# Divide by zero?

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.

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

Hurkyl
Staff Emeritus
Gold Member
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.

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

Hurkyl said:
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.

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

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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.

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

kant said:
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.
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.

Tide
Homework Helper
kant,

Intuitively specking, if an object is divide by nothing, then the result should be the object itself.
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! :)

kant said:
It is not garbage.
It is...
kant said:
I don t believe in meaningless mathematics. Perhaps it is a revelation into something deeper.
It's not...

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

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

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.

arildno
Homework Helper
Gold Member
Dearly Missed
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.

Zurtex
Homework Helper
kant said:
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.

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:

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

Hurkyl
Staff Emeritus
Gold Member
It is not garbage. I don t believe in meaningless mathematics. Perhaps it is a revelation into something deeper.
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.

HallsofIvy
Homework Helper
kant said:
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.
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!)

mathmike said:
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
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

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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.

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
If $$c > 0$$, that is... division by zero is undefined in the real numbers. There's really no arguing this.
I agree.

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shmoe
Homework Helper
Werg22 said:
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.
If x is a positive real, x/0 is not infinity, it's not an apple pie, it's simply an undefined string of symbols (likewise for x/infinity). By (lack of) definition, division by zero is undefined in the real numbers. There's really no arguing this.

What is the importance to define x/0? Really there is no mathematical importance to this. It is completly irrevelant. See it the way you want.

Icebreaker
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HallsofIvy