- #1

avemt1

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**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?

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- Thread starter avemt1
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- #1

avemt1

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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?

- #2

NateTG

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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.

- #3

HallsofIvy

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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".

- #4

Hessam

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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

- #5

HallsofIvy

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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

- #6

suyver

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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,

[tex]\lim_{x\uparrow 0}\frac{1}{x}\rightarrow -\infty[/tex]

but also

[tex]\lim_{x\downarrow 0}\frac{1}{x}\rightarrow +\infty[/tex]

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

[tex]\lim_{x\rightarrow 0}\frac{1}{x}\rightarrow -\sqrt{\frac{i\pi}{e}}[/tex]

- #7

HallsofIvy

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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!

- #8

suyver

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- #9

HallsofIvy

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

- #10

NateTG

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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.

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- #11

avemt1

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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

- #12

HallsofIvy

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"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".

- #13

avemt1

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"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.

- #14

NateTG

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For example, [tex]\intfy-\infty[/tex] and [tex]\frac{\infty}{\infty}[/tex] will cause problems.

- #15

avemt1

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

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.

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- #16

HallsofIvy

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

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!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.

- #17

avemt1

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

it is an algebraic function

I was merely visualizing the numberline as a whole.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!

I isolated the denomenatorsand 1/infinity is equal to 1/0

1/-infinity is equal to 1/0

- #18

HallsofIvy

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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.infinity divided by infinity

it is an algebraic function

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.I was merely visualizing the numberline as a whole.

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.

No, 1/infinity isand 1/infinity is equal to 1/0

1/-infinity is equal to 1/0

I isolated the denomenators

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).

- #19

avemt1

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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.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.

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).

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.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

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.

- #20

luther_paul

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

- #21

deda

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The true totally undefined expresion in math are

[tex]\frac{0}{\infty}[/tex]

and its reciprocional one. Those two expressions cannot be any real nor imaginary number but they are some phantom numbers.

- #22

suyver

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Originally posted by deda

The true totally undefined expresion in math are

[tex]\frac{0}{\infty}[/tex]

and its reciprocional one. Those two expressions cannot be any real nor imaginary number but they are some phantom numbers.

What's wrong with [tex]\frac{0}{\infty}=0[/tex] ?

- #23

avemt1

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tecnically [tex]\frac{\chi} {\infty}=0[/tex]Originally posted by dedaIn my opinion 'const/0' is defined as[tex]\infty[/tex]

. The true totally undefined expresion in math are [tex]\frac{0}{\infty}=0[/tex] and its reciprocional one. Those two expressions cannot be any real nor imaginary number but they are some phantom numbers.

(sorry first time)

theorems:

any finite digit divided by infinity is = to zero

any difined digit divided by itself is = to one

- #24

deda

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well you can write it this way: [tex]0=0\infty=1[/tex]Originally posted by suyver

What's wrong with [tex]\frac{0}{\infty}=0[/tex] ?

[tex]\frac{0}{\infty}[/tex] cannot be either [tex]\infty[/tex] cause then [tex]\infty\infty=\infty=0[/tex]

it's obvious that [tex]\frac{0}{\infty}[/tex] cannot be any real const number between zero and infinity.

- #25

suyver

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Originally posted by deda

well you can write it this way: [tex]0=0\infty=1[/tex]

I do not understand how you get to this.

Assume [tex]\frac{0}{\infty}=0[/tex]

Then [tex]\frac{0}{\infty}=\frac{0}{1}[/tex]

Thus [tex] 0 * 1 = \infty * 0 [/tex]

Thus [tex] 0 = \infty * 0 [/tex]

How did you get to [tex] 0 * \infty = 1 [/tex] ?

- #26

himanshu121

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Originally posted by deda

well you can write it this way: [tex]0=0\infty=1[/tex]

[tex]\frac{0}{\infty}[/tex] cannot be either [tex]\infty[/tex] cause then [tex]\infty\infty=\infty=0[/tex]

it's obvious that [tex]\frac{0}{\infty}[/tex] cannot be any real const number between zero and infinity.

Is [tex]0*\infty[/tex]

defined

- #27

deda

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yes it's any number except zero or infinity.Originally posted by himanshu121

Is [tex]0*\infty[/tex]

defined

as the same number over zero gives infinity.

- #28

suyver

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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.

- #29

himanshu121

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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

- #30

HallsofIvy

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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

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.

- #31

deda

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it's simple logic:Originally posted by suyver

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

[tex]0\infty=n<=>\frac{n}{0}=\infty[/tex]

if n<>0 and n<>infinity.

- #32

HallsofIvy

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

[tex]0\infty[/tex] is not equal to n and

[tex]\frac{n}{0}[/tex] is not equal to [tex]\infty[/tex]

[tex]0\infty[/tex] and [tex]\frac{n}{0} [/tex] are

- #33

suyver

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And what about [itex]0/\infty[/itex] ?

Would you agree that [itex]0/\infty=0[/itex] ?

Would you agree that [itex]0/\infty=0[/itex] ?

- #34

HallsofIvy

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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.

- #35

suyver

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

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

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