I Is Zero x Infinity Really a Real Number?

[micromass]

Just wondering...
What about infinity plus 5+3i ?
Is that infinity, but a different kind of infinity than adding a real number?
I mean, is one case an infinite number of reals and the other case forced to be an infinite number of complex numbers (some without an imaginary part)?
Or is infinity always the count despite the variation in contents?
If so, is infinity always characterized as the count in terms of integers?

There is no one thing called infinity. If you want to know that $\infty + 5 + 3i$ is then you need to specify which formalism you're working with.

 

[Georgers]

It seems to me that with regard to Hilbert's Hotel that there is something "illegal" about taking (pushing) someone out of his room temporarily to create a vacancy. One element will always be temporarily outside the set (that is, not in a room). Also doesn't this method rely on an assumption that there is actually an empty room somewhere along the line (at the end?) or that one will eventually be created? Must be one or the other I think.

It seems like another method of creating a room for the new guest in the Hilbert Hotel would be for the manager to politely ask Room 1 to move up to Room 2 and then Room 2 would politely ask Room 3 to move up and so on. They would then wait until the room above them was vacated before moving. Using this method the new guest never would get a room. Seems like it's just a matter of good manners whether this works or not!

 

[Mark44]

It seems to me that with regard to Hilbert's Hotel that there is something "illegal" about taking (pushing) someone out of his room temporarily to create a vacancy. One element will always be temporarily outside the set (that is, not in a room). Also doesn't this method rely on an assumption that there is actually an empty room somewhere along the line (at the end?) or that one will eventually be created? Must be one or the other I think.

There is no assumption about there being an empty room at the "end." By asking each person to move to the room with the next higher room number, a space for a new guest is created at room 1. Also, as the Hilbert Hotel is strictly a thought experiment, the room switches can occur simultaneously, and there is no concern about asking all those people to move.

It seems like another method of creating a room for the new guest in the Hilbert Hotel would be for the manager to politely ask Room 1 to move up to Room 2 and then Room 2 would politely ask Room 3 to move up and so on. They would then wait until the room above them was vacated before moving. Using this method the new guest never would get a room. Seems like it's just a matter of good manners whether this works or not!

 

[Ssnow]

If you admit that $0^0$ is undefined then $0^0=0^{1-1}=0^1\cdot 0^{-1}=0\cdot \lim_{x\rightarrow 0^{+}} x^{-1}=0\cdot \lim_{x\rightarrow 0^{+}}\frac{1}{x}=0\cdot (+\infty)$ (the same happen for $x\rightarrow 0^{-}$ with $-\infty$). Thus also $0\cdot \infty$ is undefined ...

Ssnow

 

[jbriggs444]

If you admit that $0^0$ is undefined then $0^0=0^{1-1}=0^1\cdot 0^{-1}=0\cdot \lim_{x\rightarrow 0^{+}} x^{-1}=0\cdot \lim_{x\rightarrow 0^{+}}\frac{1}{x}=0\cdot (+\infty)$ (the same happen for $x\rightarrow 0^{-}$ with $-\infty$). Thus also $0\cdot \infty$ is undefined ...

Unfortunately, that argument is a string of errors.

If $0^0$ is undefined then you cannot use it in an equality.
Since $0^{-1}$ is also undefined, you also cannot use it in an expression.
The law of exponents ($a^{b+c}=a^ba^c$) is only viable for strictly positive bases.
When a limit fails to exist, you cannot use it in an expression.
The limit of a product is not necessarily equal to the product of the limits. [For finite limits that exist, the result holds]

You cannot determine what a definition will say from first principles. Definitions are created by humans and are arbitrary. If I want to define $0 \cdot \infty$ as $\frac{\pi}{2}$, no argument you make can prove that definition incorrect. Argument can only show that the definition is inconsistent with other things that you want to remain true.

 

[Ssnow]

Unfortunately, that argument is a string of errors.

yes sure, as many heuristics proofs ...

I want to observe that there are mathematicians of the opinion to consider $0^0$ defined equal to $1$ ( this is not true in general ...) at least in particular cases, this happens when you think a value for the self-exponential $f(x)=x^x$ at $x=0$ ...

 

[dkotschessaa]

To clarify what Mark44 knows perfectly well and has elected not to say out loud, these are rules for an arithmetic that has been extended to include $\infty$.

Here's how Rudin puts it:

I'm not sure this is actually ever used later.

 

[dkotschessaa]

I'm still a sucker for threads about infinity.

 

[S.G. Janssens]

I'm not sure this is actually ever used later.

For example, in the extended reals every increasing sequence has a limit. This is convenient in measure theory, for instance (but not only) when formulating the monotone convergence theorem.

 

[Logical Dog]

yes I also was of op views of multiciplication being glorified addition. (primitive view)

but then I saw the sequences and their limits combined and it is surprising;

 

[dkotschessaa]

Isnt multiplication just glorified addition? so according to my primitive views,=) if so how you can keep adding zero infinitely to get zero IMO.

Sometimes primitive ideas are the most informative.

I could see defining $x * \infty = x + x+ x + \cdots$ i.e an infinite sequence. Perhaps this has already been said in this thread in a different way.

-Dave K

 

[Logical Dog]

Sometimes primitive ideas are the most informative.

I could see defining $x * \infty = x + x+ x + \cdots$ i.e an infinite sequence. Perhaps this has already been said in this thread in a different way.

-Dave K

yes but that series it would (?) seem to diverge to infinity or minus infinity, and converge to zero if x=0. IIRC

 

[Logical Dog]

this man HATES much of set theory and calculus, and real numbers...very interesting debate:

 

[dkotschessaa]

yes but that series it would (?) seem to diverge to infinity or minus infinity, and converge to zero if x=0. IIRC

Well, that's would I would expect it to do!

 

[dkotschessaa]

this man HATES much of set theory and calculus, and real numbers...very interesting debate:

Yeah, there are some.

I've considered being a finitist if only to be contrary.