# What does 1 / infinity =

1 / infinity = ?

is there anything that we can reduce this to?
(i doubt there is, but you never know)

well, it's REALLY close to 0

for all practical purposes 1 / infinity = 0

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just thought of something!

1 / infinity = x

1 = x * infinity

~~~~~ analyze this ^^

x must be a REALLY small number to get infinity back down to 1

i.e. x = .0000000000000000000001

aka an infinite number of 0's and then a 1???

or an infinite number of 0's and then a 2?

or an infinite number of 0's and then a --
3?
47?
31415926535897932?
infinity? (hah .00000000000000000infinitity)

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i suppose you can't really get anywhere when dealing with these "infinities"

im stuck

climbhi
you're trying to make too much sense out of something which you can't do it with. The only way you can really think about it as is the limit of 1/x as x approaches infinity, not this crazy other stuff that you are floating around.

Integral
Staff Emeritus
Gold Member
1/infinity =0

There is no need to play games with it.

Infinity is a carefully defined extension of the real number system. since it is not a real number all operations involving infinity and the real number system are defined.

steppenwolf
1/infinity=infinitesimal

you were right about this, an infinitesimal is an infinite number of zeros after the decimal point with a one at the end, but of course there is no end but ah well!

infinitesimals are very interesting, archemedes used them to find the volume of the sphere, newton also used them to work out velocity and related stuff.

now days it is used to explain bronwian motion and other nasty topics.

1/infinity =0

No argument.

Where infinity is an element of the extended real line (real numbers with -infinity and infinity added).

Hurkyl
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1 / infinity = ?

is there anything that we can reduce this to?
(i doubt there is, but you never know)

well, it's REALLY close to 0

for all practical purposes 1 / infinity = 0

Infinity isn't a uniquely defined concept. There are many ways to include infinity (infinities) into a number system, each with their own properties.

The most common one is the example plus gave of the extended real numbers; in that system 1 / infinity is defined simply to be zero, because that's the choice that gives the right answer for the applications for which the extended real numbers were invented.

Other number systems behave differently. 1 / infinity is undefined for any infinite quantity in the ordinal number system. In other transfinite systems each infinite value has a unique nonzero reciprical (an infinitessimal).

just thought of something!

1 / infinity = x

1 = x * infinity

In the real numbers or other nice systems (like division rings and fields!) that step is fine, but in general number systems you cannot make that step. That step is patently illegal in the extended real numbers when infinity is involved.

And as I mentioned, you're no longer working in the real numbers, so you cannot generally use decimal expansions to describe the numbers you're using (since decimal expansions always represent real numbers). There are, though, number systems which generalize decimal expansions. For example in my favorite transfinite system:

1 / infinity = 0 + 1 * infinitessimal
= ... ; ...000.000... ; ...0000.00000... | ...00001.00000... ; ...000.000... ; ...
= 0 | 1

This is kind of a "nested" decimal representation. The numbers to the left of the pipe (|) are the ordinary real digit of the transfinite number. The next grouping left are multiplied by infinity, the next grouping left are multiplied by infinity^2, etc. The grouping to the right of the pipe is divided by infinity, the next to the right is divided by infinity^2, etc.

This system will let you add, subtract, multiply, and divide infinities, but more advanced operations are not always allowed (e.g. sqrt(infinity) is not defined)

Hurkyl

1/infinity is Not equal to zero

1/infinity is not a number

however

limitx->inf1/x = 0

there's a key difference

&infinity;
&inf;
fty &fty;

Integral
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Gold Member
From
Principles of Mathematical Analysis (Thrid Edition) by Walter Rudin
on page 12 in the section entitled The Extented Real Number system He states

"The extented real number system does not form a field, but it is customary to make the following conventions:
(a) If x is real then

x + oo = oo, x - oo = -oo, x/+oo = x/-oo =0

(b) If x>0 then x * (+oo) = +oo, x * (-oo) = -oo
(c) If x<0 then x * (+oo) = -oo, x * (-oo) = +oo.
When it is desired to make the distinction between real numbers on the one hand and the symbols +oo and -oo on the other quite explicit, the former are called finite."

I have used oo to mean the sideways figure eight symbol.

Notice that this is a matter of convention and is DEFINED to be true. There is no need of limiting processes.

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arivero
Gold Member
Not so easy

The real quests are how to do a Taylor expansion of f(1/x) around x=infinity. Or also expansions of g(x) with x=infinity, or around h(1/x) with x=0... Poincare did a good deal of mathematics about this kind of questions.

Originally posted by Integral

I have used oo to mean the sideways figure eight symbol.

[/B]

If you put the oo's between brackets, you'll get the infinity symbol, but you got the point across, and that's what's important.

Integral
Staff Emeritus
Gold Member
Mentat,
New board, new rules. While we appear to have the greek alphbet, the brackets did not give me the infinty symbol

&inf;

roeighty
infinity is "not limited by any succeeding number"

1/ infinity is "infinitely near 0" ("not limited by any decreasing number")

so ..infinity or 1/infinity are not numbers but terms describing limitlessness, just as numbers describe entities (or distinct parts of).

You have to be careful with the assumption of 1/oo = 0 because that would sort of imply that 0(oo) = 1.

Integral
Staff Emeritus
Gold Member
If you will look at my quote from the Real Analysis text, ALL results of mathematical operations involving infinity are defined. If you stick to the definitons all will be well. Since infinity is NOT a real number it does not follow the same rules as the rest of the reals. It has its own set and they are definied right along with (mathematical) infinity.

Note that earlier in the section I quoted oo was definined as

oo>x for all real x>0 and -oo<x for all real x>0
in this case our x = 0 the definition of infinity does not provide the answer, we must rely on the definition of the 0 element such that 0 * anything = 0, by this definition we have 0 * oo = 0 since.

Integral
Staff Emeritus
Gold Member
Please read my posts. To use infinity with real numbers you must carefully adhere to the definitions.

0 * oo <> 1

0 * oo = 0

Hurkyl
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0 * oo is indeterminate, not 0.

(and for those who don't know, indeterminate does not mean "we're not sure"; it has a precise technical meaning related to the contexts where the extended real numbers are used)

Hurkyl

AAAHHH!!! 1/infinity=0!!! 1/infinity does not=0!!! why can't this forum make up its mind!!!

I used to think 1/infinity=0, but now im not so sure thanks to what this guy said.
You have to be careful with the assumption of 1/oo = 0 because that would sort of imply that 0(oo) = 1.

great point, i hadn't thought of that.

What is oo -oo ?
Since none is real it cannot be oo or -oo.
Is cannot be zero as well
How is it that you solve Hilbert's paradox?

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HallsofIvy
Homework Helper
AAAHHH!!! 1/infinity=0!!! 1/infinity does not=0!!! why can't this forum make up its mind!!!

I used to think 1/infinity=0, but now im not so sure thanks to what this guy said.

great point, i hadn't thought of that.
This forum "can't make up its mind" because this forum is not one single person and so has many minds. Also whether or not "1/infinity= 0" or not depends upon which "infinity" you are talking about. There is NO "infinity" in the standard real number system but you can have other, more general systems, with different kinds of "infinity".

i think everyone is mistaking the math of this. 1/∞≠0, however, lim x→∞ (1/x)=0. which is to say that 1/∞ is the closest one can get to zero without being zero.

Hurkyl
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Gold Member
i think everyone is mistaking the math of this. 1/∞≠0, however, lim x→∞ (1/x)=0. which is to say that 1/∞ is the closest one can get to zero without being zero.
What number system are you using? This is definitely not true of, for example, the extended real number system. (which is essentially the one you use when doing calculus)

The key thing to understand from what I have been told is the 'real' number system defines particular things about infinity. I think that means that 1/∞ = 0 in the 'real' number system? I hope that is correct. It is just the rules of the 'real' number system so there is no point in arguing about it and that is all that needs to be stated.

Another key point - that I have had confirmed by others - is that 'real' does not mean what we usually think by the term. It is just a label for a number system that we use for most of what we do. It does not mean that it perfectly matches the 'real' universe.
Please correct me if that is wrong but it was the conclusion that we got to.

Afterall we can not physically deal with ∞ in any way. At no point can you say you have done something infinitely or to the infinite degree (nor even infinitesimally). However for the purposes of math, and physics which uses the math system, it is perfectly adequate to make the 'real' number rules that exist. From what I am led to believe, calculus would be hampered without these rules and calculus works perfectly well for us so there is no need to diverge from it for any of our needs.

So don't worry about it. The 'real' number system works just fine for us as we don't actually have to worry about the unattainable infinite.

I think something like 2∞ = ∞ is not an acceptable statement in any number system 'real' or real real. Correct please if I'm wrong on this.

one divided by infinity from what I know can never equal zero due to the fact that zero=nothing and infinity implies something, just nothing measurable.
so 1/oo= an infinity small number but not zero.

I am currently in pre-ap algebra 2, so I believe this has a high probability of being wrong..never the less I wished to try.

Hurkyl
Staff Emeritus
In the extended real numbers -- the number system you implicitly learn when doing calculus -- $1 / (+\infty) = 0$.