# Infinite infinities

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Infinity has a lot of odd, or amazing, or impenetrable, properties.

Infinity can increase or decrease, but still be itself. Except that, even though infinity plus n (or plus infinity) is still infinite, infinity minus infinity is zero, but infinity minus n (some “ordinary” number that our brains can deal with) is still infinite. Infinity can be multiplied and divided and still be infinite. But infinity divided by infinity is 1. Infinity squared is infinite, as is infinity to any power. So infinity to the power of infinity is still infinity (to the power of 1). It's a really big huge meta-number that can't be transformed like other numbers. It's outside the normal behaviour range, or whatever. But Math would be lost without it. 'Specially those cosmologists and particle physicists.

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Infinity minus infinity is undefined, as is infinity divided by infinity. This means that you can get any result from it.

Infinity is just a tool and not a number. There are also several types of infinity which you would know if you knew counting. Other than counting, the only use I know for infinity is taking limits where things tend toward, but never reach, infinity.

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Infinity and Zero are my two favourite numbers . You can do some amazing things with them and without them number theory would not be possible as we know it. In fact a lot of today's technology (computers included) could not be created (or work) without the concept of Zero.

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Infinity minus infinity is undefined, as is infinity divided by infinity. This means that you can get any result from it.

Can you illustrate this? My limited mathematical grasp extends to the concept that something/something = 1, or something - something = 0, if the somethings are all the same symbol. But infinity doesn't do this huh?

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$0 = (1 - 1) = 1 - 1 + (\infty - \infty) = 1 + (-1 + \infty - \infty) = 1 + (\infty - \infty) = 1$ This would prove that 0 = 1. Do you now understand why $\infty - \infty$ is undefined rather than zero?

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An example that infinity/infinity=anything

let a=the total number of integers

let b=the total number of even integers

both a and b are infinite but a is twice as large as b

so, a/b=infinity/infinity=2

and b/a=infinity/infinity=0.5

The reason we get this is that there are many different ways to define infinity.

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Infinity and Zero are my two favourite numbers .

Infinity if not a number.

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Golden rule: there are no infinities in nature.

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Ah, but what is the infinite set? Is it unbounded? Is there an infinite series which is finite (has a finite result)? Or an infinite set of mathematical descriptions of infinity?

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An example that infinity/infinity=anything

let a=the total number of integers

let b=the total number of even integers

both a and b are infinite but a is twice as large as b

so, a/b=infinity/infinity=2

and b/a=infinity/infinity=0.5

The reason we get this is that there are many different ways to define infinity.

Thats like saying:

a/b=c

but with out knowing at least two of the variables, you can solve the problem, but if you have:

b = 2

c = -6

a / b = c

a / 2 = -6

-6 x 2=-12

a = -12

this can be solved, but without knowing b and c, a could be any thing, then the world would be drove insane cuz no one can figure for a.

opps! I did it again, by friends IQ just droped 5 points cuz i damaged his head with a unsovable equation.

Get my point, people only give infinity for an answer, cuz the dont realy know the answer!...or im talking crap, and have no clue what im saying...

-Iruka

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m4rc:

Actually the "number" of integers and the number of even numbers is the same!

The "number" (really called cardinality) of the integers, evens, odds, primes, even rational numbers is the same. It is commonly denoted by the hebrew letter aleph with a 0 subscript.

You should read about Cantor and set theory. It is very interesting.

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Right. I'm kinda trying to sample how far others have got with this. Infinity is one of those things, like pi or e or i that keep "popping up" or squeezing between the gaps, sort of. I know there are lots (maybe an infinite number) of infinite series that have an actual, finite result. But infinity has no "actual" value, no eigenvalue. The set of terms in an infinite series is infinite but the result is finite, is this the same as saying an infinite series is finite, though (or bounded)?

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Right. I'm kinda trying to sample how far others have got with this. Infinity is one of those things, like pi or e or i that keep "popping up" or squeezing between the gaps, sort of. I know there are lots (maybe an infinite number) of infinite series that have an actual, finite result. But infinity has no "actual" value, no eigenvalue. The set of terms in an infinite series is infinite but the result is finite, is this the same as saying an infinite series is finite, though (or bounded)?

I'm kind of suspicious of infinity in that way. Pi and e and all the other irrational numbers can be approximated as close as you like, but you will never get anywhere close to infinity. No matter how close you get, you are still infinitely far away. In more mathematical terms, you can converge to an irrational number, but infinity diverges. Does infinity have a use other than for taking limits or counting sets?

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Here is a good example of how two infinties are not equal. Consider two functions: $f(x) = \frac{1}{x}$ and $g(x) = \frac{1}{x^2}$. In the limit as x goes to zero, both of these will go to infinity. But what about the limit of f(x)/g(x)? Even though both the numerator and the denominator go to infinity as x approaches 0, the limit of f/g goes to zero, a finite number.

Same sort of thing on the other direction. $e^x$ goes to infinity faster than $x^3$ as x goes to infinity.

Nature may not have any infinities, but the concept of infinity sure is useful. We say that things are infinitely far away all the time. In order to completely 100% accurately calculate how far I hit a golf ball, the gravitational pull of every single atom in the universe should go into that calculation. Do we really calculate anything like that? No. For the purposes of this example, the Andromeda galaxy, Venus, and the moon are all infinitely far away and have no influence on the flight of the ball. In the same way, the fluid dynamic effects of the air going around every single tree on the planet have some effect -- but again, they are all infinitely far away.

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Here is a good example of how two infinties are not equal. Consider two functions: $f(x) = \frac{1}{x}$ and $g(x) = \frac{1}{x^2}$. In the limit as x goes to zero, both of these will go to infinity. [/math]NO! $\lim_{x\rightarrow{0}}\frac{1}{x}$ DOES NOT EXIST! If you approach from the positive side, you get $+\infty$, but if you approach from the left hand side, you get $-\infty$. However, $\lim_{x\rightarrow{0}}\frac{1}{x^2}=+\infty$.
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yourdad, I wasn't trying to get too technical, just showing that not all infinities are equal.

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There could even be an infinite number of such things...

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Ah, but what is the infinite set? Is it unbounded? Is there an infinite series which is finite (has a finite result)? Or an infinite set of mathematical descriptions of infinity?

Yes, there are countlessly infinite such sets. So long as the terms in the set get small enough quickly enough (including canceling), that can happen. When an infinite set is such that the farther along the set you go, the closer you get to a specific number, the set is said to converge to that number. The canonical example is $1/2 + 1/4 + 1/8 + ... = \frac{2^n-1}{2^n}$ which converges to 1 as n tends toward infinity. Many other infinite sets diverge (they get progressively bigger or smaller) or they oscillate.

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yourdad, I wasn't trying to get too technical, just showing that not all infinities are equal.

You're post addressed limits, not different infinities. These are examples of different sizes of infinities: The number of integers, the number of reals, the number of functions that map a real to a real, and so on. There are an uncountable number of different sizes of infinities.

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well if you minus infinty by infinity isnt it zero. same goes for if you minus a billion from a billion

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yourdad, I wasn't trying to get too technical, just showing that not all infinities are equal.

You're post addressed limits, not different infinities. These are examples of different sizes of infinities: The number of integers, the number of reals, the number of functions that map a real to a real, and so on. There are an uncountable number of different sizes of infinities.

In other words infinity actually has an infinite number of values, but is also actually none of them (in actuality).

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well if you minus infinty by infinity isnt it zero. same goes for if you minus a billion from a billion

No, that's pretty much the point of this thread.

Again, look at some of the examples I gave. The best one for this is probably to compare $x^3$ and $e^x$ again. As x goes to infinity, $e^x$ goes to infinity much faster. Both terms go to infinity, but in the limit as x goes to infinity, $e^x - x^3$, would still be infinity. Infinity - Infinity = Infinity, in this case.

Look what I mean:

$x$..........$x^3$..........$e^x$

1..........1.........2.72

5.........125.........148.4

10.........1000.........22026

20.........8000.........4.85*10^8

40..........64000.........2.35*10^17

You can clearly see how the $e^x$ terms starts to completely and totally dominate there after a while. It goes to infinity faster, and you can see that even though $x^3$ still goes to infinity, at x=40, the $e^x$ term is changed less than 0.000 000 000 005 percent if you would subtract the $x^3$ term.

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That does not mean these functions generate different infinities. It just means they approach the point at infinity at different rates.

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DH:

I may be wrong but I was under the impression that there are only 2 "sizes" of infinities:

aleph 0: The size of the integers, primes, rationals etc...

aleph 1: The size of the real numbers, R^n, C^n etc...

The continuum hypothesis is unproven, but generally thought to be true. It states that there is no cardinality between aleph 0 and aleph 1.

I once asked an acquaintance of mine, who has many mathematics degrees, about sets larger than aleph 1. If I understood him, he said that he could not imagine that there were any.

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and of all the topics which have been on, so many others are the ones I see contain wisdom and knowledge. This very topic renders me so emotional that I could run out of the database memory. I simply can not comment a single thing without embarking upon a sojourn of consensus ridicularum.

i even see people's replies which I always am interested in seeing what they have to say but this is just too much. enjoy the posts till it becomes degenerate and the inductions and deductions mentioned from the OP's post will become redundant.

That does not mean these functions generate different infinities. It just means they approach the point at infinity at different rates.

Sorry, just have to, cos i started reading it all instead of browsing but from bottom up i think.

I mean, the point of infinity... one specific point? lol , yes lol. I mean if you are speaking from a metaphysical point , then this becomes interesting. If you mention it as a valid mathematical definition, then there is just no sense in it.

(not the part about funtion behaviour, but how you worded 'the' point AT infinity).

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