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SolomonZelman

  • one year ago

just a very quick though about division by 0.

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  1. SolomonZelman
    • one year ago
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    It is correct to define the division by zero, as \(\Large\color{black}{ \displaystyle \lim_{{\rm n}\rightarrow~0}~\left(\frac{1}{\rm n}\right) }\) --------------------------------------------- because \(\Large\color{black}{ \displaystyle \lim_{{\rm n}\rightarrow~0}~\left(\rm n\right) }\) is zero, and dividing by zero gives (with the application of the limit properties)... \(\Large\color{black}{ \displaystyle \frac{1}{0}=\frac{1}{\displaystyle\lim_{{\rm n}\rightarrow~0}\left(\rm n\right)}=\displaystyle \lim_{{\rm n}\rightarrow~0}~\left(\frac{1}{\rm n}\right) }\) we could be dividing any number doesn't have to be a 1, it is just more convenient to see the point this way. ---------------------------------------------- Now, we know that \(\Large\color{black}{ \displaystyle \lim_{{\rm n}\rightarrow~0^+}~\left(\frac{1}{\rm n}\right) }\) diverges to \(+\infty\) and \(\Large\color{black}{ \displaystyle \lim_{{\rm n}\rightarrow~0^-}~\left(\frac{1}{\rm n}\right) }\) diverges to \(-\infty\) NOW, division by zero gets even worse, because dividing by zero is a two-sided limit. that, is: \(\Large\color{black}{ \displaystyle \lim_{{\rm n}\rightarrow~0}~\left(\frac{1}{\rm n}\right) }\) and this way, the limit diverges to \(+\infty\) and \(-\infty\) all at once. Clearly, this makes no sense. How can one point diverge to both infinities? (there is only 1 point, it can't tear apart and go both directions) the (obvious) CONCLUSION is: division by zero makes no sense.

  2. anonymous
    • one year ago
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    I wish I could say I understood this. But this reminds of me of that large "proofs of why you can't divide by 0" thread that was up a while ago.

  3. SolomonZelman
    • one year ago
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    would you agree that \(\Large\color{black}{ \displaystyle \frac{1}{~~\frac{1}{2}~~}=2 }\)

  4. anonymous
    • one year ago
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    Yeah. \[\frac{1}{\frac{1}{2}}=1*2\]

  5. SolomonZelman
    • one year ago
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    yes, and tell me what do you get if: \(\Large\color{black}{ \displaystyle \frac{1}{~~\frac{1}{3}~~}=? }\) \(\Large\color{black}{ \displaystyle \frac{1}{~~\frac{1}{4}~~}=? }\) \(\Large\color{black}{ \displaystyle \frac{1}{~~\frac{1}{5}~~}=? }\) and on....

  6. anonymous
    • one year ago
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    So 3,4,5 to infinity? Or to a limit?

  7. SolomonZelman
    • one year ago
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    don't use the word limit. yes, because the smaller number you divide by, (and this number will approach to zero as much as you would like) the bigger output you get

  8. SolomonZelman
    • one year ago
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    this way, you are going to get an infinitely large output.

  9. SolomonZelman
    • one year ago
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    now, we can do same thing from negative numbers \(\Large\color{black}{ \displaystyle \frac{1}{~~-\frac{1}{2}~~}=-2 }\) \(\Large\color{black}{ \displaystyle \frac{1}{~~-\frac{1}{3}~~}=-3 }\) \(\Large\color{black}{ \displaystyle \frac{1}{~~-\frac{1}{4}~~}=-4 }\) and so on.... so as you take \(\Large\color{black}{ \displaystyle \frac{1}{~~-\frac{1}{very~~large~~number}~~} }\) you are then going to get an output which will tend to negative infinity closer and closer the bigger "very bg number" you choose

  10. SolomonZelman
    • one year ago
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    Also: a "limit", just literally means a "limit" in an English sense of it. In other words, and expression is limited to a certain number/value. this value is called "limit" (such that you can get 5, 0.2, or any C units away from the limit)

  11. SolomonZelman
    • one year ago
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    you can't get a certain number of units close to positive or negative infinity. that is why when your limit is infinity, the limit doesn't exist. (technically speaking "it diverges")

  12. SolomonZelman
    • one year ago
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    sorry for dumping all of this on you.... u will get to this later.

  13. anonymous
    • one year ago
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    haha, it's fine. It's pretty complicated but it's real interesting stuff :)

  14. SolomonZelman
    • one year ago
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    Yeah, it is even more interesting when it comes to gamma functions cycle integrals and all that... but, anyway, ... it was a nice recap:) tnx for participation

  15. anonymous
    • one year ago
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    No problem, it would've been fun to see this in the other thread. That one had a lot of other good explanations about dividing by 0.

  16. SolomonZelman
    • one year ago
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    There is also a pizza thing. You can't divide evenly your 6 slice- pizza, to 0 of your friends, because how any will each get. (that is not a complete disprove, as that denies dividing by rational numb. as well)

  17. anonymous
    • one year ago
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    Yeah, I've seen the word representations on those before. They're a simple way of representing something so complex.

  18. SolomonZelman
    • one year ago
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    yes, my favorite one. On wiki, when I read some math I get lost at times even though the math it is about is a topic i know.

  19. anonymous
    • one year ago
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    Well anyways, thanks a bunch for the explanation. It'll probably take a little while for my head to completely wrap around the concepts though. As a final question, how did you get your latex writing to be so big? Still kinda new to writing in latex.

  20. SolomonZelman
    • one year ago
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    i used \Large in front of the latex

  21. anonymous
    • one year ago
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    Oh ok, thanks a ton for everything :)

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