Is infinity a number?

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Shredderbeam

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Re: Is infinity a number?
« Reply #30 on: May 09, 2008, 10:06:56 AM »
This isn't even a debate.  It's not a number.  It's an abstraction, or a limit.

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EvilToothpaste

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Re: Is infinity a number?
« Reply #31 on: May 09, 2008, 10:48:08 AM »
The keyword was if.

Stuff your 'if's in a sack. 

On second thought, take your 'if's out, stuff yourself in, and drown in a river. 

This isn't even a debate.  It's not a number.  It's an abstraction, or a limit.

Errr, yeah, it is a debate.  Depending on the circumstances, infinity is a number. 

For more information, read: this thread. 

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cmdshft

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Re: Is infinity a number?
« Reply #32 on: May 09, 2008, 10:52:10 AM »
The keyword was if.

Stuff your 'if's in a sack. 

On second thought, take your 'if's out, stuff yourself in, and drown in a river.

Don't be mad because I am right, and slightly more intelligent than I present myself to be on a regular basis. :P

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Username

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Re: Is infinity a number?
« Reply #33 on: May 09, 2008, 11:34:49 AM »
The key word being some. The problem is that mathematics must be treated as a universal system in order to create consistent results. You can't just go tossing certain methods out the window.

Mathematics != FET
Except we do go tossing certain methods out the window everyday depending on what we are doing.  Math is a tool, and we use it differently and in different ways depending on the task at hand.

Many things in math are mutually exclusive to each other.  There is no "universal system".
If you can't argue both sidees, you understaned neither

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fshy94

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Re: Is infinity a number?
« Reply #34 on: May 09, 2008, 01:32:12 PM »
Infinity just means something never stops or runs out. If you get an infinite ammount of apples, you will never run out of apples. Simple.

If we have an infinite number of apple trees with each tree supporting three apples from their branches, how many apples do we have?

LOL...infinite, dimwit. Even eric cannot eat them all, geddit yet? Even if eric had help from you....
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Shredderbeam

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Re: Is infinity a number?
« Reply #35 on: May 09, 2008, 03:29:30 PM »
Quote from: EvilToothpaste
Errr, yeah, it is a debate.  Depending on the circumstances, infinity is a number.

How, how, HOW is it EVER a number?

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Anteater7171

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Re: Is infinity a number?
« Reply #36 on: May 09, 2008, 11:55:03 PM »
No, it doesn't behave like one.
I don't remember anything. Well, I do, but it's really vague. Like I was on drugs the whole time.

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Username

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Re: Is infinity a number?
« Reply #37 on: May 10, 2008, 01:47:34 AM »
Quote from: EvilToothpaste
Errr, yeah, it is a debate.  Depending on the circumstances, infinity is a number.

How, how, HOW is it EVER a number?
Read this thread...
If you can't argue both sidees, you understaned neither

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Chris Spaghetti

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Re: Is infinity a number?
« Reply #38 on: May 10, 2008, 08:32:19 AM »
Infinity is an 8 pushed over

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cmdshft

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Re: Is infinity a number?
« Reply #39 on: May 10, 2008, 09:18:28 AM »
Infinity is an 8 pushed over

OMG it all makes perfect sense now.

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Beeper

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Re: Is infinity a number?
« Reply #40 on: May 12, 2008, 01:18:55 PM »
Infinity is an 8 pushed over

OMG it all makes perfect sense now.
I AM IRON MAN! DUHHH DUGG MAW MAW WAAAM DA  DAA
Easy as 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679

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Raist

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Re: Is infinity a number?
« Reply #41 on: May 12, 2008, 01:33:08 PM »
Quote from: EvilToothpaste
Errr, yeah, it is a debate.  Depending on the circumstances, infinity is a number.

How, how, HOW is it EVER a number?
Read this thread...
So divide any number by infinity. what number do you get?

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Bushido

Re: Is infinity a number?
« Reply #42 on: May 12, 2008, 05:14:09 PM »
Quote from: EvilToothpaste
Errr, yeah, it is a debate.  Depending on the circumstances, infinity is a number.

How, how, HOW is it EVER a number?
Read this thread...
So divide any number by infinity. what number do you get?

zero, except if you divide infinity, in which case it is not defined.

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cmdshft

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Re: Is infinity a number?
« Reply #43 on: May 12, 2008, 08:24:41 PM »
Infinity is an 8 pushed over

OMG it all makes perfect sense now.
I AM IRON MAN! DUHHH DUGG MAW MAW WAAAM DA  DAA

 ::)

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narcberry

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Re: Is infinity a number?
« Reply #44 on: May 13, 2008, 04:46:27 PM »
Quote from: EvilToothpaste
Errr, yeah, it is a debate.  Depending on the circumstances, infinity is a number.

How, how, HOW is it EVER a number?
Read this thread...
So divide any number by infinity. what number do you get?

zero, except if you divide infinity, in which case it is not defined.

It's only "not defined" because the infinities you are operating on are "not defined." With more useful information than "infinity" you can get quantifiable numbers. For instance:
5 / any infinity = 0
infinity / infinity = not defined
Sum of Dispersal Rate A / Sum of Dispersal Rate B = 2 | Dispersal Rate B is twice Dispersal Rate A.
Sum of Velocities A / Sum of Velocities B = Some infinity | a greater acceleration is imposed on set B over set A

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EvilToothpaste

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Re: Is infinity a number?
« Reply #45 on: May 13, 2008, 11:52:52 PM »
The answer to this topic is mostly about the definition of "number", as Username said in some other thread.  For those that didn't get it, let me elaborate set theory a little bit. 

Start with zero, the empty set: 0 = {}.  Next, the number one is the set that contains the empty set: 1 = {0}.  The number two is the set that contains the empty set and the set with one member: 2 = {0,1}.  Three is the set containing the previous sets: 3 = {0,1,2}. 

You should quickly be able to see that the counting numbers are sets that contain all the previous sequential sets.  If we then say infinity is the set containing all these counting numbers, ordered sequentially, then one can conclude that infinity is indeed a number. 

One can argue that such a set cannot be constructed and thus is not real, but a look back through the history of mathematics should persuade such an argument.  It was little more than 100 years ago that most mathematicians considered pi to NOT be a number since it cannot be represented as a fraction of counting numbers.  Such a number as pi and e and the square root of 2 were given the name irrational as a form of derision, but the name lost its intended purpose as they become useful in everyday life.  The same can be said of imaginary numbers, surd (as in absurd), and zero, too.  Even though they cannot be constructed, they are useful to us as numbers. 

Infinity exists in mathematics and is used everyday in every facet of your lives, even if you don't know it.  Based on the only consistent and fundamental definition of a "number" (as shown by Georg Cantor), infinity is a number.  Even though in conventional maths it is used as a trend or a limit, not a number, the rules regarding its behavior are consistent with set theory. 
« Last Edit: May 13, 2008, 11:55:06 PM by EvilToothpaste »

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Dead Kangaroo

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Re: Is infinity a number?
« Reply #46 on: June 06, 2008, 06:31:33 AM »

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lived_eht_asan

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Re: Is infinity a number?
« Reply #47 on: June 06, 2008, 07:47:24 AM »
Right, infinity can fit as a "number" in the set theoretic definition.

But that is only in the sense that it is a transfinite cardinal, a member of the affinely or projectively extended Reals.  It is not a member of the Real Numbers themselves.

If an axiomatic definition is constructed, infinity does not fit as a regular member of the field (addition and multiplication are not defined as other members).  To be a member of the field, it must be commutative under + and *, associative under + and *, and have a multiplicative inverse (or be 1 and 0).

So you certainly can't go about performing regular math with infinity (because it is not a regular member of the field):

∞ * 8 = ∞ * 100

Therefore 8 = 100.


The problem isn't considering infinity, it's considering infinity as a single number. Infinity is actually an uncountably infinite set of numbers, like the real numbers.

Also, infinity can be "countably" infinite.  If you can construct a 1 - 1 mapping to the natural numbers, then it is countably infinite.  All of the rational numbers between 0 and 1 (1/2,1/3,3/4,9/73 etc) are countably infinite.  However, all of the irrational numbers cannot be mapped to the naturals, and are uncountable.  There are more irrational numbers between 0 and 1 than there are integers to infinity.

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EvilToothpaste

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Re: Is infinity a number?
« Reply #48 on: June 06, 2008, 09:11:37 AM »
Narc may have been thinking of a different meaning for "countable."  As in you cannot count to infinity; you cannot construct infinity. 

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Guessed

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Re: Is infinity a number?
« Reply #49 on: June 06, 2008, 12:37:24 PM »
Narc may have been thinking of a different meaning for "countable."  As in you cannot count to infinity; you cannot construct infinity. 

You may not be able to count to it, but i believe it can be constructed, because it has a symbolic representation.
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Kasroa Is Gone

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Re: Is infinity a number?
« Reply #50 on: June 06, 2008, 02:54:27 PM »
This is just three pages of people repeating what they've been told.

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Guessed

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Re: Is infinity a number?
« Reply #51 on: June 06, 2008, 03:13:15 PM »
This is just three pages of people repeating what they've been told.

You've just described every forum ever made.
Is Dino open source?

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lived_eht_asan

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Re: Is infinity a number?
« Reply #52 on: June 06, 2008, 03:30:03 PM »
Believe it or not, there have actually been idiots who have come on here and tried to do algebraic math with infinity, as though it was a regular member of the field, so it does matter a little.

As to how you define what "a number" is in a larger sense and whether to include infinity or not... that might be irrelevant.  But it is interesting.  The difference between countably infinite and uncountably infinite is important, to be sure.  You might have a math or logic error if you assume the wrong set.

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EvilToothpaste

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Re: Is infinity a number?
« Reply #53 on: June 06, 2008, 04:29:57 PM »
Narc may have been thinking of a different meaning for "countable."  As in you cannot count to infinity; you cannot construct infinity. 

You may not be able to count to it, but i believe it can be constructed, because it has a symbolic representation.

I don't think symbolic representation should be considered a "construction".  Such as, how would one construct an infinitely long ruler. 

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General Douchebag

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Re: Is infinity a number?
« Reply #54 on: June 06, 2008, 06:43:12 PM »
Infinity divided by infinity is 1. While neither number is defined, presumably both are the same, so the answer would be one.
No but I'm guess your what? 90? Cause you just so darn mature </sarcasm>

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narcberry

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Re: Is infinity a number?
« Reply #55 on: June 06, 2008, 06:47:09 PM »
The problem isn't considering infinity, it's considering infinity as a single number. Infinity is actually an uncountably infinite set of numbers, like the real numbers.

Also, infinity can be "countably" infinite.  If you can construct a 1 - 1 mapping to the natural numbers, then it is countably infinite.  All of the rational numbers between 0 and 1 (1/2,1/3,3/4,9/73 etc) are countably infinite.  However, all of the irrational numbers cannot be mapped to the naturals, and are uncountable.  There are more irrational numbers between 0 and 1 than there are integers to infinity.

We're talking about two different things, and I completely agree with what you've said. I was speaking about infinity in the abstract, as in the set of infinities, where an uncountably infinite set of numbers (infinities) exist.

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lived_eht_asan

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Re: Is infinity a number?
« Reply #56 on: June 07, 2008, 12:57:43 AM »
Infinity divided by infinity is 1. While neither number is defined, presumably both are the same, so the answer would be one.

∞  / ∞  is undefined.

The property x / x = 1  only applies when x is a member of a field (or other less restrictive groups/rings), which ∞ is not.

If you just treat infinity like a normal number:

∞ + 1 = ∞   implies 1 = 0 by the algebraic rules of a field.

The normal way to define something like this would be to use limits.  For example:

1 / ∞  is undefined.

But lim 1 / x  = 0, with x -> ∞

Give me any number as close to 0 as you want, and I can give you x high enough to make 1 / x even closer to 0.

But when you talk about lim x / y, as x,y -> ∞ you can't specify the rate at which the numerator and denominator increase.  By choosing the rate of increase, the limit can be any number you want.



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lived_eht_asan

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Re: Is infinity a number?
« Reply #57 on: June 07, 2008, 01:33:20 PM »
The problem isn't considering infinity, it's considering infinity as a single number. Infinity is actually an uncountably infinite set of numbers, like the real numbers.

Also, infinity can be "countably" infinite.  If you can construct a 1 - 1 mapping to the natural numbers, then it is countably infinite.  All of the rational numbers between 0 and 1 (1/2,1/3,3/4,9/73 etc) are countably infinite.  However, all of the irrational numbers cannot be mapped to the naturals, and are uncountable.  There are more irrational numbers between 0 and 1 than there are integers to infinity.

We're talking about two different things, and I completely agree with what you've said. I was speaking about infinity in the abstract, as in the set of infinities, where an uncountably infinite set of numbers (infinities) exist.

Oh, I do see now.  Great posting actually.

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cmdshft

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Re: Is infinity a number?
« Reply #58 on: June 11, 2008, 03:32:58 PM »
Is it possible to have an infinite set of infinte?

 For example: {infinite, infinite, infinite,...}

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narcberry

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Re: Is infinity a number?
« Reply #59 on: June 11, 2008, 03:48:04 PM »
Yes. For some exciting reading start here:
http://en.wikipedia.org/wiki/Transfinite
branch into whichever links you fancy.