BuffettIs 1 divided by 0 undefined or infinity?

[BigStelly]

OK recently in class some of my classmates were debating whethter 1 divided by zero is undefined or infinity. I saw both points but I am wondering what everyone else out there thinks. So 1/0= undefined or infinity or does it depend on other stuff?

 

[matt grime]

Oh dear God, not again...

 

[Icebreaker]

Dividing by zero is undefined. There are ways around that using limits; however, in those cases, you are not dividing by zero, per se.

 

[Alkatran]

lim(x->0)(1/x) = undefined
lim(x->0+)(1/x) = infinity
lim(x->0-)(1/x) = -infinity

Take your pick. It all depends on the question.

 

[dextercioby]

The interesting part is that he posted a poll...He probably meant division.All operations involving infinities & dividing through 0 are undefined...

Daniel.

 

[arildno]

I voted other: 0/1=0.

 

[Icebreaker]

This reminds me; if you ever see a book called "Zero: The Biography of a Dangerous Idea" and want to read it, be careful. The author frequently uses expressions like "dividing by zero/infinity". One of the passages goes like this: "[...]dividing anything by infinity gives zero; dividing anything by zero gives infinity. We can see the relationship these two numbers have throughout[...]"

http://www.chapters.indigo.ca/item....talog=Books&N=35&Lang=en&Section=books&zxac=1

I am told, however, that the historical account of zero of the book is rather fascinating.

 

[dextercioby]

Don't read the book.Division under the reals has the property

a:b=c\Rightarrow a=b\cdot c

You'll see that in the case of 0 or infinity,it doesn't work...(multivaluedness).

Daniel.

 

[HallsofIvy]

Define your terms. If you are working in the standard real number system, division of any number is "undefined"-literally. However, some people use "infinity" as a shorthand for "undefined": saying \lim_{x->0} \frac{1}{x}= \infty is precisely the same as saying "the limit does not exist".

 

[Owen Holden]

Oh dear God, not again...

Yes, inevitably.
We must make sense out of the expression EyEx(x/0 = y).
Is it tautologous or is it contradictory?

i.e. is (1/0) a number?, or, is (0/0) a number?, or, is (2/0) a number? etc..

It seems contradictory to me.

 

[dextercioby]

As Halls said,how do you define your concepts:the concept of "number"...?

Daniel.

 

[arivero]

When x=0, the expression 1/x="multiplicative inverse of x"="the number y such that x times y equals unity" is a lot less ambiguous that 0/x, "the number y such that x times y equals zero".

 

[dextercioby]

Yes,but when u see the overall picture,they're both faulty.

Daniel.

 

[fdarkangel]

i think, this question can take place on a math-FAQ.

people who think about division with zero seem to forget the meaning of division.
steps of division are, you subtract the divisior from the divdend as long as there's something remaining dividend, ie it hits 0. for each exact subtraction, add 1 to the counter (ie, division). (i'm not covering the details for fractional parts, since my purpose in telling this is to remind the definition)

so, applying this definition to 1/0: subtract 0 from 1 as long as you have a non-0, and add 1 to the counter each time. now, if you do this for 1/0, what's your counter?

similar idea can be applied to 0/0: subtract 0 from 0 until as long as you have a non-0. now, indefinitiny approcahes at this point. no matter how you repeat this (and remember that you're adding a 1 to the result in each step), it'll be valid against the definition of division. 0/0 = 5, 0/0 = 9, 0/0 = anything.

an application can be: x.0 = 0, this equation true for every value of x, ie, it's independent from the contents of x, as long as it's a valid number. dividing two sides by 0, you get x.\frac{0}{0} = \frac{0}{0}. an equation independent from the value of x. it's pretty weird at the first glance, but if you remember what division is, the explanation reveals itself.

i'd like to emphasize that division by zero in not undefined, it's indefinite (or maybe uncertain is a better word), and there're problem-dependent methods for getting rid of the indefinition.

 

[arildno]

fdarkangel:
Euclid's algorithm has nothing to do with the fact that the multiplicative inverse of 0 is undefined.

 

[The Rev]

I saw an interesting "proof" that illustrates the problems of dividing by zero:

a=b
a²=ab
a²-b²=ab-b²
(a-b)(a+b)=b(a-b)
(a-b)(a+b)/(a-b)=b(a-b)/(a-b)
a+b=b
2b=b
2=1

The reason it comes out wrong is because (a-b)=0

Kinda Fun.

:D

The Rev

 

[jcsd]

We've had this question many times before:

It cna easily be shown in any ring that:

a*0 = a(b - b) = a(b - b) + 0 = a(b - b) + ab - (ab) = a(b - b + b) - (ab) = ab - (ab) = 0

i.e. a*0 = 0 for all 'a';

define divison by:

a = b/c iff a*c = b

let a = 1/0 therefore a*0 = 1, but a*0 = 0 for all 'a' so there is no 'a' which cna be identifed with 1/0.

 

[Owen Holden]

We've had this question many times before:

It cna easily be shown in any ring that:

a*0 = a(b - b) = a(b - b) + 0 = a(b - b) + ab - (ab) = a(b - b + b) - (ab) = ab - (ab) = 0

i.e. a*0 = 0 for all 'a';

define divison by:

a = b/c iff a*c = b

let a = 1/0 therefore a*0 = 1, but a*0 = 0 for all 'a' so there is no 'a' which cna be identifed with 1/0.

Of course.

 

[jcsd]

Of course.

Anything else to add...?

The thing to note about the proof is that it applies to all rings.

 

[fdarkangel]

fdarkangel:
Euclid's algorithm has nothing to do with the fact that the multiplicative inverse of 0 is undefined.

are you sure you've read my post?

 

[Owen Holden]

Anything else to add...?

Yes.
That which is equal to (1/0) does not exist.

1/x =df (the y: x*y=1), includes the definition of (1/0).

If x=0 then 1/0 = (the y:0*y=1)

But, there is no y such that: y*0=1.

That is, Ay(y*0=0).

Therefore, (the y:y*0=1) does not exist.

 

[jcsd]

you mean to say there does not exist any element of x in any element of ... (a certain algebra, the most general class of al;gebras you probably wnat to consider are called the rings) which meets this definition of 1/0. General, unqaulifed staements like "1/0 doesn't exist" lack meaning in a matehmatical context.

 

[Owen Holden]

you mean to say there does not exist any element of x in any element of ... (a certain algebra, the most general class of algebras which meet the two requiremnts you have stipulated are called rings) which meets this definition of 1/0. General, unqaulifed staements like "1/0 doesn't exist" lack meaning in a matehmatical context.

I don't agree.

E!x =df EF(Fx).

~E!(the x:~(x=x)), is a theorem.
All objects described by contradicory predications do not exist!
They have no properties at all.

~E!x <-> ~EF(Fx).

 

[Alkatran]

1/0 = 1 or 2 or 1.5 or 1.2 or 1 + 5i or sqr(2) or ...

Why are we doing this, again?

 

[jcsd]

so are you saying that for example in the extended complex plane 1/0 cna't be defined (hint: it can be)? 1/0 is only undefined in mathematical structures which have certain propetries.

 

[moose]

In my mind it isn't infinite because it never even approaches it. I understand how things that approach a certain point, for example how a 1/x graph gets close to the intercepts, but never touches, then yeah at infinite it will be touching, but this never gets anywhere, so...thats my point.

 

[arildno]

are you sure you've read my post?

Yes, I've read it. It is irrelevant to the fact that 1/0 cannot be defined in the real number system.
(Which follows from the fact that given any real number "a", a*0=0)

 

[fdarkangel]

It is irrelevant to the fact that 1/0 cannot be defined in the real number system.

really?

steps of division are, you subtract the divisior from the divdend as long as there's something remaining dividend, ie it hits 0. for each exact subtraction, add 1 to the counter (ie, division). (i'm not covering the details for fractional parts, since my purpose in telling this is to remind the definition)

so, applying this definition to 1/0: subtract 0 from 1 as long as you have a non-0, and add 1 to the counter each time. now, if you do this for 1/0, what's your counter?

can you see the definition now?

 

[MathStudent]

In my mind it isn't infinite because it never even approaches it. I understand how things that approach a certain point, for example how a 1/x graph gets close to the intercepts, but never touches, then yeah at infinite it will be touching, but this never gets anywhere, so...thats my point.

What point is that? In this particular case we are not concerned with limits. We are stating that 1/0 (as a real number) is not defined for the reasons which have been shown above by jcsd. There are defined properties for the real numbers ( or fields in the more general sense) which we hold true. What has been shown is that if a number such as 1/0 were to exist, then it would contradict those properties. Hence we define that no such number can exist.

 

[master_coda]

really?



can you see the definition now?

Your definition of division is not the same as the definition of division on the real numbers.

 

[fdarkangel]

Your definition of division is not the same as the definition of division on the real numbers.

i guess you're implying the irrational numbers. i was not willing to expand the algoritm for non-integer results. i have never given any irrational number input. if you mean the non-integer results.. well, here it goes:

let A and B two reel numbers. subtract B from A until A is bigger or equal to zero, and add 1 to the division at each step. after this process, if A is negative, undo the one step. multiply A with 10 (or what ever the step is) and add 10^{-1} to the result after each subraction. repeat these steps until A=0, and decrement the power of 10 (to generalize, base) by 1.

however, i'd like to point that the original question is integer-wise, and this definition was uncessary.

 

[master_coda]

i guess you're implying the irrational numbers. i was not willing to expand the algoritm for non-integer results. i have never given any irrational number input. if you mean the non-integer results.. well, here it goes:

let A and B two reel numbers. subtract B from A until A is bigger or equal to zero, and add 1 to the division at each step. after this process, if A is negative, undo the one step. multiply A with 10 (or what ever the step is) and add 10^{-1} to the result after each subraction. repeat these steps until A=0, and decrement the power of 10 (to generalize, base) by 1.

however, i'd like to point that the original question is integer-wise, and this definition was uncessary.

But you attempted to extend the algorithm to try and divide by zero. Once you do that, the algorithm is no longer giving the same results as real number division.

 

[fdarkangel]

it does, for integer results.
i expanded the algorithm to generalize. basics of both division methods are same. the initial explanations i made about 1/0 and 0/0 are still correct. please don't be so pedantic, the interger-only algorithm is consistent and sufficient enough to explain 1/0.

 

[chroot]

fdarkangel,

Again, you are wrong. Master_coda is correct. Your algorithm is, basically, junk.

Either way, this thread is just a rehash of many other similar threads here. We don't need another thread with the same arguments from the same people.

Thread closed.

- Warren