1^∞, 0^0 and others on the real projective line

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In summary, the conversation discusses the concept of the real projective line and its connection to infinity. The speaker presents their idea of a "collection" of numbers, represented by the symbol A, which includes all real numbers. They also discuss the practical applications of A, such as in arithmetic calculations. The conversation also touches on the concept of limit points and the dangers of working with multi-valued functions. The speaker also shares their personal notes and thoughts on philosophy.
  • #1
n_kelthuzad
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http://en.wikipedia.org/wiki/Real_projective_line
https://www.physicsforums.com/showthread.php?t=591892
https://www.physicsforums.com/showthread.php?t=592694
https://www.physicsforums.com/showthread.php?t=530207 [Broken]
Read these first before you criticize me.
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Working on infinity and the progress is very very slow. (maybe mainly because I am so lazy)
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under the conditions that 1/0=∞, 1/∞=0 and so on;
I assumed that 0*∞, ∞/∞ and 0/0 share the same answer;
thats answer is, not a number but a 'collection' of numbers(not a set with defined ranges yet)
I call that answer A
all the real numbers are assigned to A, in most conditions I can say it is true;
and let's go to the practical things.
----------------------------------------------------------------------
take any number from A: let's just call it n,
n^0=1
substitute 0=n/∞
n^(1/∞)=1
however if we apply basic exponentiation rule that:
x^y=z if and only if x=z^(1/y)
so n=1^(1/(1/∞))
n=1^(∞)
so that is, 1^∞=A
(in fact from this it can be also proven that for any nonzero real number n, n^∞=A;But I couldn't find that piece of paper.)
----------------------------------------------------------------------
0^0
it is true that 0=-n+n and (x^y)*(x^z)=x^(y+z)
so 0^0=0^1*0^(-1)
0^0=0*(1/0)
0^0=0*∞
or 0^0=(1/∞)*∞
=∞/∞

0^0=A
----------------------------------------------------------------------
∞^0
=(n/0)^0
=(n^0)/(0^0)
=1/A
=A
----------------------------------------------------------------------
for ∞^∞ and 0^∞; they are being worked on but in fact that (∞^∞)*(0^∞)=A
----------------------------------------------------------------------
so that's all the practical usage of A that I've worked out for now. However the definition of A is still unclear and needs a lot of work. Hope anyone can give me some pointers.

Victor Lu,
16
BHS, CHCH, NZ
 
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  • #2
thinking from the Riemann Sphere: can the real projective line be described as a circular graph?
So all the arithmetic calculations can be done via angular calculations, and 0 or infinity would have a unique angle from the axis?
 
  • #3
so if those things are true, then many of the limits can be viewed in a different perspective.
e.g.
lim->infinity (1+1/m)^m=e
u couldn't just substitute m=infinity into the equation;
however if we do that:
(1+0)^infinity=e
1^infinity=e
it makes sense now since e is a member of A.
 
  • #4
This is very strange...
∞/0
=(n/0)/0
=n*(0^-2)
=n/0
=∞

0/∞
=(n/∞)/∞
=n*(∞^-2)
=n*0
=0

And from that , ∞^∞=A and 0^∞=A
 
  • #5
Nothing particularly strange about any of that. You are simply mistaken to think that you can do arithmetic with "infinity" the same way you can with numbers.
 
  • #6
I think you're struggling towards the concept of a "limit point". A point L is a limit point of f(x) at a if and only if

[tex]\forall \epsilon > 0: \exists \delta > 0 : \exists x : 0 < |x-a| < \delta \wedge |f(x) - L| < \epsilon[/tex]

in other words, f(x) gets arbitrarily close to L infinitely often as x approaches a.

L is the limit of f(x) if and only if it is the only limit point of f(x).

The set of limit points of a function can sometimes be used usefully. For example, it streamlines the proof

[tex]\lim_{x \to +\infty} \frac{\sin(x)}{x} = 0[/tex]

the limit of the numerator doesn't exist, but its set of limit points is the interval [itex][-1,1][/itex], and we can use that in the calculation:

[tex]\lim_{x \to +\infty} \frac{\sin(x)}{x} = \frac{ [-1, 1] } {+\infty} = 0[/tex]

This is a horrendous abuse of notation, so don't take it's specific form seriously, but it is clear what the argument is.


However, for the real projective line, the set of limit points of / at (0,0) is all real projective numbers (you overlooked the fact that [itex]\infty[/itex] is a limit point too). This means there is pretty much no value whatsoever in applying the notion. The problem can be observed by paying attention to the argument you made with e:

You (correctly) observe that the limit points of
[tex]\lim_{m \to +\infty} \left(1 + \frac{1}{m} \right)^m [/tex]
are all limit points of [itex]1^\infty[/itex]. But the only thing that this observation tells us is that those limit points must be either non-negative real numbers or [itex]\infty[/itex]. There is absolutely no additional information content in that observation!




One could decide to extend / not as a function, but as a multi-valued function, and set 0/0=a for every real projective number a

However, this is a BAD IDEA.

When I say bad idea, I don't mean that one can't do it: you most surely can, and in some sense the math will work out.

Instead, what I mean is that working with multi-valued functions are a death trap for those who haven't properly studied how to work with them, and this goes doubly so for people who are already sloppy/weak in arithmetic and logic.
 
  • #7
let there be y=0*x
there are 4 value ranges on the real plane:
I. x<0 y=inf
II. x=0 y=A
III. x>0 y=0
IV. x=inf y=A
as we can see here the value of y is like a sine wave;
(Although A is not a number.)
inf --> A --> 0 --> A --> inf --> A ...
so, does A represent a intermediate range of values inbetween 0 and infinity, from both sides?
if the quantity of 0 represents 'nothing'
then is the definition of infinity 'endless', 'extreme' 'everything' or 'anything'?
or does A represent 'everything' or 'anything'?
----------------------------------------------------------------
just need to post this online so I can keep my progress, since I cannot think about philosophy when extremely tired and listening to rap music.
 
  • #8
I am so dumb. Used the hard way to do all the things.
Because 1^inf=A
then for any number n, n^inf=A
because n^inf=n^inf * 1^inf
=n^inf * A
 
  • #9
n_kelthuzad said:
just need to post this online so I can keep my progress, since I cannot think about philosophy when extremely tired and listening to rap music.
There are more appropriate tools for keeping notes. The program "notepad", for example. These days, we even have a fabulous invention called "pencil and paper".

Posting to a public discussion is not an appropriate way for keeping notes.
 
  • #10
But the problem for me is that I use computer in many different places, and I can't even read my own writing
 
  • #11
n_kelthuzad said:
But the problem for me is that I use computer in many different places, and I can't even read my own writing

Presumably you have an email account. Just write your thoughts out and email them to yourself.
 
  • #12
I can see that infinity does not 'equal' to infinity
(inf/inf=A)
but does 0 'equal' to 0?
(0/0=A)!
there are values other than 1 in A,
then is 0=0 false?
Trying to find a equation to explain it.
 
  • #13
I feel you bro, I can't read my own writing either, every time I write with pen and paper it is in French and I can't speak a word of it.

I too also need to take notes in places where everyone can see my thoughts, because I am usually listening to dubstep at really loud volumes in Starbucks with their internet. I never go to the same one because inevitably someone complains about the lack of headphones.

I got to go get some muscle milk because I am going to go to the gym and work out for a few hours. I like to get their 10 minutes before they close, I bet the employees were going to stay there anyway. They have to clean up right?

Anyway, yea, infinity and stuff.
 
  • #14
n_kelthuzad said:
I can see that infinity does not 'equal' to infinity
(inf/inf=A)
As you were already told, you are mistaken if you think you can to arithmetic with ∞.
n_kelthuzad said:
but does 0 'equal' to 0?
Of course it does? Why would you even think to ask this?
n_kelthuzad said:
(0/0=A)!
Division by zero is NOT defined!
n_kelthuzad said:
there are values other than 1 in A,
then is 0=0 false?
No.
n_kelthuzad said:
Trying to find a equation to explain it.

Explain what? Most of what you wrote here is meaningless.
 
  • #15
Ok Maybe it is time I add a useful response.

To everyone trying to respond please keep in mind the following.

1) The OP is 16. What would did you know about mathematical spaces, such as the real projective line at 16? That is how much he knows about this stuff.

2) the OP's goal is as follows: From the wiki page on The Real projective line, he noticed that certain operations are undefined. He is trying to define them. In his mind, he is defining them as a set A, that contains real numbers. He uses n to denote an element from A. This may not make any sense to you or I but that is what he is trying to do.

To n_kelthuzad:

I understand you are trying to define these values on the Real projective line. You are absolutely allowed to do that. In fact, people have studied what happens when you do end up defining them.

You can define infinity + infinity = 10 if you want, or 0*Infinity = 20. But you know what happens when you do?

The properties of arithmetic don't work anymore.

What I mean by that is that you will start seeing contradictions, strange results. I believe you have discovered some of these strange results already.

If you define these things, you can start getting results like Infinity = 7 or 2 = 5, very strange stuff indeed.

However, for this reason, this is exactly why they have not been defined. It is not that the values for these have not been "discovered", or examined. It is just that the whole system works better when you DO NOT define them.

I hope you can see from what I wrote, that when you define these values, to anything, the regular properties break. It is no longer the case that a + b = b + a, and it is not longer the case that a*(b*c) = (a*b)*c. These properties can only be true in our Real Proejective Line if and only if the values we have left undefined stay undefined.

-Diffy
 
  • #16
HallsofIvy said:
You are simply mistaken to think that you can do arithmetic with "infinity" the same way you can with numbers.
Diffy said:
: From the wiki page on The Real projective line, he noticed that certain operations are undefined. He is trying to define them. -Diffy
Hi Diffy, I think there is something more general to clarify before that. Mentioned article Real projective line says :'arithmetic operations' which are defined:
a + ∞ = ∞
a - ∞ = ∞ , ∞ - a = ∞
a * ∞ = ∞
a : ∞ = 0
---
a : 0 = ∞
if we follow these definitions, can we do arithmetic with ∞ the same way we do with numbers or not ? Can we move something, for example, from LHS to RHS?
Can we do arithmetics at all, when ∞ is involved? if so, is wiki wrong to call them 'arithmetic operations' in the first place?
 
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  • #17
logics said:
Hi Diffy, I think there is something more general to clarify before that. Mentioned article Real projective line says :'arithmetic operations' which are defined:
a + ∞ = ∞
a - ∞ = ∞ , ∞ - a = ∞
a * ∞ = ∞
a : ∞ = 0
---
a : 0 = ∞
if we follow these definitions, can we do arithmetic with ∞ the same way we do with numbers or not ? Can we move something, for example, from LHS to RHS?
I don't see why not, as long as you are using the operations as defined above.
logics said:
Can we do arithmetics at all, when ∞ is involved?
Yes, of course. The list above defines the possible arithmetic (not 'arithmetics') operations.
logics said:
if so, is wiki wrong to call them 'arithmetic operations' in the first place?
Do you understand what the term "arithmetic operations" means? These are the binary operations that involve addition, subtraction, multiplication, and division.
 
  • #18
so, is this possible? (a = 10):
a + ∞ = ∞ → ∞ -∞ = a
a - ∞ = ∞ → ∞ + ∞ = a
a * ∞ = ∞ → ∞ : ∞ = a
a : ∞ = 0 → 0 : ∞ = a, ∞ : 0 = a
it appears that also defined operations are indeterminate
then,
(a= 10 )- ∞ = ∞ (??) how can one subtract ∞ from 10?
and, more in general, in principle:

- how can one subtract from a number that which is not a number?
- is the circle a misrepresentation? if not: how can +∞ and - ∞ meet? how can a circumference (a closed, finite curve) be infinite?
Thanks
 
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  • #19
logics said:
a - ∞ = ∞

this would actually be

a - ∞ = -∞ (negative infinity) --> ∞ + (-∞) = a
 
  • #20
Infinitum said:
this would actually be

a - ∞ = -∞ (negative infinity) --> ∞ + (-∞) = a
a - ∞ = ∞, is copied from wiki
 
  • #21
Sorry, I was thinking about the extended number line. Should have paid attention to the 'real projective' title.

But it essentially means the same, I think.
 
  • #22
The same page you provided the link for also mentions several operations that are left undefined. Did you read this section? The first three of your four questions below are undefined (∞ ± ∞, ∞/∞).

For the fourth, 0/∞ = 0, not a, and ∞/0 is not listed.
logics said:
so, is this possible? (a = 10):
a + ∞ = ∞ → ∞ -∞ = a
a - ∞ = ∞ → ∞ + ∞ = a
a * ∞ = ∞ → ∞ : ∞ = a
a : ∞ = 0 → 0 : ∞ = a, ∞ : 0 = a
it appears that also defined operations are indeterminate
then,
(a= 10 )- ∞ = ∞ (??) how can one subtract ∞ from 10?
and, more in general, in principle:

- how can one subtract from a number that which is not a number?
I assume you are referring to ∞. On the real projective line, ∞ is for all intents a number.
logics said:
- is the circle a misrepresentation? if not: how can +∞ and - ∞ meet?
By bending the real line around to form a circle
logics said:
how can a circumference (a closed, finite curve) be infinite?
The finite interval [0, 1] is closed and finite, but contains an infinite number of points.
logics said:
Thanks
 
  • #23
logics said:
so, is this possible? (a = 10):
a + ∞ = ∞ → ∞ -∞ = a
a - ∞ = ∞ → ∞ + ∞ = a
a * ∞ = ∞ → ∞ : ∞ = a
a : ∞ = 0 → 0 : ∞ = a, ∞ : 0 = a
Mark44 said:
mentions several operations that are left undefined. Did you read this section? The first three of your four questions below are undefined (∞ ± ∞, ∞/∞).
I examined it carefully, tht's why I printed in red. The point is that de facto what you said is possible is impossible: we cannot move anything from one side to the other, because it becomes an unlawful, undefined operation.
The only use and purpose of ∞ seems: a/0= ∞, this makes possible division of a number by zero. Is this correct?
On the real projective line, ∞ is for all intents a number.
if so, 10 - ∞ is a normal arithmetic operation. How can I subtract ∞ from 10 and get +∞ ?
By bending the real line around to form a circle
By what principle we can bend a 'number line' ?
How can a 'for-all-intents' negative number meet or 'become' a positive number?
Thanks, Mark44:smile:
 
  • #24
logics said:
The only use and purpose of ∞ seems: a/0= ∞, this makes possible division of a number by zero. Is this correct?
Projective geometry was discovered for the purposes of geometry, and has shown itself in may cases to be the "right" way to think about geometry. This, in turn, has shown its importance in algebra.

if so, 10 - ∞ is a normal arithmetic operation. How can I subtract ∞ from 10 and get +∞ ?
It seems pretty clear that's how it works. Why do you think there's a problem?

By what principle we can bend a 'number line' ?
The projective line has a standard topology that is homeomorphic to a circle. It has a standard differentiable structure that makes it diffeomorphic to a circle as well.

How can a 'for-all-intents' negative number meet or 'become' a positive number?
What is this notion of "negative" and "positive" of which you speak? If we were talking about real numbers, it would be clear. But aren't we talking about projective real numbers?

The Euclidean line has a notion of "betweenness": we can talk about a point B being between two points A and C. The point 0 splits the line into two regions. We call a point x "positive" if 1 lies between 0 and x. We call it "negative" if 0 lies betwen 1 and x.

The projective line doesn't have this notion. However, if you choose a pair of points, we can talk about a notion of "separation" -- i.e. "the points A and B are separated by the points C and D". This notion makes sense in the Euclidean line too, by the way. 1 and 4 are separated by 0 and 3. 1 and 4 are not separated by 0 and 5. 1 and 4 are not separated by 2 and 3.

I suppose it's reasonable to define "positive" to mean that 1 and x are not separated by 0 and ∞, and "negative" to mean that 1 and x are separated by 0 and ∞. But then it's obvious that ∞ is neither positive nor negative in exactly the same way as 0.
 
  • #25
logics said:
so, is this possible? (a = 10):
a + ∞ = ∞ → ∞ -∞ = a
a - ∞ = ∞ → ∞ + ∞ = a
a * ∞ = ∞ → ∞ : ∞ = a
a : ∞ = 0 → 0 : ∞ = a, ∞ : 0 = a
Thanks, Hurkyl for your explanations,
So, those are possible 'normal' arithmetic operations: in conclusion, is that possible?
 
  • #26
logics said:
Thanks, Hurkyl for your explanations,
So, those are possible 'normal' arithmetic operations: in conclusion, is that possible?
The first three of those are ill-defined statements, since they include the expression "∞-∞", "∞+∞", and "∞/∞" respectively.

The fourth is simply false, since its hypothesis is true, but it's conclusion is false: "0/∞=0" and "∞/0=∞", and a was defined to be neither 0 nor ∞.


By the way, while it's a little strange to use ":" for division, it's especially bad in this context. We common use projective coordinates on the projective line. Projective coordinates are a pair (x:y) where x,y are real numbers, and at least one of them is non-zero. (x:y) and (s:t) denote the same point if and only if xt = ys. (i.e. if there is a c so that x=cs and y=ct)

Of course, one often embeds the real line in the projective line by sending the point x to the point with coordinates (x:1). So when y is a non-zero real number, we can interpret (x:y) as being the point x/y. But writing x:∞ would be weird.
 
  • #27
Is there an alternative symbol that can be used instead of '=', for a different logical expression? a=b means a and b are equivalent in quantity, however infinity and 0 are not ordinary quantities?
 
  • #28
n_kelthuzad said:
Is there an alternative symbol that can be used instead of '=', for a different logical expression? a=b means a and b are equivalent in quantity, however infinity and 0 are not ordinary quantities?
The symbol = is used between quantities whose numeric values are conditionally equal. E.g., 2x - 6 = 0. This equation is true only under the condition that x = 3.
The symbol ##\equiv## is used between quantities whose numeric values are identically equal. E.g., sin2(x) + cos2(x) ##\equiv## 1. This equation is true for all real numbers x. I occasionally see := used for this purpose, a symbol that comes, I believe, from computer science. It is used in Pascal and programming languages derived from Pascal, such as Modula-2 and Ada (and possibly others I'm not familiar with).
The symbol ##\Leftrightarrow## is used for logical statements that have the same truth values. E.g., 2x - 6 = 0 ##\Leftrightarrow## x = 3.
 
  • #29
ok I just got a kinda 'crazy' idea that would explain the arithmetic paradox.
if say, ∞/∞=A and 0/0=A;
pick 2 random numbers from A, just 2 and 3;
so
∞/∞=2
∞/∞=3
0/0=2
0/0=3
however 2[itex]\neq[/itex]3;
∞/∞=2[itex]\neq[/itex]3=∞/∞;
so ∞/∞[itex]\neq[/itex]∞/∞, same goes for 0/0[itex]\neq[/itex]0/0
so is it possible to say that:
∞[itex]\neq[/itex]∞, 0[itex]\neq[/itex]0
because they are not 'real' numbers.(note: that '' means I don't know how to explain it)
lets say any number n, n[itex]\in[/itex]ℝ; So |2n|>|n| [itex]\forall[/itex]n;
however this is not true for 0. 0*0 = 1*0 = 2*0 = 3*0 = n*0;
then I'd say that 0 and infinity are not just a single quantity, but a quantity with infinite different values, but one unique value in the real plane?

The cause of this should be the special properties of 0 and infinity.
that n0=0, n∞=∞.

A question: does infinity work as a intepretor between different degrees of dimentions?
 
  • #30
n_kelthuzad said:
ok I just got a kinda 'crazy' idea that would explain the arithmetic paradox.
if say, ∞/∞=A and 0/0=A;
You've already gone wrong.

This situation doesn't need crazy ideas. It just needs you to think clearly, and to stop doing crazy things like pretending that ∞/∞ is defined in the usual arithmetic of projective numbers.
 
  • #31
Hurkyl is exactly right.

I said earlier that these things are not defined, because when you define them you get contractions.

Wouldn't you agree that 0≠0 is a contradiction?

This is why we leave these things undefined.
 
  • #32
Diffy said:
Hurkyl is exactly right.

I said earlier that these things are not defined, because when you define them you get contractions.
The reason they aren't defined is because it's not useful to define them.

It's not useful to define them, because if you did, it wouldn't have many useful properties.

You only get a contradiction if you both define them and assume that the definition has useful properties.
 
  • #33
Hurkyl said:
The reason they aren't defined is because it's not useful to define them.

It's not useful to define them, because if you did, it wouldn't have many useful properties.

You only get a contradiction if you both define them and assume that the definition has useful properties.
While I agree in principle, I've sometimes found it useful to define 0^0 as 1.
 
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  • #34
MTannock said:
While I agree in principle, I've sometimes found it useful to define 0^0 as 1.
Hurkyl was referring to ∞/∞ and 0/0.

A lot of calculators agree with you on 0^0, and produce 1 as a result. This doesn't have anything to do with the real projective line or the extended real number line, though.
 
  • #35
Actually, 0^0 is relevant here. It's also a wonderful illustration of some of the relevant problems.

On the one hand, is not useful to give a value to 0^0. On the other hand, 0^0 is not only equal to 1, but it's not even a special case.

What changed from one side to the other is that ^ refers to the continuous exponentiation operator or its continuous extensions to things like the extended and projective real lines. ^, however, is being used to some sort of algebraic exponentiation operator; examples have domains including things like the base from any ring and exponent from the natural numbers. The exponentiation operation appearing in a power series is ^.
 

1^∞

1. What is the value of 1 raised to the power of infinity?

The value of 1 raised to the power of infinity is undefined. This is because infinity is not a real number and therefore, cannot be used as an exponent in mathematical operations.

0^0

2. What is the value of 0 raised to the power of 0?

The value of 0 raised to the power of 0 is also undefined. This is because there are conflicting mathematical definitions for this expression and it can lead to different results depending on the context in which it is used.

Real Projective Line

3. What is the real projective line?

The real projective line is a mathematical concept that extends the real number line by adding a point at infinity. This allows for the representation of points at infinity and simplifies certain geometric concepts and calculations.

4. How is the real projective line different from the real number line?

The real projective line differs from the real number line in that it includes a point at infinity and has a different topology. This means that certain properties, such as continuity and convergence, may behave differently on the real projective line compared to the real number line.

5. What are some applications of the real projective line?

The real projective line has applications in various fields of mathematics, such as geometry, topology, and complex analysis. It is also used in computer graphics and computer vision for representing points at infinity and creating panoramic images. Additionally, the real projective line has applications in physics, particularly in the study of projective geometry and projective transformations.

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