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View Poll Results: Is x/0 infinitesimal?  
Yes! Completely agree!  0  0%  
Possibly agree...  0  0%  
Don't agree.  2  100.00%  
I have proof against it.  0  0%  
I don't know.  0  0%  
Multiple Choice Poll. Voters: 2. You may not vote on this poll 
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January 28th, 2016, 08:47 PM  #1 
Newbie Joined: Jan 2016 From: Canada Posts: 2 Thanks: 0  Dividing by Zero Theories
I recently watched a numberphile video on youtube about the problems with zero, and it got my thinking. Looking at the linguistics, dividing means putting into a given number of groups, and if you have zero groups, then every For lack of a better term 'piece' would be by itself, and each piece would be infinitesimally small. So by that thinking, x/0=0.000...01, but since you cant write that, in a finite space, we would need a new symbol for it.

January 28th, 2016, 08:53 PM  #2 
Newbie Joined: Jan 2016 From: Canada Posts: 2 Thanks: 0  Please share your opinion!
I'm only in grade 9, and don't know much about this stuff, so if anyone more experienced is completely facepalming right now, please don't hate, explain it to me. And use simple language Last edited by Chichenwin; January 28th, 2016 at 08:58 PM. Reason: Afterthought 
January 28th, 2016, 09:45 PM  #3 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 
Below is the graph of $y = \dfrac{a}{x}$. Can you see what happens near $x = 0$? The right side of the graph shoots up towards infinity and the left side of the graph shoots down towards negative infinity. That's certainly not infinitesimal. 
January 29th, 2016, 12:03 AM  #4 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,161 Thanks: 734 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
Another fun case is: $\displaystyle y = \frac{\sin x}{x}$ as x gets closer and closer to 0 (i.e. closer and closer towards 0/0). Here's a plot of the function: https://www.google.co.uk/webhp?sourc...%20x%20%2F%20x y = 1 at x = 0! The function above is well known and is called sinc x. Weird no? The point of this is that the divide by zero operation really does give weird behaviour... all functions that have a possible divide by zero in it, such as $\displaystyle y = \frac{a}{x}$ and $\displaystyle y = \frac{\sin x}{x}$ could have different behaviours as the divisor gets closer and closer to zero. So what do you do to get around this? In practise what you would do is look to $\displaystyle limit theory$ and evaluate the limit of the function as the dividend tends to 0. 
January 29th, 2016, 05:08 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra 
On a technical note $y={\sin x \over x}$ does not have $y(0)=1$. Instead $y(0)$ is undefined  it has no value. This is precisely because the operation of dividing by zero is not well defined. On a more general mathematical note: mathematical results are not a matter of opinion determined by the result of a poll. A statement is either true, false or undecidable. Sometimes you can get different results in different mathematical systems, but no one mathematical system is more true than any other: each is simply a product of the axioms and definitions that underpin it. Some systems are considered more standard than others, and if you don't specify a particular system, most people will assume you mean one of those. Last edited by v8archie; January 29th, 2016 at 05:23 AM. 
February 2nd, 2016, 03:53 AM  #6 
Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2 
To me, the many problems with zero are an indication of serious flaws with the fundamentals of mathematics. It is stated that zero has no multiplicative inverse and this is used as an excuse to treat zero differently to all other numbers. Divide by zero is just one of many zerorelated problems. Another obvious one is the belief that zero times anything equals zero. If I have zero apples this is not the same as there being zero universes, but I can equate these two because they both supposedly equal 0. I have also read that zero to the power zero can supposedly have different values in different situations. In base 10 there are lots of division operations that are problematic, not just divide by zero. A simple example is 1 divided by 3. The problem is that the division algorithm (for long/short division) has no defined way it can end, it cannot complete. It is a huge copout to avoid the issue by calling it a ratio, or to simply assert that using the words ‘infinitely many’ will somehow mysteriously cause it to finish. To solve problems like divide 1 by 3, I produced my own algorithm for division in any given base where an end point is always achieved. My new algorithm works for any integer values, including divide by zero. For example, it evaluates 0 divided by 0 to be 1, which just happens to support the argument that sin(x)/x = 1 at x=0 (as opposed to being undefined). Sometime soon I hope to write a blog article on this subject at Extreme Finitism. The introduction of complex numbers made many problems solvable which were previously not solvable using algebra. The treatment of zero should have a similarly elegant solution; it should not involve lots of annoying rules that say what we can and cannot do when rearranging algebraic expressions to avoid hitting problems with zero. My response to the OP is that the question makes no sense because I do not accept ‘infinitesimal’ to be well defined. Last edited by Karma Peny; February 2nd, 2016 at 04:32 AM. 
February 2nd, 2016, 04:25 AM  #7  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra  Quote:
Most of what you wrote is either wrong or right, but not for the reasons you give.  

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