Is infinity a constant or a variable ?

[aaryan0077]

Infinity can't be a constant because ∞ ± k = ∞ , but a constant changes (its) value when something is added or subtracted.
But infinity can't be variable because the definition of variable is
"A variable is a symbol that stands for a value that may vary" or stating in simple terms
"In mathematics, a changing quantity (one that can take various values) is variable"
But infinity is not defined, so it can't vary with wrt to anything.
So what is it?

 

[Nick89]

I don't think it is either. Infinity is not a number and cannot be treated like one, it is a concept.

 

[HallsofIvy]

Nick89 is completely correct. "Infinity" is not a number, it shorthand for a limit. You can talk about the "limit as x goes to infinity but you never put "infinity" into a formula.

 

[diazona]

Nick89 is completely correct. "Infinity" is not a number, it shorthand for a limit. You can talk about the "limit as x goes to infinity but you never put "infinity" into a formula.

I would have said exactly the same thing Infinity is only useful in limits (including the limits of sums or integrals). And if you ever do plug infinity into a formula - like e^{-\infty} - it's shorthand for taking a limit \lim_{x->\infty} e^{-x}

 

[Hurkyl]

All this misinformation, and some from people who should know better!

Like any other mathematical term such as 3, plus, or local, many contexts (precisely) define a term named "infinity". For example, the extended real number system contains two very useful numbers it calls +\infty and -\infty.


Don't make random speculation about what properties something called "infinity" might have (along with other mathematical terms), such as what's in the opening post. You are far more likely to confuse yourself than you are to understand something. Either use/learn the thing properly or don't use it at all.

 

[HallsofIvy]

Had this been posted in a mathematics section, I might have made refence to the extended real number system, etc. However, this was posted in the "General Physics" section so I think it is reasonable to respond to the assumption that "infinity" is a number in the real number system.

 

[mathman]

This discussion does belong in the math section. Infiity as a concept is fairly complicated, particularly when looking at infinite cardinal or ordinal numbers. The simplest kind of example is the comparison between the cardinality of the integers and the cardinality of the reals.

 

[bmxkid]

Sorry to steal your thread but I didn't feel this was worthy of its own. Why does a fraction raised to an infinite power equal zero. Example: (1/3)^(infinity) = 0

 

[CRGreathouse]

But infinity is not defined, so it can't vary with wrt to anything.

It is defined, where do you get the idea it's not?

So what is it?

As an extended real number, it's a constant just like 7 or pi.

 

[slider142]

Sorry to steal your thread but I didn't feel this was worthy of its own. Why does a fraction raised to an infinite power equal zero. Example: (1/3)^(infinity) = 0

You must be referring to \lim_{x\rightarrow \infty} \frac{1}{3}^x. As such, it is easy to see that as x increases without bound, the term decreases without positive bound, and is never negative. This argument can be made rigorous with the epsilon-delta formulation of limits. Or you can do a simple proof by contradiction, ie., assume h is the greatest lower bound for the set of numbers satisfying the form inside the limit and that h > 0. It is easy to exhibit a number in the set less than h.
This limit is just a complicated way of noting that if 0 < x < 1, then 0 < rx < x for any real r > 0.
If you meant something else by an infinite power, feel free to elaborate.

 

[slider142]

Infinity can't be a constant because ∞ ± k = ∞ , but a constant changes (its) value when something is added or subtracted.
But infinity can't be variable because the definition of variable is
"A variable is a symbol that stands for a value that may vary" or stating in simple terms
"In mathematics, a changing quantity (one that can take various values) is variable"
But infinity is not defined, so it can't vary with wrt to anything.
So what is it?

Unfortunately, mathematics defines many types of infinite numbers. As such this question is a bit vague; it is akin to asking "Is finite a constant or a variable?"

 

[bmxkid]

Sorry for not knowing LATEX.

Given problem:
Find the limit of the sequence or show that it diverges.

The Limit as n approaches infinity of the sequence (e^n + 3^n)/(5^n).

As I was doing the algebra, I broke the equation into two separate parts and applied the law of exponents and ended up with this: (e/5)^(n) + (3/5)^n. I immediately thought that substituting infinity for n would give me (infinity - infinity). However checking this step in a CAS yields that a fraction (e/5)^(infinity) yields the answer 0. My question is why? Maybe I missed something in grade school.

 

[slider142]

Sorry for not knowing LATEX.

Given problem:
Find the limit of the sequence or show that it diverges.

The Limit as n approaches infinity of the sequence (e^n + 3^n)/(5^n).

As I was doing the algebra, I broke the equation into two separate parts and applied the law of exponents and ended up with this: (e/5)^(n) + (3/5)^n. I immediately thought that substituting infinity for n would give me (infinity - infinity). However checking this step in a CAS yields that a fraction (e/5)^(infinity) yields the answer 0. My question is why? Maybe I missed something in grade school.

Are those fractions greater than 1 or less than 1 ? See the argument made in my post.

 

[bmxkid]

oh... so I'm using a squeeze theorem in which the value rn is squeezed in between 0 and 1 thus would tend to 0. Thank you very much slider you don't how confusing this had made me.

 

[HallsofIvy]

Had this been posted in a mathematics section, I might have made refence to the extended real number system, etc. However, this was posted in the "General Physics" section so I think it is reasonable to respond to the assumption that "infinity" is a number in the real number system.

No fair moving this to "General Mathematics" so I look like a fool!

 

[aaryan0077]

It is defined, where do you get the idea it's not?



As an extended real number, it's a constant just like 7 or pi.

Okay, so how will you define infinity?
Except that it is undefined, or say
Something without upper bound?
Though second definition looks to be defining infinity, it itself implies the non-definitive nature of infinity.


Also ,you said like 7, pie, next you will say e, though they are not "exactly" defined as
e = 2.2.7182818... approx
pi = 3.1415926.. approx
But you can say
∞ = something... approx

 

[aaryan0077]

Unfortunately, mathematics defines many types of infinite numbers. As such this question is a bit vague; it is akin to asking "Is finite a constant or a variable?"

Okay, so "Is finite a constant or a variable?".
Yes even this question can't be answered, but we can take one (or more) particular value(s) from this "finite" thing, and we can ask lower this form that set of "finite" (which is infinite in itself) to a particular element and than it would be answerable.
But not for infinite, one can't take a subset or element from this "infinite" to constrain this question upto that subset/element only and than make the question answerable.

 

[aaryan0077]

Sorry to steal your thread but I didn't feel this was worthy of its own. Why does a fraction raised to an infinite power equal zero. Example: (1/3)^(infinity) = 0

Okay so you can answer all the things like ∞⁰ and ∞/∞ and all this stuff.
Right?

 

[aaryan0077]

Sorry for not knowing LATEX.

Given problem:
Find the limit of the sequence or show that it diverges.

The Limit as n approaches infinity of the sequence (e^n + 3^n)/(5^n).

As I was doing the algebra, I broke the equation into two separate parts and applied the law of exponents and ended up with this: (e/5)^(n) + (3/5)^n. I immediately thought that substituting infinity for n would give me (infinity - infinity). However checking this step in a CAS yields that a fraction (e/5)^(infinity) yields the answer 0. My question is why? Maybe I missed something in grade school.

I am not talking about any ratio ( or fraction ) times itself when till the limit of times becomes ( tend to go ) unbound i.e. infinity.
I am talking about that unbound thing itself, not something else when that thing goes unbound

 

[HallsofIvy]

Okay, so how will you define infinity?
Except that it is undefined, or say
Something without upper bound?
Though second definition looks to be defining infinity, it itself implies the non-definitive nature of infinity.


Also ,you said like 7, pie, next you will say e, though they are not "exactly" defined as
e = 2.2.7182818... approx
pi = 3.1415926.. approx
But you can say
∞ = something... approx

He said "as an extended real number". You would first have to define "extended real numbers system". You can see that at
http://en.wikipedia.org/wiki/Extended_real_number_line

 

[CRGreathouse]

Okay, so how will you define infinity?

How would you define 1?

There is a symbol \infty and a series of rules for manipulating it, just like there is a symbol 7 and a series of rules for manipulating it.

A formal definition (Peano arithmetic => rationals => Dedekind cuts => extended reals) would probably be too difficult for you -- but essentially all of the complexity is in defining the real numbers. Once you have the real numbers, it's pretty easy to get the extended numbers: infinity is a much simpler concept than real numbers.

 

[ice109]

whats with all this dedekind cuts stuff?

infinity is the number larger than any positive integer and I'm pretty sure that's a rigorous definition.

 

[dx]

No, that's not a meaningful definition for the following reason: There is no number β in R that satisfies β > x for all x in R. That's a theorem about the reals. The only way to introduce ∞ as a number is to extend R, i.e. extend the notion of 'number'.

 

[ice109]

No, that's not a meaningful definition for the following reason: There is no number β in R that satisfies β > x for all x in R. That's a theorem about the reals. The only way to introduce ∞ as a number is to extend R, i.e. extend the notion of 'number'.

show me this theorem

 

[CRGreathouse]

show me this theorem

Here's an exceedingly rigorous proof:
http://us.metamath.org/mpeuni/arch.html

Here's a slightly less formal proof:
http://planetmath.org/encyclopedia/AxiomOfArchimedes.html

Here's the Wikipedia article:
http://en.wikipedia.org/wiki/Archimedean_property

 

[slider142]

Okay, so "Is finite a constant or a variable?".
Yes even this question can't be answered, but we can take one (or more) particular value(s) from this "finite" thing, and we can ask lower this form that set of "finite" (which is infinite in itself) to a particular element and than it would be answerable.
But not for infinite, one can't take a subset or element from this "infinite" to constrain this question upto that subset/element only and than make the question answerable.

You can do this in the same fashion that you did it for finite numbers. Each infinite number is a constant. Which one are you interested in? A few examples are Aleph_Null, the Continuum/Bet, Epsilon₀, any limit ordinal, complex infinity, and extended real number infinity. Some of these are different types of infinitude, while others measure different magnitudes of infinity.

 

[CRGreathouse]

whats with all this dedekind cuts stuff?

A Dedekind cut is a formal way to define real numbers from rational numbers.

infinity is the number larger than any positive integer and I'm pretty sure that's a rigorous definition.

dx gave you one reason that might not be a good definition. I'll give you another: there's no reason to think that there's only one infinity. What if there are two, one bigger than the other? What if there are infinitely many infinities? This actually happens in set theory, where aleph_0 < aleph_1 < aleph_2 < ...

Here's a third: What if it turns out that there's no good way to compare the size of different numbers? What if the numbers loop around, so you can't really tell if infinity is less than negative ten billion or more than 7? This also actually happens with the projective line or the Riemann sphere.

 

[ice109]

A Dedekind cut is a formal way to define real numbers from rational numbers.



dx gave you one reason that might not be a good definition. I'll give you another: there's no reason to think that there's only one infinity. What if there are two, one bigger than the other? What if there are infinitely many infinities? This actually happens in set theory, where aleph_0 < aleph_1 < aleph_2 < ...

Here's a third: What if it turns out that there's no good way to compare the size of different numbers? What if the numbers loop around, so you can't really tell if infinity is less than negative ten billion or more than 7? This also actually happens with the projective line or the Riemann sphere.

meh the cardinals and the reals are 2 different sets. I'm not going to translate knowledge about into intuition about the other. and again the riemann sphere and the projective line aren't R1. but anyway you make a good case.

and i knew what dedekind cuts were, i meant they weren't necessary for the definition of inf.

 

[CRGreathouse]

meh the cardinals and the reals are 2 different sets.

So are the reals and the extended reals. What's your point?

and i knew what dedekind cuts were, i meant they weren't necessary for the definition of inf.

Dedekind cuts are needed* to formally construct the real numbers. Once you have a construction of the real numbers it's very easy to define the extended reals.

* Actually, this isn't true. There are many other ways of constructing the reals; the next most popular is probably axiomatically constructing a real closed field and then adding an axiom stating that the field is Dedekind-complete (and maybe one other property?)... but perhaps this isn't really that different.

 

[Russell Berty]

In keeping with the original question of this post, I believe there are some other concepts that need clarifying (not merely “infinity”.) What is a constant? What is a variable? These also need to be understood from context. From the question it sounds like the underlying context is the set of Real Numbers.

So, to be more clear, I will emphasize “real-valued”.

Infinity can't be a constant because ∞ ± k = ∞ , but a constant changes (its) value when something is added or subtracted.

A real-valued constant changes when a nonzero number is added. The constant changes? Hmm... What is meant by this, I think, is that “The sum of a real-valued constant and a nonzero number is different than the value of that constant.”

Hence:
Infinity is not a real-valued constant. [Even when the set of Reals is extended to make the “Extended Reals”, infinity is still not a real-valued constant.]

But infinity can't be variable because the definition of variable is
"A variable is a symbol that stands for a value that may vary" or stating in simple terms
"In mathematics, a changing quantity (one that can take various values) is variable"

A real-valued variable is a variable that takes on various real-values.

Infinity never takes on a real-value.

Hence:
Infinity is not a real-valued variable.

But infinity is not defined, so it can't vary with wrt to anything.
So what is it?

“not defined” is a bad choice of words. As has been mentioned before in this post, the context is important for knowing which “infinity” is being used; which definition is being used.

When dealing with real-valued numbers, and their functions, the term infinity is used to describe a process in which a varying real-value increases without bound.

“As x grows without bound” [Notation: x\rightarrow \infty ]

That is, for any real number b, x is “eventually” larger than b (and stays larger.) There is an implied process going on, namely that of x changing in value.


Infinity is also used in interval notation to represent an unbounded interval.

“All real numbers greater than 4” [Notation: \left(4,\infty\right) ]

Notice in the two examples above, the word “infinity” is not needed, nor is the symbol needed. It is for convenience so we need not always write “unbounded growth” or “unbounded interval”.

When dealing with sets other than the Real Numbers, the term “infinity” might not be used in these ways.

 

[aaryan0077]

How would you define 1?

There is a symbol ∞ and a series of rules for manipulating it, just like there is a symbol 7 and a series of rules for manipulating it.

A formal definition (Peano arithmetic => rationals => Dedekind cuts => extended reals) would probably be too difficult for you -- but essentially all of the complexity is in defining the real numbers. Once you have the real numbers, it's pretty easy to get the extended numbers: infinity is a much simpler concept than real numbers.

Yes one is defined, we have a set of operations and rules to perform over it, also 7 is a symbol with same thing as 1, just 7 times of it.
But for ∞ we have very limited rules, say just evaluating some limits in which something tends to ∞ we have certain rules, or in physics we have some process of renormalization (I just know the name, nothing else) which can remove certain infinities by inserting some more infinities.
So it is certainly not as "expressible", rather say "explicit" as, say "the symblol 7".

 

[aaryan0077]

whats with all this dedekind cuts stuff?

infinity is the number larger than any positive integer and I'm pretty sure that's a rigorous definition.

If ∞ were just a "positive number" there won't have been any confusion about it.
Also its definition would be "explicit"

 

[aaryan0077]

No, that's not a meaningful definition for the following reason: There is no number β in R that satisfies β > x for all x in R. That's a theorem about the reals. The only way to introduce ∞ as a number is to extend R, i.e. extend the notion of 'number'.

I think I should agree with you.

 

[aaryan0077]

You can do this in the same fashion that you did it for finite numbers. Each infinite number is a constant. Which one are you interested in? A few examples are Aleph_Null, the Continuum/Bet, Epsilon₀, any limit ordinal, complex infinity, and extended real number infinity. Some of these are different types of infinitude, while others measure different magnitudes of infinity.

Okay, I don't know anything about this Aleph_Null and all, so I'll check them then I'll reply.
Anyway, thanks for sharing this thing with me.

 

[aaryan0077]

A Dedekind cut is a formal way to define real numbers from rational numbers.



dx gave you one reason that might not be a good definition. I'll give you another: there's no reason to think that there's only one infinity. What if there are two, one bigger than the other? What if there are infinitely many infinities? This actually happens in set theory, where aleph_0 < aleph_1 < aleph_2 < ...

Here's a third: What if it turns out that there's no good way to compare the size of different numbers? What if the numbers loop around, so you can't really tell if infinity is less than negative ten billion or more than 7? This also actually happens with the projective line or the Riemann sphere.

AWESOME! I never knew that, thanks for making me know, but I am still not sure I get it completely or not, I think I got to have a sleep before any more reasoning.

 

[HallsofIvy]

Okay, so how will you define infinity?
Except that it is undefined, or say
Something without upper bound?
Though second definition looks to be defining infinity, it itself implies the non-definitive nature of infinity.


Also ,you said like 7, pie, next you will say e, though they are not "exactly" defined as
e = 2.2.7182818... approx
pi = 3.1415926.. approx
But you can say
∞ = something... approx

pi and e are exactly defined. The fact that they require an infinite number of symbols in some specific numeration system has nothing to do with their definition.

 

[CRGreathouse]

Yes one is defined, we have a set of operations and rules to perform over it, also 7 is a symbol with same thing as 1, just 7 times of it.
But for ∞ we have very limited rules, say just evaluating some limits in which something tends to ∞ we have certain rules, or in physics we have some process of renormalization (I just know the name, nothing else) which can remove certain infinities by inserting some more infinities.
So it is certainly not as "expressible", rather say "explicit" as, say "the symblol 7".

You haven't defined one for me, nor have you defended your bare assertion that ∞ is not expressible or explicit. I also have no idea what, if anything, you mean by "very limited rules".

The renormalizations of physics have little or nothing to do with math.

 

[CRGreathouse]

AWESOME! I never knew that, thanks for making me know, but I am still not sure I get it completely or not, I think I got to have a sleep before any more reasoning.

Feel free to ask any questions you like. Someone can probably address them -- maybe me, maybe someone else.

Edit: By the way, the "Aleph_Null" slider mentioned is the same as the "aleph_0" I mentioned.

 

[aaryan0077]

Okay thanks everyone for you help, but how does this ends?
I mean what's the conclusion.

 

[CRGreathouse]

Okay thanks everyone for you help, but how does this ends?
I mean what's the conclusion.

Something like
"There are lots of kinds of infinities, none of which are variables."

 

[aaryan0077]

Say,
Is infinity like what Nick89, HallsofIvy, & diazona mentioned that it is rather a concept than a number.

Or as what CRGreathouse said, that it's a symbol ∞, and has got some rules to maniupalate it.

Or is it like CRGreathouse said later that what if numbers were to loop, and we cannot figure out if infinity is less than negative ten billion or more than 7?

Or as said by Russell Berty, that it's is not a real-valued constant. [Even when the set of Reals is extended to make the “Extended Reals”, infinity is still not a real-valued constant.]
and Infinity is not a real-valued variable, or as he says in end that it's just an interval notation to represent an unbounded interval.

What's the final result?
How should this thread end?

 

[aaryan0077]

Something like
"There are lots of kinds of infinities, none of which are variables."

Can you explain it a bit, and what's the meaning of "none of which are variables." does it means they are constant?
Why are you confusing me?

 

[matt grime]

I think you're probably confused because you have asked an ill-defined question - one which cannot have a satisfying answer.

The symbol \infty[/tex] and the notion of infinity or something being infinite can have different connotations in different contexts, that is what you should have learned in this thread. The best attempt one can make at answering &#039;is it a constant or variable&#039; is that they (the different meanings) are neither and nor is it a sensible question to ask in the first place. My guess as to what you mean by &#039;constant or variable&#039; is that you need some physical quantity or model of it which is &#039;infinity&#039;. For example, position and time will be variables to you, and the ratio of an (idealized) circle&#039;s circumference and its diameter is pi, a constant.

 

[HallsofIvy]

Can you explain it a bit, and what's the meaning of "none of which are variables." does it means they are constant?
Why are you confusing me?

What do you see as the distinction between "variable" and "constant"? What are the definitions?

 

[CRGreathouse]

The best attempt one can make at answering 'is it a constant or variable' is that they (the different meanings) are neither and nor is it a sensible question to ask in the first place.

Although I don't describe the various infinities as constants, I think that's really what they are. So I'll differ from you on this point.

But when summing up for the OP I did avoid that, sticking to the well-agreed-upon "they're not variables".

 

[CRGreathouse]

Is infinity like what Nick89, HallsofIvy, & diazona mentioned that it is rather a concept than a number.

It's not a "real number". As used in high-school calculus, it's not a number at all but a concept.

Or as what CRGreathouse said, that it's a symbol ∞, and has got some rules to maniupalate it.

In R*, "+∞" is an "extended real number" just like any other, and has rules to manipulate it.

In P¹, "∞" is a "projective real number" and has rules to manipulate it.

In C* (the Riemann sphere), "∞" is an extended complex number and has rules to manipulate it.

In ZF, "\aleph_0" is a cardinal and has rules to manipulate it.

...

There are lots of infinities.

Or is it like CRGreathouse said later that what if numbers were to loop, and we cannot figure out if infinity is less than negative ten billion or more than 7?

I was talking about the ∞ in C* (or the ∞ in P¹) when I said that. ∞, in that context, can be thought of as the "greatest and the least element". It has the largest size, but it's not sensible to think of it as positive or negative.

Or as said by Russell Berty, that it's is not a real-valued constant. [Even when the set of Reals is extended to make the “Extended Reals”, infinity is still not a real-valued constant.]
and Infinity is not a real-valued variable, or as he says in end that it's just an interval notation to represent an unbounded interval.

None of the "real numbers" (but one of the extended complex numbers, two of the extended real numbers, and infinitely many of the cardinals) are infinite, so whatever kind of infinity you mean it isn't real-valued.

 

[matt grime]

Although I don't describe the various infinities as constants, I think that's really what they are. So I'll differ from you on this point.

As we're both guessing what the OP means by 'constant' and 'variable', I don't think that we differ, or agree. I mean, what if I fixed a field k, and formed the polynomial algebra k[some infinite cardinals] with the rules of cardinal arithemetic? Surely they're now 'variables'? Of course if we ascribe the 'physical' meaning of infinite cardinals as equivalence classes of sets, then aleph_0 is always the cardinality of the the integers (and let's avoid any set theoretic issues, which might imply that the cardinal number of a set may 'change' and be 'variable'), so it's 'constant' now... It's a truly pointless debate.

 

[Hurkyl]

I know I was thinking more along the lines of formal logic, with 'constant' meaning a nullary function.

 

[AUMathTutor]

In mathematics, which is referentially transparent, I don't think there's much of a difference between the idea of "variable" and "constant". Perhaps the closest I could come up with is that a variable is an undetermined constant. I really have no idea how mathematics formally distinguishes between the two.

In CS, it's a lot easier. It's a constant if it's a bit pattern. It's a variable if it's a reference to a memory location (which is inherently changeable). These concepts rely on a concept of "state" which you really don't have in mathematics.

Even given the CS definition, however, the idea of infinity (or infinities) seems to correspond more closely (in a semantic sense) to constants rather than variables. If I give you a certain instance of infinity (some kind of infinity, say, aleph-nought or something) that's what it is, and it can be encoded somehow (say, using its definition). Since this definition can be encoded as a sequence of bits, and this sequence isn't a placeholder for anything (it is what it is), then it seems to me that (depending on what is meant by infinity) that infinity is a constant, not a variable.

Unless you want to get into actual vs potential infinities, in which case I'm peace out, yo.

 

[Hurkyl]

In CS, it's a lot easier. It's a constant if it's a bit pattern. It's a variable if it's a reference to a memory location (which is inherently changeable). These concepts rely on a concept of "state" which you really don't have in mathematics.

I'm not really sure at what you're thinking, but this is fairly inaccurate if applied to programming language syntax.

For example, in C, even if we decide to insist that "const" is different from what you mean by constant (despite the standard specifying that certain const variables are compile-time constants), you still have things like:
. Macros (and their arguments) have nothing to do with the abstract machine model, let alone memory locations on actual computers
. String literals are generally put into memory locations, despite being constants

Of course (IMHO) it's saner to include what C calls "const" to be considered a constant -- the terms "constant" and "varaible" are defined by the formal language.



State is relatively easy to treat mathematically; you just make everything a function defined on some abstract "state space" (which is often just a set, although it might have other properties defined for it).