# Is infinity a constant or a variable ?

## Main Question or Discussion Point

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?

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

HallsofIvy
Homework Helper
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
Homework Helper
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
Staff Emeritus
Gold Member 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
Homework Helper
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.

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
Homework Helper
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.

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.

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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?"

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.

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.

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.

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HallsofIvy
Homework Helper
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!

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

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.

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?

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
Homework Helper
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

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CRGreathouse
Homework Helper
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.

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
Homework Helper
Gold Member
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'.

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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