Question

just a very quick though about division by 0.

Answer 1

It is correct to define the division by zero, as

\(\Large\color{black}{ \displaystyle \lim_{{\rm n}\rightarrow~0}~\left(\frac{1}{\rm n}\right) }\)

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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.
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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.

Answer 2

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.

Answer 3

would you agree that

\(\Large\color{black}{ \displaystyle \frac{1}{~~\frac{1}{2}~~}=2 }\)

Answer 4

Yeah. \[\frac{1}{\frac{1}{2}}=1*2\]

Answer 5

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....

Answer 6

So 3,4,5 to infinity? Or to a limit?

Answer 7

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

Answer 8

this way, you are going to get an infinitely large output.

Answer 9

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

Answer 10

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)

Answer 11

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")

Answer 12

sorry for dumping all of this on you.... u will get to this later.

Answer 13

haha, it's fine. It's pretty complicated but it's real interesting stuff :)

Answer 14

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

Answer 15

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.

Answer 16

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)

Answer 17

Yeah, I've seen the word representations on those before. They're a simple way of representing something so complex.

Answer 18

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.

Answer 19

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.

Answer 20

i used \Large in front of the latex

Answer 21

Oh ok, thanks a ton for everything :)