful as whether there are unicorns and to which species they belong. It is not "forbidden", it is simply not existent. Now one can ask, if we can extend the multiplicative group by the zero of the additive group, to which the answer is no, as we will run into contradictions:
$$
0^{-1}\cdot 0 \stackrel{(1)}{=} 1 \text{ and } 0^{-1}\cdot 0 \stackrel{(2)}{=}0^{-1}\cdot (1+(-1))\stackrel{(3)}{=}0^{-1}\cdot 1 + (- 0^{-1}\cdot 1)\stackrel{(4)}{=}0
$$
  1. definition of the multiplicative inverse
  2. definition of the additive neutral
  3. distribution law
  4. definition of the additive inverse
Hence the answer is: The question is nonsense, and any attempt to attach a meaning is necessarily in vain.
 
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  • #9
Atlas3 said:
If you do the multiplication to check like you would after a long division and ask yourself HOW MANY zeroes you end up with you have one. Same with zero into zero like I was taught to think, how many times does two go into four, well twice.

I think the argument is as follows:

Because ##0 = 1 \times 0## then ##\frac{0}{0} = 1##

One problem with this argument is that we also have:

Because ##0 = 2 \times 0## then ##\frac{0}{0} = 2##

Etc.
 
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  • #10
Atlas3 said:
If you do the multiplication to check like you would after a long division and ask yourself HOW MANY zeroes you end up with you have one. Same with zero into zero like I was taught to think, how many times does two go into four, well twice.
No. That division process goes on forever, with no stopping point. Division By Zero is meaningless and is undefined.
 
  • #11
Thank you for the better topic Title. I appreciate the moderation.
 
  • #12
PeroK said:
I think the argument is as follows:

Because ##0 = 1 \times 0## then ##\frac{0}{0} = 1##

One problem with this argument is that we also have:

Because ##0 = 2 \times 0## then ##\frac{0}{0} = 2##

Etc.
This is exactly why the expression ##\frac 0 0## is meaningless. We in mathematics are picky about arithmetic -- we like any division problem to have only one answer.
 
  • #13
Well I don’t know the formatting markup to display such nice notations. I know it’s pretty stupid to conversation about an undefined. As far as one answer for any division problem. What about the converse? 2 x 0 = 0 3 x 0 = 0 has many many problems and the same answer. I’m not contesting anything. Just rattling around on a few thoughts.
 
  • #14
Atlas3 said:
Well I don’t know the formatting markup to display such nice notations. I know it’s pretty stupid to conversation about an undefined. As far as one answer for any division problem. What about the converse? 2 x 0 = 0 3 x 0 = 0 has many many problems and the same answer. I’m not contesting anything. Just rattling around on a few thoughts.
##a \cdot 0 = 0## is due to the distributive law in a ring. It combines multiplication and addition, but is does not extend the multiplicative structure by the additive neutral! We just have ##a \cdot 0 = a \cdot (1+(-1))=a\cdot 1 + a\cdot (-1)=a + (-a) = 0##. The element ##0^{-1}## is still undefined, it does not exist.
 
  • #15
Mark44 said:
This is exactly why the expression ##\frac 0 0## is meaningless. We in mathematics are picky about arithmetic -- we like any division problem to have only one answer.
These kind of questions must have bored the crap out of the posters/moderators with such strange notions. However, I did ask a simple thing. Is it useful? Not one comment to the question. I have been schooled instead. I think the title is useful if you knew where to put it. I also received replies stating the state of the nature of the question. Comments followed up by phrases of military parlance of urinating in a vertical fashion are amusing. I suppose it is hard to have an option about things that were left undefined since the 1500's. Thanks to those that took the time to distribute the proofs. It read a bit condescending but I can live with it. Thank you.
 
  • #16
Atlas3 said:
What about the converse? 2 x 0 = 0 3 x 0 = 0 has many many problems and the same answer.
It is all right for many different expressions to all yield the same result. A bit inevitable, perhaps. There are so many expressions and so few results.

It is not all right for one expression to yield many different results. We normally require each expression [with no free variables] that produces a result must produce the same result every time it is evaluated.
 
  • #17
I think you have got more answers than the question deserved, esp. as there are already dozens of such threads which can easily be found by a forums search. The short answer is: ##\dfrac{0}{0} =1## is not useful, because it will result in logical contradictions. Instead you have been explained why these contradictions will arise. I bet you would had been equally unsatisfied by a simple "No!" which is a complete answer and barely better than not one comment to the question, that wouldn't had schooled you, and didn't contain phrases of military parlance. Furthermore, this
Atlas3 said:
I suppose it is hard to have an option about things that were left undefined since the 1500's.
doesn't make sense either, as there is no such option.

I consider your criticisms unfair, because - as I pointed out - seemingly there will be no answer which would have pleased you. But I will respect your wish: next time I will try to answer your questions by a simple yes, no, correct or false. It will save me a lot of time, I only wished you would had said this right from the beginning.

The question itself is as if you had asked: "What if I fly a plane under water, will it increase gas mileage?" Please ask yourself how you would have answered to such a question.
 
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  • #18
Atlas3 said:
I could not find a proper group for number theory in this forum. I think from what I know of Discrete Mathematics, it is a truth.
The category is neither number theory nor discrete mathematics -- it's just plain old arithmetic. And no, it's not true that 0/0 = 1, for reasons already given.

Atlas3 said:
However, I did ask a simple thing.
Yes, and it has been answered.

Atlas3 said:
Is it useful?
Not at all, as has been amply pointed out in the responses.

Atlas3 said:
Not one comment to the question.
This is post #18 -- you have received lots of good responses.

fresh_42 said:
think you have got more answers than the question deserved
Indeed.

Thread closed.
 

1. Why is it important to investigate common causes?

Investigating common causes helps us understand the underlying reasons for why certain events or phenomena occur. This knowledge can be used to prevent or mitigate negative outcomes, and improve our understanding of the world around us.

2. What steps are involved in investigating a common cause?

The first step is to clearly define the problem or phenomenon being investigated. Next, data must be collected and analyzed to identify potential causes. Then, experiments or observations are conducted to test these potential causes and determine their significance. Finally, conclusions are drawn based on the results and further investigation may be needed if the cause is not definitively determined.

3. How do we determine the most likely common cause?

The most likely common cause is determined through a process of elimination. By ruling out other potential causes and conducting thorough investigations, the most likely cause can be identified. Additionally, statistical analysis and expert knowledge can also play a role in determining the most likely cause.

4. Can common causes change over time?

Yes, common causes can change over time. As our understanding and technology advances, new causes may be identified or previous causes may become less relevant. It is important to continuously investigate and reassess common causes to stay current and adapt to changing circumstances.

5. What are some limitations of investigating common causes?

Some limitations of investigating common causes include limited resources, incomplete data, and the complexity of certain phenomena. Additionally, there may be multiple causes or interactions between causes, making it difficult to determine a single common cause. It is important to acknowledge and address these limitations in order to accurately interpret and apply the results of an investigation.

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