# Divide by zero?

1. Sep 17, 2005

### kant

For s=p/q

What happen when q is actually o? We are taugh in school mathematics that such propositive is 'undefine', but what is it really mean by 'undefine'? To me, this is unsatisfactory answer. Perhaps people can help me clearify this matter for me.

What i accept:

Imagine a object subdivide into many equal parts, then the size of each parts approach zero.

The converse is not true.

Mathematcally, if q is a number approaching 0, but not actually 0, then s would approach infinity, which to me is a contradiction of our normal intuition. if you divide a finite size object into an infinitly fewer and fewer equal parts, each parts would approach infinity. This would mean the parts of an object is greater then the object itself. This is a contradiction of intuition, so it must not be true.

Another way to look at it is this:

If q is actually 0, then s should be p.
Intuitively specking, if an object is divide by nothing, then the result should be the object itself.

2. Sep 17, 2005

### mathmike

take 1 / .1 = 10

1 / .01 = 100

1 / .00...1 = 100...0

the closer trhe denominator gets to zero the closer the answer gets to infinity

therefore, if the den. is 0 the the answer is infinity

3. Sep 17, 2005

### Hurkyl

Staff Emeritus
I remember a time (I think I was around 2) when I threw a tantrum because I thought 8*0 = 8.

Anyways, consider the following string:

"Blue English gazorninplat purple fruitcake is"

It's absolutely gibberish, isn't it?

That's what undefined means in mathematics. You've put symbols together that simply aren't allowed to be put together, thus the result is entirely meaningless.

In the specification for division, it says that the numerator shalt be a real number and the denominator shalt be a nonzero real number.

So if you try to put a 0 in the denominator, you get meaningless garbage.

Actually, if q is a variable denoting a real number, you're not even allowed to say "p/q"!!! In order for "p/q" to be a valid mathematical phrase, q has to be a variable denoting a nonzero real number.

4. Sep 17, 2005

### kant

^ this is of course obvious, and i mentioned it. it is unsatisfatory.

5. Sep 17, 2005

### kant

It is not garbage. I don t believe in meaningless mathematics. Perhaps it is a revelation into something deeper.

Last edited: Sep 17, 2005
6. Sep 17, 2005

### tmc

Take any a in R
a/0=b
a=b*0
a=0
Obviously false, since not all numbers in R are equal to 0.

7. Sep 17, 2005

### es

Also, if we accepted the definition s=p/0 then division would not be 1-1.
s=p/0=p/1

8. Sep 17, 2005

### ComputerGeek

I created an unordered co-set of numbers called the undefined set. it was just a goofy thing I did during my modern algebra class because I was board. I have a PDF of the description. since no one has done anything in it I made up my own Axioms and definitions. it was kind of weird because in this set, 0 is the multiplicative inverse of 1/0 and I denote 1/0 to be "u". It was kind of fun.

9. Sep 18, 2005

### Tide

kant,

No. The correct question would become "how many times does nothing go into the object?" That is what is meant by "divide by." Since you cannot specify the number of times 0 goes into p the division is undefined! :)

10. Sep 18, 2005

### benjamincarson

It is...
It's not...

11. Sep 18, 2005

### Manchot

Yes, it is meaningless. If I asked you what the prime factorization of pi was, would you not call me crazy? Prime factorization is only defined on the integers, so it doesn't make sense for you to find the factorization of a non-integer. Likewise, the multiplicative inverse of an element in a field is only defined for non-zero elements, and so division must be undefined for divisors which are zero.

Now, the question you are probably thinking is, "Why skirt the issue like this? Isn't that creating a hole in our mathematical theory?" While you can create a logically-consistent field with division by zero (such as with Riemann spheres), it would not have the same basic properties that our common notions of numbers have. Therefore, when mathematicians seek to define fields that apply to our world (such as the integers, the reals, and the complex numbers), it makes a lot more sense to leave division by zero undefined.

12. Sep 18, 2005

### rachmaninoff

The question does go very deep, as you'll soon come across studying complex analysis, or modern algebra, or what have you. To the point: it's not just meaningless mathematics; it's not mathematics to say "what is 'p/0'"? Start from the algebraic definition of a/b; we define

c = a / b = iff b*c = a

But since 0*c is 0, clearly to say 0*c=1 or 0*2=1 or whatever has no meaning. So 1/0 or 2/0 have no meaning either. Division is a man-made concept, and we've only defined it over nonzero denominators. We like it that way; it's consistent with everything else we like. You could try to make a new "division" if you want; but it'll be ugly and conflict with many important axioms. You won't get the same "0" additive identity, for instance.

To the moderators - I offer this thread as further evidence that we need some new stickies in this subforum.

13. Sep 18, 2005

### kant

I am not going to reply to everyones post. It would just be a waste of my time. Thank you for all your replies. With that being said, i cannot accept any one of the above answers. I know very well that for mathematics to be consistent, you simply cannot divide by zero, but my main concern is not the consistency of mathematical statments.

14. Sep 18, 2005

### arildno

kant:
Evidently you live in some fantasy world where you believe that "division" has some truer, deeper meaning than how mathematicians DEFINE it.
I would strongly suggest you change your attitude; that's the only way to learn real maths.

15. Sep 18, 2005

### Zurtex

You come to a mathematical board, asking a question about why a statement is mathematically undefined and seemingly you do not want an answer that explains it.

1/0 in the field $\left( \mathbb{R}, +, * \right)$ (real numbers with addition and multiplication) is completely gibberish. Just as this would be:

$$++-11*/5^+1$$

16. Sep 18, 2005

### Hurkyl

Staff Emeritus
1/0 isn't meaningless mathematics: it's a meaningless string of symbols. As I stated, to be mathematics, the denominator of a real number division calculation has to be a nonzero real number. If the denominator is not known to be a nonzero real number, then it's not mathematics.

17. Sep 18, 2005

### HallsofIvy

Staff Emeritus
Okay, so you prefer mysticism to mathematics! I can live with that.
(Like Zurtex, I find it peculiar that you come to a mathematics board to annouce that you don't like mathematics!)

18. Sep 18, 2005

this is not true. $$\lim_{x{\to}0^+}\frac{c}{x}=+\infty$$.....x is never actually 0

edit: that limit is a good way to show why c/0 isn't infinity

$$\lim_{x{\to}0^+}\frac{c}{x}=+\infty$$
$$\lim_{x{\to}0^-}\frac{c}{x}=-\infty$$
$$\lim_{x{\to}0^+}\frac{c}{x}{\not{=}}\lim_{x{\to}0^-}\frac{c}{x}$$
therefore, $$\lim_{x{\to}0}\frac{c}{x}$$ does not exist

Last edited: Sep 18, 2005
19. Sep 19, 2005

### Werg22

Simple proof (by definition): imagine you have triangle with an angle theta. Now imagine the angle grows constantly. When it will reach 90 degree you will have two parallel sides, and both are infinite. Since we know that cos(90)=0, then x/infinity=0, where x is any positive integrer. If x/infinity=0, then x/0=infinity.

20. Sep 19, 2005

### bomba923

If $$c > 0$$, that is...

I agree.

Last edited: Sep 19, 2005