Fun with Infinity

1+1=2

1+1=2

0.9 recurring = 1.

Prove me wrong. I dares ya.

According to the bible  1 man and 1 woman equals 1.
Soooooooooooooooo  1+1 can = 1

hehehehehe

Quote from: "The_Earth_Is_Round"

1+1=2

0.9 recurring = 1.

Prove me wrong. I dares ya.

OK. Here goes....


0.9999999... = 1-(∞^-1) You can't argue with THAT can you?
That means 1 minus the smallest possible number (infinity to the power of negative one, which is the same as 1 over infinity), which is the same as 0.9 reccuring. Both are ALMOST but not quite one.
therefore 1-(∞^-1) = 1 ?
Sorry but math and common sense doesn't allow for things to magically dissappear. Even if they are very very very small.

Therefore:


0.9999... = 1-(∞^-1) ≠ 1

Quote from: "The_Earth_Is_Round"

1+1=2

0.9 recurring = 1.

Prove me wrong. I dares ya.

Err... you can't prove illogical numbers using a logical system, so you can't prove it with mathematics.

Illogical numbers, indeed all numbers, are not a THING but a CONCEPT. The concept of 0.9 recurring is the idea of a number which is almost one but not quite by an infinitely small amount.

The very concept of 0.9 recurring is that it is not quite 1. Therefore, 0.9 recurring is not 1.

Err... you can't prove illogical numbers using a logical system, so you can't prove it with mathematics.

I'm sorry, can you define illogical numbers for us please?

The very concept of 0.9 recurring is that it is not quite 1. Therefore, 0.9 recurring is not 1.

*Actually*, the "very concept" of the decimal representation of any number is an infinite sum.  Any time you write blah.blah, you're not saying something atomic, but rather an abbreviated form of a longer description.

May I ask, do you believe that 1/3 = 0.33333...?

-Erasmus

sphere fans seem to think all items of mass are drawn to eachother, but cannot explain why. Newton and Einstein couldn'

Incorrect.  Newton couldn't, Einstein could (distortions in the four-dimensional spacetime, a geometry effect).

0.9 recurring = 1.

Prove me wrong. I dares ya.

Let
X=.11111...
10X = 1.111111111....
Therefore,
10X-X=1
9X=1

However, since we defined X to be .111.....

9X=.99999999....


Yep, both are true.
Its a limit thing.
(Just an aside.  Its a fun proof.  I know its kinda shocking to get and grasp, but it requires to you fully embrace the idea of an infinite sum an a limit.)
Besides, math is correct by definition.  Its an analytical truth.

01:003 And God said, Let there be light: and there was light.

Hm, allow me to correct you here:

"And God said:

"

;-)

According to the bible  1 man and 1 woman equals 1.
Soooooooooooooooo  1+1 can = 1

hehehehehe

that's multiplication, silly

1 x 1 = 1

flatties - answer please:
if the earth is flat and travelling upwards (why not downwards? why not side ways? why not hip-shakin' shimmyin' this way and that way?) why don't we break orbit with the sun? in fact, if there is no gravity how do we orbit the sun? if we don't orbit the sun how do we get seasons? if there is no gravity with the moon and shit how do tides work?

Quote from: "humanbean"

According to the bible  1 man and 1 woman equals 1.
Soooooooooooooooo  1+1 can = 1

hehehehehe

that's multiplication, silly

1 x 1 = 1

flatties - answer please:
if the earth is flat and travelling upwards (why not downwards? why not side ways? why not hip-shakin' shimmyin' this way and that way?) why don't we break orbit with the sun? in fact, if there is no gravity how do we orbit the sun? if we don't orbit the sun how do we get seasons? if there is no gravity with the moon and shit how do tides work?

They will just say that the sun orbits the Earth. Not sure what they'll say about seasons and tides.

Quote

sphere fans seem to think all items of mass are drawn to eachother, but cannot explain why. Newton and Einstein couldn'

Incorrect.  Newton couldn't, Einstein could (distortions in the four-dimensional spacetime, a geometry effect).

Quote

0.9 recurring = 1.

Prove me wrong. I dares ya.

Let
X=.11111...
10X = 1.111111111....
Therefore,
10X-X=1
9X=1

However, since we defined X to be .111.....

9X=.99999999....


Yep, both are true.
Its a limit thing.
(Just an aside.  Its a fun proof.  I know its kinda shocking to get and grasp, but it requires to you fully embrace the idea of an infinite sum an a limit.)
Besides, math is correct by definition.  Its an analytical truth.

Quote

01:003 And God said, Let there be light: and there was light.

Hm, allow me to correct you here:

"And God said:

"

;-)

It's not difficult to grasp unless you don't understand math. 1=.9~, but 2=Two. You cannot use infinity in math, because it is not a number. Also, repeating decimals are real numbers, and there is no such thing as an "illogical number", which happens to be an oxymoron. Anyway, .9~=1, but it doesn't mean anything. At all.

It's not difficult to grasp unless you don't understand math. 1=.9~, but 2=Two. You cannot use infinity in math, because it is not a number. Also, repeating decimals are real numbers, and there is no such thing as an "illogical number", which happens to be an oxymoron. Anyway, .9~=1, but it doesn't mean anything. At all.

What is going on here?

let us clear this up 1=1.0... and .9...=.9... now while the the differance may be infintesimally small it is still a differance.

It's already been posted, but I'll do it again.  Here's a fairly simple proof to show that .999...=1.

x = 0.999...
10x = (10)(0.999...)
10x = 9.999...
10x - x = 9.999... - x
10x - x = 9.999... - 0.999...
9x = 9
9x / 9 = 9 / 9
x = 1
0.999... = 1

If you want a proof that is even more iron clad, go study the epsilon-delta proof of limits.

how does 10 x .999...... - .999..... = 9 x .999......?

how does 10 x .999...... - .999..... = 9 x .999......?

They are equal by the axiom of distributivity, which asserts that

  ac + bc = (a + b) c

for all numbers a, b, and c.  In this case, a=10, b=-1, and c=0.999....

You are permitted to disbelieve that this is the case, although you would be hard pressed to give a good reason why.

-Erasmus

It's already been posted, but I'll do it again.  Here's a fairly simple proof to show that .999...=1.

x = 0.999...
10x = (10)(0.999...)
10x = 9.999...
10x - x = 9.999... - x
10x - x = 9.999... - 0.999...
9x = 9
9x / 9 = 9 / 9
x = 1
0.999... = 1

If you want a proof that is even more iron clad, go study the epsilon-delta proof of limits.

I stared at this for about 10 minutes till I found the mistake.
Watch

x = 0.999... Okay
10x = (10)(0.999...) Still Okay
10x = 9.999...Yes
10x - x = 9.999... - x Yes
10x - x = 9.999... - 0.999...
9x = 9 HERE IS THE MISTAKE
 You see 9.999.... should have one less 9 than the original x since it was multiplied by 10. THus 9.999....-x will not be 9 and the whole thing falls to pieces.
At least I think so
An bemused
Cinlef

No, thats the proof.

It works, and there a no mistakes in it.

So then about what I said how is that explained?
An puzzled
Cinlef

Well.
When you say it should have 'one less nine than the original x'

No, because the nines recur for infinity.

Infinity minus one is still infinity

Hmmmmm

Hmmmmm

HiveLord is correct.  By the definition of "infinity", removing any proper subset of the list of 9s leaves you with the same number of 9s that you started with.

Don't forget that "abcd.efg" (where a..g are some decimal digits) is just a convenient notation for representing the following sum:

1000*a + 100*b + 10*c + 1*d + e/10 + f/100 + g/1000

If you have an infinitely repeating decimal, like 0.99999...., it represents

1*0 + 9/10 + 9/100 + 9/1000 + 9/10000 + ...

If you multiply that by 10, you get

10*0 + 1*9 + 9/10 + 9/100 + 9/1000 + ...

But you haven't lost any terms; you've just multiplied each one by 10.  So all the negative powers of ten that were there previously are still there.  So if you subtract the first series from the second one, you get

10*0 + 1*(9-0) + (9-9)/10 + (9-9)/100 + (9-9)/1000 + (9-9)/10000 + ...

The only term there that isn't 0 is

         1*(9-0)

which is 9.

-Erasmus

Oh I see
sorry
An contrite
Cinlef

Infinity is so much fun.

You run a hotel with an infinite number of rooms, numbered 1, 2, 3, etc.  Each room can only hold one person.  On this particular night, every room is full.  It is just your luck that a man walks in and demands a room.  What do you do?

Infinity is so much fun.

You run a hotel with an infinite number of rooms, numbered 1, 2, 3, etc.  Each room can only hold one person.  On this particular night, every room is full.  It is just your luck that a man walks in and demands a room.  What do you do?

You give him the room. Of course to make sure that this doesn't happen again, you can divide every room in half which doubles the amount of rooms...or does it?

Every room is full, so you have no room to give him.  You can only put one person in one room.  Dividing the rooms counts as multiple people in each room.

You run a hotel with an infinite number of rooms, numbered 1, 2, 3, etc.  Each room can only hold one person.  On this particular night, every room is full.  It is just your luck that a man walks in and demands a room.  What do you do?

Give the guest in room 1 a message saying:

Please move to the next room; then, pass this message to the guest occupying that room.  Thankyou!  p.s. you can check out any time you like, but you can't never leave.

Then put the new guest in room 1.

-Erasmus

nice
but theres one floor if all the rooms are full and no one can leave then poeple will be perpetually move rooms and die of exuation

nice
but theres one floor if all the rooms are full and no one can leave then poeple will be perpetually move rooms and die of exuation

No individual person will perpetually move; each guest only has to walk the distance from his door to the next door, which is presumably finite.  An infinite number of people move, yes, and it takes infinite time for them all to move (since each one starts a finite time after the previous one), unless you can broadcast the message to all rooms at once.

If you can broadcast, each leaves his room at the same time, and each arrives at his new room at the same time.  Finite-time broadcast to infinite distances violates information theory, but whatever, so do infinite hotels :)

Anyway every guest is guaranteed to be able to move into the next room, so nobody dies of exhaustion.

-Erasmus

Infinity is so much fun.

You run a hotel with an infinite number of rooms, numbered 1, 2, 3, etc.  Each room can only hold one person.  On this particular night, every room is full.  It is just your luck that a man walks in and demands a room.  What do you do?

Wait, if an infinite number of rooms are full, there must be infinite guests.

Now correct me if I'm wrong, but infinity + 1 = infinity (??)

Therefore there are still infinite rooms and infinite guests, so the everyone has a room and the hotel is now full.

Wait, if an infinite number of rooms are full, there must be infinite guests.

Now correct me if I'm wrong, but infinity + 1 = infinity (??)

Therefore there are still infinite rooms and infinite guests, so the everyone has a room and the hotel is now full.

That's all true.  The question is, if all the rooms are full, where do you put the new guy, and how do you put him there?

The answer is given above.

-Erasmus