My Math Forum Why isn't dividing by zero = zero?

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April 4th, 2019, 04:03 AM   #31
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Quote:
 Okay, but how do you respond to somebody who claims that division by zero should have meaning, if not by looking at examples where imposing such a meaning gives erroneous results? Is there a better application or use case where errors would clearly be obtained? The apples example provided earlier clearly indicates an issue with division by zero, but doesn't seem to indicate a definitive error if someone forcefully overrides existing knowledge with their own definition of 0/0. After all, the OP responded to the apples example in a way that seems to suggest that they don't care that 0/0 is undefined.

One of the issues you need to consider in assigning a meaning to 0/0 is the deeply philosophical question

Why is

$\displaystyle \frac{{dy}}{{dx}} \ne \frac{0}{0}$

For instance consider Bolzano's example

$\displaystyle {y^3} = a{x^2} + {a^3}$

$\displaystyle {\left( {y + dy} \right)^3} = a{\left( {x + dx} \right)^2} + {a^3}$

So

$\displaystyle \frac{{dy}}{{dx}} = \frac{{2ax + adx}}{{3{y^2} + 3ydy + d{y^2}}}$

set dy, dx = 0

$\displaystyle \frac{{dy}}{{dx}} = \frac{{2ax}}{{3{y^2}}}$

April 4th, 2019, 05:32 AM   #32
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 Originally Posted by Micrm@ss We have that $n/m = k$ if and only if $mk = n$. But we have $0\cdot 1 = 0$ and $0\cdot 2=0$. This means that $0/0 = 1$ and $0/0 = 2$. So $0/0$ is not a unique answer. This is problematic.
I think this example is a better indication about the thornery of division by zero. I'll use this from now on instead of the sinc x example.

 April 4th, 2019, 06:34 AM #33 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,663 Thanks: 2642 Math Focus: Mainly analysis and algebra Surely the point here is that the OP proposed the idea that since $\displaystyle \lim_{x \to 0^+} \frac1x = +\infty$ and $\displaystyle \lim_{x \to 0^-} \frac1x = -\infty$, taking the "average" of these two "yields zero" for division by zero. Obviously, there are many problems with his reasoning, and so there are many ways to counter his argument, but one perfectly valid one is to demonstrate that there are other limits involving "division by zero" that do not result in the same value. The fact that you prefer a different rebuttal doesn't make everybody else wrong, and implying so simply exhibits an unfortunate arrogance on the part of the poster making that claim. There is space on this forum for many people to demonstrate their knowledge and understanding, each contributing to the knowledge and understanding of the OP.
 May 1st, 2019, 11:09 AM #34 Newbie   Joined: Apr 2019 From: North Carolina Posts: 2 Thanks: 0 8/2 = 4 is means that 4 is the unique number such that 4 times 2 =8. 8/0 is undefined because there does not exist any number n such that n times 0 equals 8. 0/0 is undefined because any number times zero equals zero, so there is no unique number n such that n times zero equals zero (emphasis on the word unique).
May 1st, 2019, 04:04 PM   #35
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 Originally Posted by Micrm@ss How many times do I need to give you 0 apples in order for you to have 1 apple.
A countably infinite number of times, as though partitioning (0, 1] into a countably infinite number of disjoint Vitali sets.

May 1st, 2019, 11:23 PM   #36
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 Originally Posted by AplanisTophet A countably infinite number of times, as though partitioning (0, 1] into a countably infinite number of disjoint Vitali sets.
What do Vitali sets have to do with this???

May 2nd, 2019, 08:27 AM   #37
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Quote:
 Originally Posted by Micrm@ss What do Vitali sets have to do with this???
Their measure is undefined, not “0”. How many times do you need to give me 0 apples before I have 1 apple?: 1/0 is also undefined, not 0, as is the focus per the OP. So I said (somewhat jokingly), that you would need to give me 0 apples a countably infinite number of times as though equating the measure of a Vitali set to 1/0.

May 2nd, 2019, 10:24 AM   #38
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Quote:
 Originally Posted by AplanisTophet Their measure is undefined, not “0”. How many times do you need to give me 0 apples before I have 1 apple?: 1/0 is also undefined, not 0, as is the focus per the OP. So I said (somewhat jokingly), that you would need to give me 0 apples a countably infinite number of times as though equating the measure of a Vitali set to 1/0.
But the measure of the Vitali set has nothing to do with 1/0?

May 2nd, 2019, 02:57 PM   #39
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 Originally Posted by Micrm@ss But the measure of the Vitali set has nothing to do with 1/0?
It must be greater than 0, but also must be less than all real numbers greater than 0, so why not compare 1/0 ~ 1/[the measure of a Vitali set] ? It’s not a lot unlike considering the limit as x approaches 0 of 1/x, as was done earlier in the thread, even if the result of 1/0 is seemingly independent of anything involving limits.

May 2nd, 2019, 03:56 PM   #40
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Quote:
 Originally Posted by AplanisTophet It must be greater than 0, but also must be less than all real numbers greater than 0, so why not compare 1/0 ~ 1/[the measure of a Vitali set] ? It’s not a lot unlike considering the limit as x approaches 0 of 1/x, as was done earlier in the thread, even if the result of 1/0 is seemingly independent of anything involving limits.
Since I'm making some assumptions here and have the urge to stray a little off topic, I started a new thread:

Some Crank Talking About Vitali Sets

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