1. SolomonZelman

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)

2. SolomonZelman

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) $$\large\color{black}{ \displaystyle 1\div 1=1 }$$ $$\large\color{black}{ \displaystyle 1\div \frac{1}{2}=2 }$$ $$\large\color{black}{ \displaystyle 1\div \frac{1}{3}=3 }$$ $$\large\color{black}{ \displaystyle 1\div \frac{1}{4}=4 }$$ $$\large\color{black}{ \displaystyle 1\div \frac{1}{5}=5 }$$ $$\large\color{black}{ \displaystyle 1\div \frac{1}{\rm n}=\rm n }$$ $$\large\color{black}{ \displaystyle 1\div \lim_{{\rm n}\rightarrow0^+}\frac{1}{n}=\infty }$$

3. SolomonZelman

Now, as divisor approaches zero from the left side, $$\large\color{black}{ \displaystyle 1\div -1=-1 }$$ $$\large\color{black}{ \displaystyle 1\div \frac{1}{2}=-2 }$$ $$\large\color{black}{ \displaystyle 1\div \frac{-1}{3}=-3 }$$ $$\large\color{black}{ \displaystyle 1\div \frac{-1}{4}=-4 }$$ $$\large\color{black}{ \displaystyle 1\div \frac{-1}{5}=-5 }$$ $$\large\color{black}{ \displaystyle 1\div \frac{-1}{\rm n}=\rm -n }$$ $$\large\color{black}{ \displaystyle 1\div \lim_{{\rm n}\rightarrow0^+}\frac{-1}{n}=-\infty }$$

4. SolomonZelman

oh the last line should say $$\large\color{black}{ \displaystyle 1\div \lim_{{\rm n}\rightarrow0^-}\frac{1}{n}=\infty }$$ (minus by 0)

5. SolomonZelman

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

6. SolomonZelman

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

7. SolomonZelman

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

8. SolomonZelman

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

9. freckles

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

10. SolomonZelman

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

11. freckles

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

12. SolomonZelman

no, I am dividing by 1/n

13. freckles

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

14. SolomonZelman

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.

15. SolomonZelman

u r right actually

16. SolomonZelman

should just be $$\large\color{black}{ \displaystyle\lim_{n\rightarrow0}~\frac{C}{n} =\left\{{\rm a,~b}\right\}}$$

17. SolomonZelman

that is how I should put it

18. freckles

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

19. SolomonZelman

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

20. SolomonZelman

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)

21. freckles

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 $\lim_{n \rightarrow -\infty}(1 \div \frac{1}{n})$

22. SolomonZelman

Oh, yeah I made that technical error. Apologize.

23. SolomonZelman

I should have used infinity, or I could have said $$\large\color{black}{ \displaystyle C\div \color{orangered}{\lim_{n \rightarrow 0}\left(n \right)}}$$

24. SolomonZelman

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

25. ybarrap

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?

26. SolomonZelman

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.)

27. SolomonZelman

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

28. SolomonZelman

I am even disregarding the fact that a and b diverge

29. ikram002p

im founding this very interesting, good job !!

30. SolomonZelman

Tnx ikram...

31. Hero

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.

32. SolomonZelman

33. SolomonZelman

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.

34. SolomonZelman

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

35. Hero

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

36. SolomonZelman

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))

37. ikram002p

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

38. SolomonZelman

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. )))

39. SolomonZelman

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 ?

40. Hero

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.

41. SolomonZelman

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

42. anonymous

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.

43. SolomonZelman

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.)) |dw:1434667396117:dw| 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)

44. Hero

Actually, it's not just because of the fact that it would break algebra rules. You don't even need to use algebra rules to show that division by zero is not possible. If you think of division as it relates to subtraction, then you can describe it this way: If you wanted to know how many times you could take 2 away from 12, the shortcut way to do it is to just divide. 12/2 = 6. You could take 2 from 12 six times before you have nothing left. To show this same thing using long subtraction, you have: 12 - 2 - 2 - 2 - 2 - 2 - 2 = 0 And you could do this with almost any division as long as the divisor isn't zero. Suppose we tried this same approach with zero as the divisor. We want to know how many times we can take zero from twelve before we have nothing left. In other words, 12/0 = what? Well, if we try long subtraction, here's what would happen: 12 - 0 - 0 - 0 - 0 - 0 - 0 - 0 ... We would keep subtracting zero from 12 infinitely but we would never get to zero. We'd remain at 12. Division is nothing but a shortcut for long subtraction. So when we say we can't divide a number by zero, what we're really saying is, We can subtract zero from a number as many times as we won't but we will never be able to determine exactly how many times it would take us to subtract before we have nothing left because we will always have the same amount we started with.

45. SolomonZelman

beautiful.....

46. anonymous

Division is an abstract concept that can be interpreted as repeated subtraction for cases when the divisor divides the dividend, but otherwise the interpretation falls short. You know that for 1/3 this would happen 1 - 3 - 3 - 3 ... and it will never get to zero, either. The next step is to consider values beyond the integers. The only way to really argue whether or not an operation or value deserves a definition is by considering how many algebra rules it breaks and how many it obeys.

47. Hero

Actually, you can demonstrate it for 1/3 as well when I apply the shorthand version of the long subtraction. In other words, each time I subtract 2, I multiply by 1 which signifies how many times I subtracted the number. The demonstration is here: 12 - 2(6) = 0 If I want to do the same for 1/3, then it would be 1 - 3(1/3) = 0 So it is possible to subtract 3 to get to zero, you just have to subtract 1/3 of 3 to get there.

48. Hero

What I showed immediately above is the way I meant to demonstrate it initially. To show the subtraction version of division it's dividend/divisor = quotient dividend MINUS divisor TIMES quotient = 0 The long hand version of the same thing I showed above is 1 - 1/3 - 1/3 - 1/3 = 0

49. anonymous

That's not really an actual definition. You're using subtraction and multiplication properties of equality to manipulate an equation. Also you had to create a class a numbers (non-integral rational numbers) for it to work.

50. Hero

All I was demonstrating was that division is a shortcut for long subtraction.

51. anonymous

And I agree that it makes sense when the divisor perfectly divides the dividend, but I disagree that it can be used to justify not defining zero division

52. Hero

I demonstrated also using long subtraction why zero division makes no sense. That's the first thing I did.

53. SolomonZelman

C / 0 = X C - 0 - 0 - ... - 0 = 0 \_____________/ X number times.

54. SolomonZelman

I like Hero's idea:)

55. SolomonZelman

well, for all non zero C's

56. SolomonZelman

and for C=0 you get X=anything

57. SolomonZelman

I want to know one thing. I previously asked it, but maybe not explicitly. 0/0=X 0 -0-0-0...-0 =0 \___________/ X times In this case, because X can be absolutely anything, does that make 0/0 undefined, or, 0/0 does make sense?

58. anonymous

I don't think it is satisfactory. It really just leads to the conclusion that $$12/0 = \infty$$, which is fine in the context presented; however, it is still a definition.

59. anonymous

An indeterminate form is an operation where all values satisfy it given the right context.

60. SolomonZelman

12/0 is not infinity if you go by the subtraction idea. Or not necessarily infinity.

61. SolomonZelman

this is why division by doesn't exist, because then 1=0

62. Hero

63. anonymous

There is no context in which 12/0 = 12

64. SolomonZelman

yes=X is mistake

65. SolomonZelman

oops

66. ybarrap

12/0=X equivalent to 12 - 0 -0-0 ... -0 = 0 \______________/ X times $$12 - X\times 0 = 0$$ Not defined

67. SolomonZelman

tnx for correcting. 12/0=X 12 - 0 -0 -0 ... -0 = 0 \______________/ X times X= DNE goes quite well then.....

68. Hero

@wio, I understand why you are not really impressed with my demonstration. I explained an abstract concept using basic mathematics.

69. SolomonZelman

that was a shift. I assert that:) jk

70. anonymous

If we say $$12/0=x$$, we don't get to be presumptuous about how $$x$$ works. For example, you can't presume that the subtraction property of equality works for $$x$$. Consider for example $$\sqrt{-1}=i$$. If we have the inequality $$2<5$$, can we multiply both sides by $$i$$?$2i<5i$We can't do this because it presumes $$i$$ is positive. Just because $$i$$ doesn't follow all the properties of real numbers doesn't make it undefined. It's a matter of how many algebra rules the concept forces us to qualify. If it doesn't provide much insight and makes add asterisks to a lot of rules, then it's better off not beind defined.

71. SolomonZelman

The subtraction approach is not like this. There is a huge difference. for any a>b you can't multiply both sides time i. And in this case, subtraction method only doesn't work when divisor is zero. This is not an evidence for invalidity of the method, but rather, it is a genuine evidence to why division by 0 doesn't exits.

72. SolomonZelman

This is my opinion, of course.