Fun with infinity - Printable Version

Fun with infinity - Printable Version

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Fun with infinity - shouri - 2009-03-11

The infinite hotel problem.

So just imagine that there's this hotel with an infinite number of rooms. One day a person walks into the hotel and asks for a room. The attendant looks to him and responds, "My dearest apologies, but we're completely booked."

The person is disappointed at first but gets a brilliant idea. "Um... what if you move the person from room 1 to room 2 and then move the person from room 2 to room 3 ... and so forth. And I'll take room 1."

The attendant looks at him and tells him, "Here's your key."

------------
Well thankfully the attendant was able to squeeze in one more person. Good thing this hotel has an infinite number of rooms right?

BUT! Little did he expect this to happen:

A countably infinite number of guests show up!

The attendant freaks out a bit, "Oh no, what are we going to do. I might be able to fit one of you, maybe two, but an infinite number of you?!" Luckily for that man, one of the guests had studied abstract algebra and found a great solution to his dilemma.

"What if you do this: Take the guest in room 1 and place him in room 2. Take the guest in room 2 and put him in room 4. Take the guest in room 3 and put him in room 6. And do that for all of the guests. All of us out here will take the odd rooms."

The attendant breathed a sigh of relief, "Enjoy your stay."

===============
I both love and hate being math major. The above is a not so intuitive proof for the reason why there are just as many even numbers as there are natural numbers. Most people would intuitively think, "Oh there are obviously twice as many counting numbers as there are even numbers." But that's not the case ;3
I both

Fun with infinity - Dusk - 2009-03-11

Remind me never to book at a hotel with infinite vacancies and an attendant that isn't bright enough just to add everyone to the lowest number not already filled.

Fun with infinity - KajitiSouls - 2009-03-11

If they were any smarter, one of those guests would have shown the attendant how to use magic and exploit the metaverse, as well as schewing the laws as we know it, so they can accommodate infinite more guests without much hassle.

Seriously, there's an infinite number of guests. One of them's bound to know it.

Spoiler

Fun with infinity - Nikkey - 2009-03-11

KajitiSouls Wrote:If they were any smarter, one of those guests would have shown the attendant how to use magic and exploit the metaverse, as well as schewing the laws as we know it, so they can accommodate infinite more guests without much hassle.

Seriously, there's an infinite number of guests. One of them's bound to know it.

Spoiler

I would like to see the raw math behind your "Infinite Hotel Problem". I might be able to get a better picture.

It's not that hard to understand, really. You have infinite natural numbers. In those infinite natural numbers, there are infinite even numbers, and infinite odd numbers.

Fun with infinity - Salguod - 2009-03-11

My friend told me the same story. Well, I guess the attendant in his story was smarter since he just thought of the solution to the two circumstances without help. I just nodded okay back then.

I guess I get it but math proofs like that just seem so abstract to me (I guess that's where they get the name) that I feel like I didn't get it at all.

Fun with infinity - Spaz - 2009-03-11

For those who don't know, this is Hilbert's Hotel.

Fun with infinity - GummyBear - 2009-03-12

Oooh Ahhh, playing with infinity!!! FUN FUN...

Let S be an arithmetic sequence as follow:
S = 1 + 2 + 4 + 8 + ... + 2n + ... {to infinity}

As we can see, S is a positive sequence, therefore
S > 0

Multiply S by 2
2S = 2 + 4 + 8 +...+ 2n + ... {to infinity}

Subtract S by 1 (by bringing the first term to the left hand side)
S - 1 = 2 + 4 + 8 + ... + 2n + ... {to infinity}

Oooohhh Aaahhhh!!! What do you know,
S - 1 = 2S

Now, let subtract S from both sides
-1 = S

But wait ..... we have already confirmed that S > 0, therefore
-1 > 0

qed!

Fun with infinity - KajitiSouls - 2009-03-12

Devil's Sunrise Wrote:It's not that hard to understand, really. You have infinite natural numbers. In those infinite natural numbers, there are infinite even numbers, and infinite odd numbers.

He said there were just as many even numbers as there are natural numbers. If limit calculus has taught me anything, there's different degrees of so-called "infinity".

GummyBear Wrote:Oooh Ahhh, playing with infinity!!! FUN FUN...

Let S be an arithmetic sequence as follow:
S = 1 + 2 + 4 + 8 + ... + 2n + ... {to infinity}

As we can see, S is a positive sequence, therefore
S > 0

Multiply S by 2
2S = 2 + 4 + 8 +...+ 2n + ... {to infinity}

Subtract S by 1 (by bringing the first term to the left hand side)
S - 1 = 2 + 4 + 8 + ... + 2n + ... {to infinity}

Oooohhh Aaahhhh!!! What do you know,
S - 1 = 2S

Now, let subtract S from both sides
-1 = S

But wait ..... we have already confirmed that S > 0, therefore
-1 > 0

qed!

You sir, are crazy xD

Fun with infinity - xLeviathan - 2009-03-12

GummyBear Wrote:Oooh Ahhh, playing with infinity!!! FUN FUN...

Let S be an arithmetic sequence as follow:
S = 1 + 2 + 4 + 8 + ... + 2n + ... {to infinity}

As we can see, S is a positive sequence, therefore
S > 0

Multiply S by 2
2S = 2 + 4 + 8 +...+ 2n + ... {to infinity}

Subtract S by 1 (by bringing the first term to the left hand side)
S - 1 = 2 + 4 + 8 + ... + 2n + ... {to infinity}

Oooohhh Aaahhhh!!! What do you know,
S - 1 = 2S

Now, let subtract S from both sides
-1 = S

But wait ..... we have already confirmed that S > 0, therefore
-1 > 0

qed!

If S > 0, it's -1 > 1? o_o;

Fun with infinity - Nikkey - 2009-03-12

GummyBear Wrote:Oooh Ahhh, playing with infinity!!! FUN FUN...

Let S be an arithmetic sequence as follow:
S = 1 + 2 + 4 + 8 + ... + 2n + ... {to infinity}

As we can see, S is a positive sequence, therefore
S > 0

Multiply S by 2
2S = 2 + 4 + 8 +...+ 2n + ... {to infinity}

Subtract S by 1 (by bringing the first term to the left hand side)
S - 1 = 2 + 4 + 8 + ... + 2n + ... {to infinity}

Oooohhh Aaahhhh!!! What do you know,
S - 1 = 2S

Now, let subtract S from both sides
-1 = S

But wait ..... we have already confirmed that S > 0, therefore
-1 > 0

qed!

infinity minus infinity? That's either negative or positive infinity, so your calculation can't be done Tongue

KajitiSouls Wrote:He said there were just as many even numbers as there are natural numbers. If limit calculus has taught me anything, there's different degrees of so-called "infinity".

Infinity isn't countable, remember. With that being said, infinity/2 equals to infinity.

Fun with infinity - shouri - 2009-03-12

Actually there IS a thing such as countably infinite. The set of natural numbers are countably infinite. So are the rational numbers. In fact, there are just as many natural numbers as rational numbers.

And to Kajiti, the infinity for even numbers is the same exact infinity for natural numbers. There are exactly as many even numbers as there are natural numbers. The only set of numbers that are more infinite are the set of real numbers.

in short the degrees of infinity for the different sets of numbers is as follows: Natural = Even = Rational < Real

Fun with infinity - GummyBear - 2009-03-12

Devil's Sunrise Wrote:infinity minus infinity? That's either negative or positive infinity, so your calculation can't be done

Otherwise, we'll be in a pretty big mess!!

Fun with infinity - Nikkey - 2009-03-12

^ I haven't worked a lot with infinity though, but infinity minus infinity doesn't make zero, does it? Tongue

shouri Wrote:Actually there IS a thing such as countably infinite. The set of natural numbers are countably infinite. So are the rational numbers. In fact, there are just as many natural numbers as rational numbers.

I'll be a meanie and state that there are infinite irrational numbers as well. Goggleemoticon

Fun with infinity - GummyBear - 2009-03-12

Mathematically speaking, infinity is a figure of speech, it's not a number, thus standard math operations cannot be applied to it, otherwise, you'll end up getting into the mess I posted above.

Fun with infinity - KajitiSouls - 2009-03-13

shouri Wrote:And to Kajiti, the infinity for even numbers is the same exact infinity for natural numbers. There are exactly as many even numbers as there are natural numbers. The only set of numbers that are more infinite are the set of real numbers.

in short the degrees of infinity for the different sets of numbers is as follows: Natural = Even = Rational < Real

What's the line of logic behind that? Also, just so I have clarity, what's the difference between natural and real numbers? Natural numbers are integers, whereas real numbers are any non-imaginative number, including decimals? (bleh I'm BSing, I don't really know the difference)

And Devil's Sunrise, what I mean by different degrees of infinity can be demonstrated by limit problems.

So say you have f(x) = x^2, and g(x) = x^3. Say we try to measure both f(x) and g(x) as x goes to infinity. They would both be infinite right? Say we define h(x) = f(x) / g(x). We try to measure h(x) as x goes to infinity. Does h(x) go to infinity, or does it go to zero? Since f(x) / g(x) = x^(-1), we can establish that we can pick apart different types of infinity.

infinity - infinity = undefined, since infinity "isn't an exact finite quantity".

Code:
Assume ∞ - ∞ = 0

∞ - ∞ + 1 = 1

Since ∞ + 1 = ∞,

∞ - ∞ = 1

1 = 0

Whoops!

Fun with infinity - GummyBear - 2009-03-13

There 2 types of infinities, convergence and divergence (sp?).

Convergence is where you see something like
0.9999999999999 {repeats}, which converges to 1

Divergence (as Celine Dion puts: "my heart will go on and on and on and on and on and on and on ..... ") is something that expands forever, eg the sequence I posted. Diverging limits cannot have mathematical operations done to them.

As for the number system, if you're to remove one set of numbers, eg improper fractions, then the amount of numbers is still just as dense. Why? because there is an unlimited amount of combination of numbers.

Fun with infinity - shouri - 2009-03-13

KajitiSouls Wrote:What's the line of logic behind that? Also, just so I have clarity, what's the difference between natural and real numbers? Natural numbers are integers, whereas real numbers are any non-imaginative number, including decimals? (bleh I'm BSing, I don't really know the difference)

And Devil's Sunrise, what I mean by different degrees of infinity can be demonstrated by limit problems.

So say you have f(x) = x^2, and g(x) = x^3. Say we try to measure both f(x) and g(x) as x goes to infinity. They would both be infinite right? Say we define h(x) = f(x) / g(x). We try to measure h(x) as x goes to infinity. Does h(x) go to infinity, or does it go to zero? Since f(x) / g(x) = x^(-1), we can establish that we can pick apart different types of infinity.

infinity - infinity = undefined, since infinity "isn't an exact finite quantity".

Code:
Assume ∞ - ∞ = 0 ∞ - ∞ + 1 = 1 Since ∞ + 1 = ∞, ∞ - ∞ = 1 1 = 0

Whoops!

The line of logic behind there being just as many even numbers as natural numbers:

We make a mapping of the natural numbers to the even numbers

1 -> 2
2 -> 4
3 -> 6
and so on.

Every natural number will have a corresponding even number associated with it. And since the natural numbers are countably infinite (<-its part of the definition for the set of natural numbers), then we can say that the natural numbers and even numbers have the same degree of infinity as each other.

And no, the two "types" of infinity are:

1) Countably infinite: rational numbers, natural numbers, integers
2) Uncountably infinite: Real numbers

Devil's Sunrise, the reason there are an infinite amount of irrational numbers is because they're what's needed to create the distinction between rational and real numbers.

Rational numbers (countable) +irrational numbers (uncountable) = real numbers (uncountable)

Fun with infinity - Nikkey - 2009-03-13

GummyBear Wrote:0.9999999999999 {repeats}, which converges to 1

It doesn't converge to 1, it is 1, ryt? Sadly, infinity is a badass thing which is hard to understand sometimes.

Fun with infinity - shouri - 2009-03-13

Yes its been proven and agreed upon by the mathematical community that .999.... <-an infinite repetition of 9's is in fact equal to 1.

Fun with infinity - KajitiSouls - 2009-03-13

Devil's Sunrise Wrote:It doesn't converge to 1, it is 1, ryt? Sadly, infinity is a badass thing which is hard to understand sometimes.

Here's a simple "stupid" proof:

Code:
1/3 = .33333333333333...

3 * 1/3 = 3/3 = 1

3 * .333333333333333... = .9999999999999...

3 * .333333333333333... = 3 * 1/3 = 1

I agree with Devil's Sunrise, you can't really say a static number "converges" to 1. A function on the other hand...

And I generally don't like to work with infinity lol.