# Operations with infinites and zero and etc.

Born2Perform
hi i'm guessing if anybody knows some work of some matematician about operations with infinites, not just the 0/0=inf/inf=inf*0 and so, but something advanced, like a deep study in infinites and zero math.

i tried with google scholar, in the math sites, on the forums but i couldn't find anything significant. i don't want to pass my days studing things that surely have alredy made, please give me some links.

## Answers and Replies

Homework Helper
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What is a "deep" study in infinities and zero maths??

wherever did you get the meaningless expessions 0/0=inf/inf=0*inf from??
From "deep" studies?

Born2Perform
arildno said:
What is a "deep" study in infinities and zero maths??

wherever did you get the meaningless expessions 0/0=inf/inf=0*inf from??
From "deep" studies?

thx i appreciate your help :uhh:
its not meaningless why should be meaningless?

however did somebody in the history of math study all types of operation beetween infinite, zero and real numbers? i mean an algebra that includes extreme cases. thx

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Since you evidently don't know a thing about indeterminate forms, I suggest you pick up a textbook in maths and start learning and understanding something, rather than fantasize in totally wrong directions.

Born2Perform
since you are the mathlord here you should have understood that i need exactly what u sayd. if a young guy does not know the name of algebra, he asks about algerbra showing a+b=c.
i don't know a thing of indeterminate forms and i asked some link.
where can i learn those thing in the web(without limits)? thx.

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Homework Helper
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without limits???

learn the concept of limits!

Born2Perform
whaaaat??? i gave calculus 1 3 weeks ago and i can demonstrate around 20 limits theorems, i know exactly what you mean for indeterminate form as limit result. i kn this licterature very well.

my question was...did somebody go deep inside into the infinite and zero operations?

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Obviously, you don't know what limits are, since you are able to set up a meaningless expression as 0/0=0*inf

Born2Perform
that thing was an example to give a name at what i asked.

and however would you explain me why that relation should be meaningless?

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Born2Perform said:
my question was...did somebody go deep inside into the infinite and zero operations?

Yes. For example, the set of integers Z with the operation of addition '+' is a group. So, the following holds:

(i) a + b is Z, for every a and b, so $$a + \infty$$ is in Z.

(ii) a + (b + c) = (a + b) + c, for every a, b and c, and so a + (b + $$\infty$$) = (a + b) + $$\infty$$.

(iii) there exists 0 in Z such that a + 0 = 0 + a = a, for every a in Z, so $$\infty$$ + 0 = 0 + $$\infty$$ = $$\infty$$

(iv) for every a in Z there exists -a such that a + (-a) = (-a) + a = 0, and so $$\infty$$ + $$-\infty$$ = $$-\infty$$ + $$\infty$$ = 0

No offence, but does this look silly enough to you to end this discussion? Last edited:
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Born2Perform said:
and however would you explain me why that relation should be meaningless?
a) You can't divide by zero. Hence 0/0 is NOT a proper representation of a number, it is an indeterminate form.
b) Infinity is not a real number. Hence, you cannot multiply it with a real number using real number multiplication. Thus, 0*inf is NOT a proper representation of a number it is an indeterminate form.

c) Considered as just symbolizing an "end result" of a particular limiting process, no two indeterminate forms can be related to each other by the equality relation; indeed, we cannot even say 0/0=0/0

that would be a triple meaninglessness by the way.

Born2Perform
Yes. For example, the set of integers Z with the operation of addition '+' is a group. So, the following holds:

(i) a + b is Z, for every a and b, so $$a + \infty$$ is in Z.

(iv) for every a in Z there exists -a such that a + (-a) = (-a) + a = 0, and so $$\infty$$ + $$-\infty$$ = $$-\infty$$ + $$\infty$$ = 0

infinity is in Z? infinity is not a number, if you include it to z shouldn't you ask what comes before or after it? infinity is not in a group

inf + (-inf) = 0??? this is an indeterminate form. are you treating it as a number?

--- sorry arildno you said infinity is not a real number and you can't multiply it as a real number, but you can sobstitute it with a couple of real numbers (n/0 for example) that you can multiply as real numbers making no mistakes.

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Homework Helper
Born2Perform said:
infinity is in Z? infinity is not a number, if you include it to z shouldn't you ask what comes before or after it? infinity is not in a group

Infinity is not a number? Really? You must be kidding me.  [:uhh:]

Born2Perform said:
...but you can sobstitute it with a couple of real numbers
(n/0 for example) that you can multiply as real numbers making no mistakes.

Since you're so convinced about dividing by zero, maybe you could bring out your new theory. Last edited:
Born2Perform
(iv) for every a in Z there exists -a such that a + (-a) = (-a) + a = 0, and so $$\infty$$ + $$-\infty$$ = $$-\infty$$ + $$\infty$$ = 0

lol seem here you are treating infinite as a real number. then you tell me it's not??

arildno said:
You can't divide by zero. Hence 0/0 is NOT a proper representation of a number, it is an indeterminate form.

i can't divide by zero? infinite is alredy divided by zero. and sobstituting n/0 to it you can threat infinites as real numbers with real num properties.
thx however i realized that actually i know nothing of math, but you dont know a &%@£ of how to manage infinites.

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Born2Perform said:
lol seem here you are treating infinite as a real number. then you tell me it's not??

I w-a-s j-u-s-t k-i-d-d-i-n-g.

Homework Helper
Just out of curiosity, what in the world did you mean by
Born2Perform said:
i gave calculus 1 3 weeks ago
?

Born2Perform said:
i can't divide by zero? infinite is alredy divided by zero. and sobstituting n/0 to it you can threat infinites as real numbers with real num properties.
Now you can't divide by 0. I have no idea what you mean by "infinite is already divided by zero".l You cannot treat infinities as real numbers, "infinities" are not real numbers and do not have "real number properties".

You might be interested in "hyperreals". Look it up in google.

You might look up the extended reals or the projective real number line.

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Born2Perform said:
i can't divide by zero? infinite is alredy divided by zero. and sobstituting n/0 to it you can threat infinites as real numbers with real num properties.
thx however i realized that actually i know nothing of math, but you dont know a &%@£ of how to manage infinites.

You cannot assert that n/0 is infinity, and can thus be treated like a real number, with real number properties.

Firstly, in the real numbers if a is not zero and ab=ac, then b=c. Now try thinking about *your* definition of infnity, to see what happens?

You are free to defein whatever it is you care in maths, and to see what follows from the definition. However, just because you can do something doesn't make it reasonanble to do it. THere are perfectly good constructs in mathematics that utilise a symbol that is called infinity, or several symbols all called infinity: the extended real numbers, the extended complex plane, cardinals and ordinals. Google will give you information about all of them. I would advise you not to just throw symbols around without declaring what they mean and asseriting things that you have not justified, like n/0 is infinity and behaves like any other real number.

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You should do well in your "study" of infinities to actually understand a couple of properties of sets of "infinite size" that are well-known:

1. The "size" of a set is said to be "infinite" if there exists no bijections between that set and any finite subset of the naturals.

Note here that an almost defining property of "infinity" is a negative one:
"That which is not finite, we call infinite."

2. If there exists a bijection between a set A and a strict subset B of itself, then the "size" of the set A is "infinite".
This can be regarded as a constructive, or "positive" property of "infinity"

Born2Perform
n/0 cant be multiplied as a real number but what i meant is that n and 0 can.
right or wrong sobstituting n/0 to the infinites you can do any type of real operation. for example saying that "inf * inf = inf" is intuitive but i can verify it sobstituting, it gives me "n²/0 = inf"

can i have an explaination of why n/0 is not infinite? all reasons say yes, starting with limits, in which when we get a n/0 there is an asyntot.
then algerbically n/0.1; n/0.001; n/0.00001......gives infinite, at least the idea.
why n/0 can't be considered the definition of infinite?

Staff Emeritus
Gold Member
can i have an explaination of why n/0 is not infinite?
Yes. If / is intended to be "division of real numbers", then (n, 0) is not in the domain of /. So, the expression n/0 is utterly meaningless. Formally, it's no different than asking for the value of "3/{chicken sandwich}".

Have you looked up the "extended reals", as someone suggested earlier? If / means "division of extended real numbers", then (n, 0) is still not in its domain. But it is true that, when n > 0:

$$\lim_{x \rightarrow 0^+} \frac{n}{x}$$

converges to the extended real number $+\infty$.

Have you looked up the "projective real number line", as someone suggested earlier? If / means "division of projective reals", then $n/0 = \omega$ for any nonzero n.

edit: typo fixed

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$$\lim_{x\to{0}^{+}}\frac{n}{x}$$