What is the Meaning of 1/0? Exploring a Puzzling Question

[avemt1]

undifined 1/0 ?!

why is anything over zero undefined?
This is a question I have faced for a while and have not found an answer.
Could you please help me?

 

[NateTG]

It depends on how you define division.

Usually
a/b=x
means that
x*b=a

Now, let's take a look at the case where b=0 and a=0;
x*0=0.

Clearly any x works.

And in the case where b=0 and a is not zero:
x*0=a
Clearly no x works.

Either way, there is no unique x so that the equation works, so it remains undefined.

 

[HallsofIvy]

By the way, because of the distinction that NateTG noted,
any x satisfies x*0= 0 but no x satisfies x*0= b for b non-zero,

it is common to say that 0/0 is "undetermined" while b/0, for b non-zero, is "undefined".

 

[Hessam]

i thought it was...

because... 1/ .01 = 100 and 1/ .00000001 = 1000000000 and so forth... and thus as the numbers get smaller on the bottom, the answer becomes higher... so much that 1/0 = undefined

 

[HallsofIvy]

posted by Hessam
I thought it was...because... 1/ .01 = 100 and 1/ .00000001 = 1000000000 and so forth... and thus as the numbers get smaller on the bottom, the answer becomes higher... so much that 1/0 = undefined

You were mistaken. If I define f(x) to be 1/x if x is not 0 and 1 if x=0, then it is also true that f(.1), f(.01), f(.00000001), etc get larger and larger but f(0) is not "undefined" (f is merely "discontinuous" at 0). 1/0 is "undefined" because there is no way to define it that does not violate some basic property of the real numbers and the definition of "/".

 

[suyver]

Originally posted by Hessam
because... 1/ .01 = 100 and 1/ .00000001 = 1000000000 and so forth... and thus as the numbers get smaller on the bottom, the answer becomes higher... so much that 1/0 = undefined

At the same time, I can say that:
because... 1/ -.01 = -100 and 1/ -.00000001 = -1000000000 and so forth... and thus as the numbers get smaller on the bottom, the answer becomes lower... so much that 1/0 = undefined


More formally,
$$\lim_{x\uparrow 0}\frac{1}{x}\rightarrow -\infty$$
but also
$$\lim_{x\downarrow 0}\frac{1}{x}\rightarrow +\infty$$

That's bad news! For me, that's the simplest argument for saying that 1/0 is undefined. Note that I can also get other answers, if you'd like I can probably find a limit such that
$$\lim_{x\rightarrow 0}\frac{1}{x}\rightarrow -\sqrt{\frac{i\pi}{e}}$$

 

[HallsofIvy]

Why in the world would that be simpler? The original question is about basic arithmetic and has nothing to do with limits.

1/0 is undefined because 1/0= x is equivalent to 1= 0*x which is not true for any x.

I don't see how using limits on some specific sequence would be simpler than that!

 

[suyver]

I guess it's a question of mindset and what you're used to. I find my argument simpler / clearer than yours, HallsofIvy, though I directly acknowledge that (many) others might disagree. That is why I included this argument: if people might have problems with your or NateTG's argument, then maybe the one I provided will give some them insight. If not, then no harm done. However, if you think it's stupid, then please feel free to remove my posts.

 

[HallsofIvy]

Well, if you find limits easier than multiplication, you have a remarkable mind!

 

[NateTG]

Originally posted by HallsofIvy
Well, if you find limits easier than multiplication, you have a remarkable mind!

In some sense, limits are a much simpler concept, even if the formalisms associated with it are a bit more involved.

On a conceptual level, I would say that limits are much simpler than multiplication.

In practice, multiplication is much more usefull.

 

[avemt1]

because... 1/ .01 = 100 and 1/ .00000001 = 1000000000 and so forth... and thus as the numbers get smaller on the bottom, the answer becomes higher... so much that 1/0 = undefined

and 1/infinity is equal to 1/0
1/-infinity is equal to 1/0
if i am incorrect please correct me

 

[HallsofIvy]

No, I wouldn't say you are incorrect but you are not very precise (and in mathematics precision is essential!).

"infinity" is not a standard real number and if you are talking about the real numbers, "1/infinity= 0" and "1/(-infinity)= 0" are short hand for "the limit of 1/x as x goes to infinity is 0" and "the limit of 1/x as x goes to -infinity is 0".

If you are using one of the several "extended number systems" in which infinity is defined, then you should say so.

That is one reason why it is not very good mathematics to say "1/0= infinity".

 

[avemt1]

"infinity" is not a standard real number and if you are talking about the real numbers, "1/infinity= 0" and "1/(-infinity)= 0" are short hand for "the limit of 1/x as x goes to infinity is 0" and "the limit of 1/x as x goes to -infinity is 0".

So if zero is the equivalent of infinity and -infinity then the numberline as a whole is not just a straight line, but a loop that resembles the sign for infinity. (KUNUNDRUM?)

If you are using one of the several "extended number systems" in which infinity is defined, then you should say so.

I am treating infinity as a variable, and the number system explained and described above.

 

[NateTG]

It's quite possible to do math with $$\infty$$ but it lacks some of the properties that are normally associated with numbers.

For example, $$\intfy-\infty$$ and $$\frac{\infty}{\infty}$$ will cause problems.

 

[avemt1]

It's quite possible to do math with but it lacks some of the properties that are normally associated with numbers.

I see no problem with using negative infinity, but i do see what you mean for the tricky infinity, because if you multiply infinity by its tricky one then it will turn out as the tricky one.
infinity multiplied by itself commes out as infinity. This is the problem with working within the boudaries of infinities, you have no room to move. That is why infinities do not work to explain the world around us.
I do understand this consept, as you can see, but i do not understand using negative infinity as a problem.

 

[HallsofIvy]

Would you mind telling us what in the world you mean by the "tricky infinity"?

By the way, you copied what I said about infinity not being in the standard real numbers and then said

So if zero is the equivalent of infinity and -infinity then the numberline as a whole is not just a straight line, but a loop that resembles the sign for infinity.

How in the world did you get that "zero is the equivalent of infinity and - infinity" from what I said? No, zero is not the equivalent of infinity in any sense!

 

[avemt1]

sorry-- I didn't explain

Would you mind telling us what in the world you mean by the "tricky infinity"?

infinity divided by infinity
it is an algebraic function

By the way, you copied what I said about infinity not being in the standard real numbers and then said

So if zero is the equivalent of infinity and -infinity then the numberline as a whole is not just a straight line, but a loop that resembles the sign for infinity.

I was merely visualizing the numberline as a whole.

How in the world did you get that "zero is the equivalent of infinity and - infinity" from what I said? No, zero is not the equivalent of infinity in any sense!

and 1/infinity is equal to 1/0
1/-infinity is equal to 1/0

I isolated the denomenators

 

[HallsofIvy]

infinity divided by infinity
it is an algebraic function

No, it not a function at all. If you are talking about the regular real numbers then it is just a shorthand method of talking about a the limit of a quotient in which numerator and denominator each go to infinity. The result might be any number depending on the specific form of the numerator and denominator functions.

I was merely visualizing the numberline as a whole.

I wasn't complaining about your saying talking about the "figure eight", I was talking about you quoting ms saying that infinity and -infinity not being real numbers and then saying "So if zero is the equivalent of infinity and -infinity" which is pretty much the opposite of what I said in the quote.

Actually there are 2 standard ways of "extending" the real numbers. One adds a single infinity that is lies at both "ends" of the number line. That (called the "one point compactification") makes the extended number line geometrically equivalent to a circle.
The other (the "Stone-Cech" compactification), more common, method adds +infinity at one end and -infinity at the other and makes the extended number system geometrically equivalent to a close line segment.

and 1/infinity is equal to 1/0
1/-infinity is equal to 1/0

I isolated the denomenators

No, 1/infinity is not equal to 1/0 and -1/infinity is not equal to 1/0.

Sometimes you will see "1/infinity= 0" (not 1/0 !) or "1/0= infinity" but those are again "shorthand" for more complicated limit statements (and which I prefer not to use because they are so easily misunderstood).

 

[avemt1]

No, it not a function at all. If you are talking about the regular real numbers then it is just a shorthand method of talking about a the limit of a quotient in which numerator and denominator each go to infinity. The result might be any number depending on the specific form of the numerator and denominator functions.

I do understand the mathematical deffinition of infinity now, and I do realize that infinity can not be treated, in any way like a variable.

Sometimes you will see "1/infinity= 0" (not 1/0 !) or "1/0= infinity" but those are again "shorthand" for more complicated limit statements (and which I prefer not to use because they are so easily misunderstood).

Originally posted by Hessem
because... 1/ .01 = 100 and 1/ .00000001 = 1000000000 and so forth... and thus as the numbers get smaller on the bottom, the answer becomes higher... so much that 1/0 = undefined

Somewhere down the line the denomenator would become infinitely small, and never reach 1/0,and the solution would be infinitely large, and not be able to be defined by any logical means. That means that 1/0 is not able to be defined by any progressive means. Other than the limit statements you were quoting about.
I think that is a competent solution to the information given.

I still have to do more research.
I need to see those limit statements.

 

[luther_paul]

Lets try arguing in not mathematical terms but logical reasoning. if a number x is divided by any number n, it means giving every individual n a single part from x... if our number is zero how are we going to divide x to zero individuals...? we have a x objects to be divided among no one... either the number stays as it is or a the number is gone. but the number cannot be gone because it was distributed to no one.. how is this?

Please do not insult my argument, I am just a high school student sharing my views in life!

 

[deda]

In my opinion 'const/0' is defined as $$\infty$$.
The true totally undefined expresion in math are
$$\frac{0}{\infty}$$
and its reciprocional one. Those two expressions cannot be any real nor imaginary number but they are some phantom numbers.

 

[suyver]

Originally posted by deda
The true totally undefined expresion in math are
$$\frac{0}{\infty}$$
and its reciprocional one. Those two expressions cannot be any real nor imaginary number but they are some phantom numbers.

What's wrong with $$\frac{0}{\infty}=0$$ ?

 

[avemt1]

Originally posted by deda In my opinion 'const/0' is defined as$$\infty$$
. The true totally undefined expresion in math are $$\frac{0}{\infty}=0$$ and its reciprocional one. Those two expressions cannot be any real nor imaginary number but they are some phantom numbers.

tecnically $$\frac{\chi} {\infty}=0$$
(sorry first time)
theorems:
any finite digit divided by infinity is = to zero
any difined digit divided by itself is = to one

 

[deda]

Originally posted by suyver
What's wrong with $$\frac{0}{\infty}=0$$ ?

well you can write it this way: $$0=0\infty=1$$
$$\frac{0}{\infty}$$ cannot be either $$\infty$$ cause then $$\infty\infty=\infty=0$$
it's obvious that $$\frac{0}{\infty}$$ cannot be any real const number between zero and infinity.

 

[suyver]

Originally posted by deda
well you can write it this way: $$0=0\infty=1$$

I do not understand how you get to this.

Assume $$\frac{0}{\infty}=0$$

Then $$\frac{0}{\infty}=\frac{0}{1}$$

Thus $$0 * 1 = \infty * 0$$

Thus $$0 = \infty * 0$$

How did you get to $$0 * \infty = 1$$ ?

 

[himanshu121]

Originally posted by deda
well you can write it this way: $$0=0\infty=1$$
$$\frac{0}{\infty}$$ cannot be either $$\infty$$ cause then $$\infty\infty=\infty=0$$
it's obvious that $$\frac{0}{\infty}$$ cannot be any real const number between zero and infinity.

Is $$0*\infty$$

defined

 

[deda]

Originally posted by himanshu121
Is $$0*\infty$$

defined

yes it's any number except zero or infinity.
as the same number over zero gives infinity.

 

[suyver]

Originally posted by deda
yes it's any number except zero or infinity.
as the same number over zero gives infinity.

Where did you learn this? I certainly never saw it in a textbook on conventional math.

 

[himanshu121]

Originally posted by deda
yes it's any number except zero or infinity.
as the same number over zero gives infinity.

It doesn't exist

 

[HallsofIvy]

Originally posted by deda
yes it's any number except zero or infinity.
as the same number over zero gives infinity.

Definitions don't work that way. A definition cannot be ambiguous. Either it is a specific number or it is not defined. Since we cannot assert that 0*infinity is a specific number, it is undefined.

Some textbooks use the term "undetermined" to distinguish between "undefined because there is no number having that property" (called simply "undefined") and "undefined because there are many numbers having that property" (called "undetermined") but they is no definition in either case.

 

[deda]

Originally posted by suyver
Where did you learn this? I certainly never saw it in a textbook on conventional math.

it's simple logic:
$$0\infty=n<=>\frac{n}{0}=\infty$$
if n<>0 and n<>infinity.

 

[HallsofIvy]

Another result of simple logic:

"Toledo is a nation in South America" <=> "The sun will rise in the west tomorrow".

$$0\infty$$ is not equal to n and
$$\frac{n}{0}$$ is not equal to $$\infty$$

$$0\infty$$ and $$\frac{n}{0}$$ are not defined.

 

[suyver]

And what about $0/\infty$ ?

Would you agree that $0/\infty=0$ ?

 

[HallsofIvy]

The point that has been made repeatedly is that $\infty$
is not a standard real number. Before it is possible to answer that question, you have to specify which of the several extensions to the real number system you are working in.

 

[suyver]

(Sorry to keep bugging you, but I just want to understand this.)

So, you are saying that, depending on the extension to the real number system I am working in, $0/\infty$ could mean different things?

Just out of curiousity, can you show a kind of extension that would give $0/\infty\neq 0$ ?