Circle with infinite circumferance

raah!

I was wondering if anyone here knew what this was all about. At first it makes sense, but then when you think about it it stops working

heh, it's that "a circle with an infinte circumferance is an infinite straight line". Now this is fine, as when the circumferance tends to infinity the slope tends to 0. But if it's infinite, isn't it not a circle?. Don't circles have to join up and be circles?

I couldn't find anything last time I looked this up on the internet but a link to another forum. Wasn't very helpful.

Gurgle

Try the philosophy forum.

Its like parallel lines meeting at infinity.
They dont really meet, but thats ok because infinity doesn't really exist.

In a *similar* way, a straight line will meet it's own origon at infinity, making it, by definition, a circle.

Now, if you start with a circle:
As a circle gets bigger, the line gets straighter.
As it gets infinitely big, the line gets infinitely straight.

The only practical use for infinity is as a denominator.

raah!

Would not the maths forum be more suitable for a mathematics question?

Gurgle

raah! wrote: »

Would not the maths forum be more suitable for a mathematics question?

Yeah, of course.
Maths annoys me when infinity gets involved, it turns into philosophy.

raah!

Yes this is why this particular circle thing bugged me too.

"In a *similar* way, a straight line will meet it's own origon at infinity, making it, by definition, a circle."

This is very interesting though. Is this due to the universe being spherical or something?

Thomas_S_Hunterson

raah! wrote: »

This is very interesting though. Is this due to the universe being spherical or something?

No, maths is completely unrestricted by the universe.

As far as I know, it's to do with the fact that a line stretches to infinity at both ends.

raah!

would these two points then not just continue to go further away from each other as they expanded infinitely?. Or is infinity a point now all of a sudden?

Thomas_S_Hunterson

But can you really differentiate between things that don't really exist...

How can you say conclusively that one infinity is different from another?

MathsManiac

Firstly, who told you that a circle with an infinite circumference is a straight line? And what context were they making that statement within?

Such a statement would not stand up in any axiomatisation of regular Euclidean geometry.

If somebeody made such a statement, then they would have to define what sort of geometry they were talking about, including appropriate definitions and properties to allow them to make specific statements about infinity, (certainly not the common-or-garden school geometry). They would have to give their definition of a circle in this geometry, and then show that under this definition in this geometry, their statement is justifiable.

The normal definition of a circle in regular Euclidean geometry is that it is the set of points whose distance from a given point C (the centre) is equal to a given fixed value r (the radius). Under this definition and in theis geometry, there is certainly no such thing as a circle of infinite circumference.

In short, given the natural assumption that one is talking about ordinary geometry, you are quite right to regard the statement as daft. (And yes, in ordinary geometry, circles do have to "join up" to be circles.)

Gurgle

raah! wrote: »

Or is infinity a point now all of a sudden?

Infinity as a mathematical concept is the inverse of zero, which means it is:
> Where parallel lines meet
> Where the ends of a straight line meet
> Where a circle becomes indistinguishable from a straight line
> The energy required to accelerate beyond the speed of light (:D)

and much more besides.

Gurgle

raah! wrote: »

"In a *similar* way, a straight line will meet it's own origon at infinity, making it, by definition, a circle."

This is very interesting though. Is this due to the universe being spherical or something?

Its more to do with infinity not actually existing, more or less by definition.

twasantis

Gurgle wrote: »

Infinity as a mathematical concept is the inverse of zero, which means it is:
> Where parallel lines meet
> Where the ends of a straight line meet
> Where a circle becomes indistinguishable from a straight line
> The energy required to accelerate beyond the speed of light (:D)

and much more besides.

this can become a very mind bending topic of discusion...

parallel lines meeting in theory mean they aint parallel!!

its like trying to find the the tan of 90' ..it dont exsist because it never reaches the point..

so a circle is not a circle if its two points dont meet!!!! so an "infinite" circumferance is not plausable!! its a line!!!!!!!!!!!!!!!

so when its said infinity is the inverse of zero is just not the case....the amount of nothing in something is not possible to calculate!!

rjt

Gurgle wrote: »

Infinity as a mathematical concept is the inverse of zero, which means it is:
> Where parallel lines meet
> Where the ends of a straight line meet
> Where a circle becomes indistinguishable from a straight line
> The energy required to accelerate beyond the speed of light (:D)

and much more besides.

I'm far from an expert, but I've never, ever seen infinity defined as "the inverse of 0". Generally the reals are defined/constructed as a field - where 0 has no inverse. Infinity only enters the scene when we're talking about the extended reals, which is needed for analysis - things like limits. Ie. if n tends to infinity (gets arbitrarily large, roughly speaking) does a sequence a_{n} get arbitrarily close to some number L (and stay there!). Certainly, we don't say that infinity is the inverse of 0.

As for parallel lines and the ends of straight lines meeting, going back to what MathsManiac said, if we're talking about Euclidean geometry (what we're all used to), these points (pardon the pun) are certainly false. In other systems however, lines can meet themselves and such (eg. geometry on the surface of a sphere rather than a plane).

Gurgle

The two ends of a line go in opposite directions, how far do they have to travel before meet: An infinite distance, so they meet at infinity.

twasantis wrote: »

its like trying to find the the tan of 90' ..it dont exsist because it never reaches the point..

90° is a real, and very common, angle.
tan of 90° = sin 90° / cos 90° = 1 / 0 = infinity

twasantis wrote: »

so a circle is not a circle if its two points dont meet!!!! so an "infinite" circumferance is not plausable!! its a line!!!!!!!!!!!!!!!

yes... a line becomes a circle at infinity, a circle becomes a line at infinity.

twasantis wrote: »

:so when its said infinity is the inverse of zero is just not the case

Trust me, it is.
If you have 1 sweet and you give 0 sweets to each person, how many people will get zero sweets before you run out?
Divide 1 by 0: 1/0 -> Infinity

rjt wrote:

Infinity only enters the scene when we're talking about the extended reals, which is needed for analysis - things like limits.

Glad you brought limits into it:
1/10 < 1/9 < 1/8 < 1/7 < 1/6 < 1/5 < 1/4 < 1/3 < 1/2 < 1/1 < 1/0

Try Excel: plot the first 9 fractions and you will see that the result is a curve that goes infinitely high as you approach Zero (Excel won't let you plot 1/0 of course)

Or try it logarithmically to really show the limit, plot these numbers:
1/1000
1/100
1/10
1/(0.1)
1/(0.01)
1/(0.001)
1/(0.0001)
1/(0.00001)
1/(0.000001)
1/(0.0000001)
1/(0.00000001)
1/(0.000000001)

rjt

Gurgle wrote: »

The two ends of a line go in opposite directions, how far do they have to travel before meet: An infinite distance, so they meet at infinity.

90° is a real, and very common, angle.
tan of 90° = sin 90° / cos 90° = 1 / 0 = infinity

yes... a line becomes a circle at infinity, a circle becomes a line at infinity.

Trust me, it is.
If you have 1 sweet and you give 0 sweets to each person, how many people will get zero sweets before you run out?
Divide 1 by 0: 1/0 -> Infinity



Glad you brought limits into it:
1/10 < 1/9 < 1/8 < 1/7 < 1/6 < 1/5 < 1/4 < 1/3 < 1/2 < 1/1 < 1/0

Try Excel: plot the first 9 fractions and you will see that the result is a curve that goes infinitely high as you approach Zero (Excel won't let you plot 1/0 of course)

Or try it logarithmically to really show the limit, plot these numbers:
1/1000
1/100
1/10
1/(0.1)
1/(0.01)
1/(0.001)
1/(0.0001)
1/(0.00001)
1/(0.000001)
1/(0.0000001)
1/(0.00000001)
1/(0.000000001)

You've failed to show anything really, simply explain why *intuitively* it might make sense for 1/0 to be infinity. The limit example simply indicates that lim 1/x as x->0 is 0. This is true, but doesn't imply 1/0 = infinity (if we can assume that f(x)=1/x is defined and continuous at 0 then you're right, it does mean 1/0 = 0; unfortunately f isn't definied at 0).

Anyway, this is maths, so there's always a way to resolve arguments. If you say the inverse of 0 is infinity, then I say prove it. Make it clear what you're assuming.

twasantis

Gurgle wrote: »

The two ends of a line go in opposite directions, how far do they have to travel before meet: An infinite distance, so they meet at infinity.

90° is a real, and very common, angle.
tan of 90° = sin 90° / cos 90° = 1 / 0 = infinity

yes... a line becomes a circle at infinity, a circle becomes a line at infinity.

Trust me, it is.
If you have 1 sweet and you give 0 sweets to each person, how many people will get zero sweets before you run out?
Divide 1 by 0: 1/0 -> Infinity



Glad you brought limits into it:
1/10 < 1/9 < 1/8 < 1/7 < 1/6 < 1/5 < 1/4 < 1/3 < 1/2 < 1/1 < 1/0

Try Excel: plot the first 9 fractions and you will see that the result is a curve that goes infinitely high as you approach Zero (Excel won't let you plot 1/0 of course)

Or try it logarithmically to really show the limit, plot these numbers:
1/1000
1/100
1/10
1/(0.1)
1/(0.01)
1/(0.001)
1/(0.0001)
1/(0.00001)
1/(0.000001)
1/(0.0000001)
1/(0.00000001)
1/(0.000000001)

i am sorry but this really is a lot of rubbish u just said...unless u are going to tell me u live in some 1,000,000th dimension were both time and space are folded and seperated:D:D:D:D:D:D

could u please tell me:

1)how can u put a value on something that has an endless value??to call something a point or a place is to value it...where exactly is infinity(be careful here since u say it is the inverse of zero)(dont trip up)


2)please tell me how to find the tan of 90' ????

cuz if something dont reach a point it nots possible to calculate,and dont say the point is infinity!!cuz i still rofl @ parellel lines meeting at infinity as u said earlier:p:p

Thomas_S_Hunterson

twasantis wrote: »

i am sorry but this really is a lot of rubbish u just said...unless u are going to tell me u live in some 1,000,000th dimension were both time and space are folded and seperated:D:D:D:D:D:D

could u please tell me:

1)how can u put a value on something that has an endless value??to call something a point or a place is to value it...where exactly is infinity(be careful here since u say it is the inverse of zero)(dont trip up)


2)please tell me how to find the tan of 90' ????

cuz if something dont reach a point it nots possible to calculate,and dont say the point is infinity!!cuz i still rofl @ parellel lines meeting at infinity as u said earlier:p:p

/head buried in hands

twasantis

Sean_K wrote: »

/head buried in hands

lol..thank you:D

yes i may be a little sarcy in what i said.:p

but i totally disagree with any notion of things meeting at infinity!!
in simple terms...if something is infinitive it is endless yes/no????

therefore how can we have infinity???


if somebody would like to prove where i am going wrong in all this i would appreciate that...but not so say to me its the inverse of zero or anything like that PLEASE

Thomas_S_Hunterson

Infinity is a concept, not a magnitude/number/value

http://en.wikipedia.org/wiki/Infinity

twasantis

Sean_K wrote: »

Infinity is a concept, not a magnitude/number/value

http://en.wikipedia.org/wiki/Infinity

exactly..is this not what i have been "trying" to say???

when i asked how can we have infinity..i should of said..how can we have infinity as a point ie.where anything can meet or join or do anything else..

Gurgle

Sean_K wrote: »

Infinity is a concept, not a magnitude/number/value

http://en.wikipedia.org/wiki/Infinity

Purely for sh1ts and giggles, I would like to point out that infinity can be a real physical measurement:

The conductivity of a material in it's superconducting state is....
... the inverse of its resistivity ...
... resistivity being zero...
... conductivity is...

*** Infinity ***

Thomas_S_Hunterson

Gurgle wrote: »

Purely for sh1ts and giggles, I would like to point out that infinity can be a real physical measurement:

The conductivity of a material in it's superconducting state is....
... the inverse of its resistivity ...
... resistivity being zero...
... conductivity is...

*** Infinity ***

As I understand it, the resistance of a semiconductor is only exactly zero in the complete absence of magnetic fields.

Given that the range of magnetic fields is infinite, there could be no place in the universe where a superconductor would have infinite conductivity. It will be extremely high however.

But yea, with no other factors present, the conductivity of a superconductor would be infinity.

raah!

Having an infinite range would also be a physical measurement for infinity.

If you apply the same logic to many other laws of maths as infinity then none of those exist either. I never ever saw it as anything other than a concept, and yet nearly 99% of the replies were "infinty isn't real".

And if it is not real then why do we see it everywhere in maths, like 10/3 equals 3.3333333333..... now if we were to say that infinity doesn't exist at all in anyway then you are suggesting that those threes end somewhere, which they can't. To say otherwise is faulty mathematical logic.

Thomas_S_Hunterson

raah! wrote: »

Having an infinite range would also be a physical measurement for infinity.

If you apply the same logic to many other laws of maths as infinity then none of those exist either. I never ever saw it as anything other than a concept, and yet nearly 99% of the replies were "infinty isn't real".

And if it is not real then why do we see it everywhere in maths, like 10/3 equals 3.3333333333..... now if we were to say that infinity doesn't exist at all in anyway then you are suggesting that those threes end somewhere, which they can't. To say otherwise is faulty mathematical logic.

No one's saying that things can't be infinite, i think it's more that infinite things can't be measured. Saying something is infinite is not giving it a measure because infinity is not a measure.

raah!

Well people were in fact saying "infinity doesn't exist". Which suggests that even as a concept it doesn't work

Sherifu

It sounds better than saying "something really, really big..."

breadmonkey

I remember my maths teacher going to great lengths to impress upon the class that

1/0 != infinity

Fremen

The concept of infinity is subtly different from the way most lay people seem to understand it. Instead of thinking of infinity as the "biggest" number, think of it as "some arbitrarily large number".

Mathematics uses the notion of infinity to express what happens in extreme circumstances. 1/x approaches +infinity as you substitute increasingly small positive values for x, but approaches -infinity as you substitute increasingly small negative values.

We say that 1/0 is undefined because if you try to do algebra with it, you can end up with some crazy contradiction like 1 = 2 (which I can show if anyone's interested).

As for the circle question, I would say that if you fix some point on the circumference of the circle, and let the distance between that fixed point and the center tend to infinity by defining new circles with successively larger radii, the curvature would tend to zero. You would probably find the equation of a line popping out somewhere, though that's a guess. I'd try to work it out but it's way past my bedtime.

breadmonkey

We say that 1/0 is undefined because if you try to do algebra with it, you can end up with some crazy contradiction like 1 = 2 (which I can show if anyone's interested).

Go for it.

Fremen

Go for it.

Right... let

x = y

then,

x^2 = xy

adding x^2 - 2xy to both sides, we get

2x^2 -2xy = x^2 - xy

2(x^2 - xy) = x^2 - xy

dividing by x^2 - xy (i.e. dividing by zero!)

2 = 1

---

I was thinking about the circle thing a bit last night, I have a proof of sorts.

We consider the set of all circles passing through the origin with centers at (a,b), i.e. circles of the form

(x - a)^2 + (y - b)^2 = a^2 + b^2

This is equivalent to
x^2 + y^2 - 2ax - 2by = 0

Set a/b = m and hold it constant
then we have

(x^2 + y^2)/b - 2mx - 2y = 0

now let a and b tend to infinity, giving

2mx - 2y = 0
which is a line which passes through the origin.


You have to be a bit careful in how you let a and b tend to infinity: their ratio needs to stay constant or you will not necessarily get convergence.

Gurgle

Fremen wrote: »

Right... let

x = y

then,

x^2 = xy

adding x^2 - 2xy to both sides, we get

2x^2 -2xy = x^2 - xy

2(x^2 - xy) = x^2 - xy

dividing by x^2 - xy (i.e. dividing by zero!)

2 = 1

^^ And there we have the difference between mathematicians and engineers

token56

Gurgle wrote: »

^^ And there we have the difference between mathematicians and engineers

LOL


Oh and I have another thing you guys can fight over.
That 0.999999~ = 1

judas101

I think there's too much reading into this going on. But agree its a difficult concept.

When i did my degree we used circles of infinite radius to encompass all point in the positive y plane when performing complex integrations.