# Infinity plus 1

• Oct 17th 2012, 03:55 AM
astartleddeer
Infinity plus 1
I have no idea what I'm attempting to get at with this one.

Anyway, the following must give a real number of some sort.

$\displaystyle \infty - 1$

$\displaystyle \infty + 1$

$\displaystyle (-\infty) - 1$

$\displaystyle (-\infty) + 1$

Thus, that real number will always be $\displaystyle \infty + 1$ for an example.

But this now leads me to the conclusion, that any number can be made greater than infinity.

$\displaystyle \infty + 2$

$\displaystyle \infty + 3$

$\displaystyle \infty + \infty = (2)(\infty)$

What is going on at this end of the spectrum for the real number line?
• Oct 17th 2012, 04:22 AM
astartleddeer
Re: Infinity plus 1

If I assume the distance to the edge of space is infinity and I travel a distance $\displaystyle \infty + 1$ and I turn around - What would I see?

Do I exist at a distance $\displaystyle \infty + 1?$

Ok, let's assume I don't. So now I travel a distance of $\displaystyle \infty - 1$ and I remain looking forward - What would I see now?

I must continue to exist at this distance if I have not surpassed $\displaystyle \infty$
• Oct 17th 2012, 07:57 AM
Plato
Re: Infinity plus 1
Quote:

Originally Posted by astartleddeer
I have no idea what I'm attempting to get at with this one.
Anyway, the following must give a real number of some sort.
$\displaystyle \infty - 1$
$\displaystyle \infty + 1$
$\displaystyle (-\infty) - 1$
$\displaystyle (-\infty) + 1$

Why in the world you think that?

$\displaystyle \infty$ is not a real number.
Or as my favorite philosophy professor would say "infinity is where mathematicians hid their ignorance".
• Oct 17th 2012, 02:59 PM
johnsomeone
Re: Infinity plus 1
What you're speculating about is either:

1) whether you actually won your grade school argument when, after that brat said "Well I call 'first' infinity times!", you replied with the devastating "Fine - then I call 'first' infinity plus one times!" (The answer is you did win!)

or

2) the ordinal numbers.

I won't even try to explain it. Just google "ordinal number".

The gist is this: Instead of thinking of the natural numbers as being about "counting", so like "number of things in a set", it's thinking of the natural numbers as being about "what's the next one greater than this one?". The former are cardinals ("the cardinality of a set"), and the later are called ordinals. These two notions are the same for finite numbers, but once things become infinite, they're very different.
• Oct 27th 2012, 05:56 AM
HallsofIvy
Re: Infinity plus 1
Quote:

Originally Posted by astartleddeer
I have no idea what I'm attempting to get at with this one.

Anyway, the following must give a real number of some sort.

Since $\displaystyle \infty$ is not a real number, this basic assumption is not true.

Quote:

$\displaystyle \infty - 1$

$\displaystyle \infty + 1$

$\displaystyle (-\infty) - 1$

$\displaystyle (-\infty) + 1$

Thus, that real number will always be $\displaystyle \infty + 1$ for an example.

But this now leads me to the conclusion, that any number can be made greater than infinity.

$\displaystyle \infty + 2$

$\displaystyle \infty + 3$

$\displaystyle \infty + \infty = (2)(\infty)$

What is going on at this end of the spectrum for the real number line?
• Oct 27th 2012, 06:41 AM
Deveno
Re: Infinity plus 1
Quote:

Originally Posted by astartleddeer
I have no idea what I'm attempting to get at with this one.

Anyway, the following must give a real number of some sort.

why do you assert this?

Quote:

$\displaystyle \infty - 1$

$\displaystyle \infty + 1$

$\displaystyle (-\infty) - 1$

$\displaystyle (-\infty) + 1$

Thus, that real number will always be $\displaystyle \infty + 1$ for an example.

But this now leads me to the conclusion, that any number can be made greater than infinity.

$\displaystyle \infty + 2$

$\displaystyle \infty + 3$

$\displaystyle \infty + \infty = (2)(\infty)$

What is going on at this end of the spectrum for the real number line?
the general rule for binary operations (like "+") is:

a thing + another same kind of thing = still the same kind of thing (one apple + one orange equals how many bananas? hmm?)

real numbers are FINITE: that is, every real number is less than some integer (we can think of unit lengths "covering" a line segment whose length is a given real number).

infinity is...erm, not finite (hence the name). there's no point on the real line where we can say "the real numbers end here".

but let's pretend that the far-off horizon we can never reach is "out there, somewhere".

the only thing that makes sense is:

∞+1 = ∞. or, in general:

∞+r = ∞-r = ∞, for any (non-infinite) real number r.

one might suspect (and perhaps might be able to prove) that if r > 0, r*∞ = ∞, as well. opinions differ as to whether we should distinguish between -∞ and ∞ (the reasons are complicated).

but now we have a curious problem, what should 0*∞ be? the temptation is to say 0*∞ = 1, but this leads to more problems that you can imagine.

another problem comes when we try to imagine what ∞-∞ might be.

what happens is: as soon as you throw ∞ into the mix, it breaks the ALGEBRA of the real numbers. so ask yourself:

which is more useful: the power of algebra for modelling and solving problems, or the ability to say, "we can use infinity now!" (just how many things have you ever encountered that ARE infinite, anyway?).
• Dec 10th 2012, 04:05 PM
FelipeAbraham
Re: Infinity plus 1
Hi.

Well, infact there is a bunch of number theories that encompasses infinite and infinitezimal numbers.
Try Ordinal numbers, Hiperreal numbers and Surreal numbers, for example.

With my best regards.
• Dec 18th 2012, 03:55 AM
astartleddeer
Re: Infinity plus 1
Hi,