Question

I have already asked, but I want to ask your opinion about my statement (I will prove my statement)

Answer 1

Divide by 0 doesn't exist because the result would split and converge to \(+\infty\) and \(-\infty\) all at once.

(that makes no sense how a point can break into two, and certainly that both way it diverges, and this is why division by zero doesn't exist).

(hold... I will post my reason)

Answer 2

lets see what happens when the divisor (the number you divide by) approaches 0 from the right. (I am using 1 is dividend for convenience, could be be any C)

\(\large\color{black}{ \displaystyle 1\div 1=1 }\)

\(\large\color{black}{ \displaystyle 1\div \frac{1}{2}=2 }\)

\(\large\color{black}{ \displaystyle 1\div \frac{1}{3}=3 }\)

\(\large\color{black}{ \displaystyle 1\div \frac{1}{4}=4 }\)

\(\large\color{black}{ \displaystyle 1\div \frac{1}{5}=5 }\)

\(\large\color{black}{ \displaystyle 1\div \frac{1}{\rm n}=\rm n }\)

\(\large\color{black}{ \displaystyle 1\div \lim_{{\rm n}\rightarrow0^+}\frac{1}{n}=\infty }\)

Answer 3

Now, as divisor approaches zero from the left side,

\(\large\color{black}{ \displaystyle 1\div -1=-1 }\)

\(\large\color{black}{ \displaystyle 1\div \frac{1}{2}=-2 }\)

\(\large\color{black}{ \displaystyle 1\div \frac{-1}{3}=-3 }\)

\(\large\color{black}{ \displaystyle 1\div \frac{-1}{4}=-4 }\)

\(\large\color{black}{ \displaystyle 1\div \frac{-1}{5}=-5 }\)

\(\large\color{black}{ \displaystyle 1\div \frac{-1}{\rm n}=\rm -n }\)

\(\large\color{black}{ \displaystyle 1\div \lim_{{\rm n}\rightarrow0^+}\frac{-1}{n}=-\infty }\)

Answer 4

oh the last line should say

\(\large\color{black}{ \displaystyle 1\div \lim_{{\rm n}\rightarrow0^-}\frac{1}{n}=\infty }\)

(minus by 0)

Answer 5

Division by zero is a division by two sides limit, and thus

\(\large\color{black}{ \displaystyle 1\div \lim_{{\rm n}\rightarrow0}\frac{1}{n}=\left\{ -\infty,~+\infty\right\} }\)

so would be true for any number C, (at least if C isn't zero)

\(\large\color{black}{ \displaystyle C\div \lim_{{\rm n}\rightarrow0}\frac{1}{n}=\left\{ -\infty,~+\infty\right\} }\)

Answer 6

((division of a number by number gave two points. ))

Answer 7

i have a typo when I am showing the left sides limit. second line in that reply should be -1/2

Answer 8

(if you want I can remove the mass and repost it entirely)

Answer 9

\[\text{ do you mean to talk about } \frac{1}{\frac{1}{0}} \text{ or } \frac{1}{0}?\]
And when I put the 0 on bottom just pretend that means n approaches 0.

Answer 10

Do you mean to ask if my definition of 0 is \(\large\color{black}{ \displaystyle \lim_{{\rm n}\rightarrow0}\frac{1}{n} }\) or \(\large\color{black}{ \displaystyle \lim_{{\rm n}\rightarrow0}{\rm n}}\) ?

Answer 11

\[\lim_{n \rightarrow 0}(1 \div \frac{1}{n})=\lim_{n \rightarrow 0} (1 \cdot n)=0\]

Answer 12

no, I am dividing by 1/n

Answer 13

why does 1/infty or 1/-infty = infty or -infty respectively?

Answer 14

My idea is that

\(\large\color{black}{ \displaystyle {\rm C}\div0 =}\)

\(\large\color{black}{ \displaystyle {\rm C}\div\lim_{n\rightarrow0}~\frac{1}{n} =\left\{{\rm a,~b}\right\}}\)

(that is, it is equivalent to a set of 'a' and 'b', equivalent to two points.)

point 'a' diverges to -infinity
point 'b' diverges to +infinity.

Answer 15

u r right actually

Answer 16

should just be

\(\large\color{black}{ \displaystyle\lim_{n\rightarrow0}~\frac{C}{n} =\left\{{\rm a,~b}\right\}}\)

Answer 17

that is how I should put it

Answer 18

ok I think I'm cool then
you are basically using the graph of f(x)=1/x on (-inf.inf)
to say why we shouldn't divide by 0
because it has a break at x=0
one side goes positive large while the other side goes negative large

Answer 19

yup, it is the graph of 1/x .....

Answer 20

I saw the idea of 1/(1/2) , 1/(1/3) , 1/(1/4) ... 1/(1/n) => infinity
so then I though, about the negatives.

the essence and a very good demonstration is the graph of 1/x
(or any f(x)=C/x)

Answer 21

you know what in your work about each time you got closer to the end of one of your posts it looked like you were making the n larger (and not closer to 0)
So maybe we could have wrote:
\[\lim_{n \rightarrow \infty}(1 \div \frac{1}{n})\]
or negative large
\[\lim_{n \rightarrow -\infty}(1 \div \frac{1}{n})\]

Answer 22

Oh, yeah I made that technical error. Apologize.

Answer 23

I should have used infinity, or I could have said

\(\large\color{black}{ \displaystyle C\div \color{orangered}{\lim_{n \rightarrow 0}\left(n \right)}}\)

Answer 24

or \(\large\color{black}{ \displaystyle 1\div \color{orangered}{\lim_{n \rightarrow 0}\left(n \right)}}\)

Answer 25

So division by 0 is undefined because \(\cfrac{1}{0}\) means there is an \(a\) such that
$$
0\times a=1
$$
But \(0\times \text{Anything}=0\)

Would that be sufficient?

Answer 26

I was proposing it (division by 0) is undefined, because when you take any Real number C and divide by 0, you get a set of 2 values {a,b} where a diverges to -infinity and b diverges to + infinity.

division by 0 gives a, where a diverges to -infinity = that is the left sided limit of 1/n
division by 0 gives b, where b diverges to +infinity = that is the right sided limit of 1/n

division by 0 (the two sides limit) gives you the set of a and b
(wher a diverges to neg. inf. and b diverges to pos. inf.)

Answer 27

C / D = E
but not
C / D = {set of more than 1 value}

Answer 28

I am even disregarding the fact that a and b diverge

Answer 29

im founding this very interesting, good job !!

Answer 30

Tnx ikram...

Answer 31

People talk about dividing by zero as if it is such a mystery. Division implies breaking a whole into parts. The term "dividing by zero", when you really think about it, is an oxymoron. If you divide something by 3, you divide it into three parts. If you divide something by 2, you divide it into 2 parts. If you divide something zero times, you divide it into zero parts. The whole remains unchanged. In other words, you've done nothing. You haven't divided anything at all. The process of "division" never took place.

Answer 32

nice fading on the pic

Answer 33

Hero, that is like a pizza/cake reasoning.
You can't split a pizza into 0 friends.

But, you can't split a pizza into p/q friends or into any x friends if x is not a natural number.

Answer 34

acc to that reasoning division by non-integers is not applicable

Answer 35

If you take the common sense approach, you won't have to prove division by zero doesn't exist.

Answer 36

Well, I knew it doesn't exist ever since I learned that in 3rd grade in Russia.... but people reason it differently... I was just trying to offer that division by zero gives a result of two points ((and not only you have a division of a number by another another that gives you two answers that makes no sense, but also these elements of my answer are not tangible they are infinities))

Answer 37

well, as long as @SolomonZelman took the limit notation i see no harm using the word "division" in fact the idea of divide on zero with limit notations show u the tiny value u can divide by for example what if i wanna divide on such SMALL number near to zero ?
and yes x/0 approaches to infinity its UNDEFINED value means you cant express it in number system terms that ppl deals with and count, but if ur a good mathematician you would use it and know the value behind it, unlike 0/0 this term is indeterminate which mean it can ANYTHING they ar such mathematical proofs shows its 2,3,4,.... or anything else

Answer 38

Yeah, elementary level:

if C is a none-zero number,
C / 0 = x
checking:
x times 0 = 0 (and can't equal to non-zero number C)
So, C/0 = no solution (or undefined) if C is not zero.

0 / 0 = x
checking
x * 0 = 0

x can literally be any number. I am not sure if 0/0 is really invalid from that standpoint.
((( likewise, 0/x=0 and x can be any number. )))

Answer 39

Just an infinite number of solution doesn't necessarily indicate that there is no solution, or perhaps when I said 0/x , then there is no division, and that is why there is an infinite number of solutions?

Perhaps, 0/x == u r not even dividing. just like saying 0=0
well, then why wouldn't 0/0=0 be like 0=0 ?

Answer 40

You divided by values that were close to zero and realized you could do that infinitely for both positive and negative values. But you never actually divided by zero. The term dividing by zero has no mathematical meaning. Dividing by zero won't result in two points. You don't know what will happen when you divide by zero because no one has yet to actually do it. When they say dividing by zero doesn't exist, there's a reason for it.

Answer 41

yes;) ...
i think i got another reason

Answer 42

The reason why division by zero is not defined is because if we allow it to be defined, many algebra rules are broken. Consider if we let \(1/0 = z\) and say \(c/0 = cz\).
The object \(z\) would not obey most algebra rules.

Answer 43

just that I need to express in a better way.

Will first start from this definiton
\(\color{blue}{\rm Dividend \div Divisor = Quotient} \)
and will name them,
\(\rm A \div B = C \)

A=Dividend
B=Divisor
C=Quotient
\(\rm (this~~is~~for~~ convenience)\)

What is dividing A by B mean?
(lets choose a unit of a meter, and assume this same unit for all A B and C)
((Will just look at flat piece of size A, and a flat piece of size B.))

and the C here, is the number of B-sized pieces that A can contain.

There is certain amount/number C of B's in the A.
(hope this phrase is not abstruse)

If your B is 0, then A can contain as many piece as you want.
And from there you get very large results when you divide by small numbers (by small B's).

(just another thought)