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#1 2005-11-09 12:59:17

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Infinity comparisons

Which is greater, infinity or 2infinity?

I like to think of infinity not as an infinitly large number but as a number that increases continuously. If you think of the two infinity terms as two trains traveling on a number line really fast, starting from the same point, their distance from the origin would always remain equal. But no matter how far the trains go, twice the distance of one train will be greater then the distance of the other train.

If you were to write the fraction:

2infinity/infinity

would infinity cancel out? Creepy!

Does infinity = infinity?

If so, then the additive and multiplicitve property of equality should apply and we can multiply or divide an equation by infinity. Add or subtract infinity from both sides. Also we should be able to multiply or divide any fraction by infinity over infinity without changing its value.

And how about division by zero? Is it infintiy? As a denominater gets smaller, the value of the fraction gets larger, but does zero work the same way?

In grade school arithmatic, we divide by placing the dividend in a box and the diviser to the right of it, for instace 5/2 the first step is to think, "how many times will 2 fit into 5? 2.5 times. If we use the same method for division by zero, 3/0, whats the most amount of times 0 can fit into 3? An infinite number of times! By this method infinity seems like the answer.

If we think in terms of fractions, 1/2, 1/3, 5.   1/2 is one half of the whole, 1/3 is one third of the whole, 5 is 5 wholes, 1/0 is one zeroith of the whole, or one NONE of the whole. One none of the whole should be zero!

If we use algebra 5 * 0 = 0. Then does we could say, 0 = 0/5. So division by zero could equal zero.

We could also rearrange it to find, 0/0 = 5.  We could also use 7, 9 or any number in place of 5 and find that 0/0 equals any number and every number!


A logarithm is just a misspelled algorithm.

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#2 2005-11-09 17:05:58

ryos
Member
Registered: 2005-08-04
Posts: 394

Re: Infinity comparisons

If so, then the additive and multiplicitve property of equality should apply and we can multiply or divide an equation by infinity. Add or subtract infinity from both sides. Also we should be able to multiply or divide any fraction by infinity over infinity without changing its value.

I tried that once, and the grader docked me points.


El que pega primero pega dos veces.

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#3 2005-11-09 23:31:52

ganesh
Moderator
Registered: 2005-06-28
Posts: 14,537

Re: Infinity comparisons

Go to this page or this one.


Character is who you are when no one is looking.

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#4 2005-11-17 14:53:29

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: Infinity comparisons

What if we were to use a completely gigantic but fixed number. We could treat it as any real number, we could use any of the laws of equality, (multiplicitive, additive) but it wouldn't have the infinite property where it can't really be measured. Might be usefull.

For instance, if we were to think of a number in groups of  999's, 100 times. The number would be:

999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999"!" FACTORIAL! :-D

If we were to call this number "w00t", then we could say "w00t^w00t" which are both factorials of the above. Note, not only is the base a factorial of the above number, but the exponent is also a factorial of the above number. Now thats a tyranasaurus number! Especially if you realize that 10!^10! is too big to fit on a 10 digit calculator. :-O We could now define "Uber" to mean "w00t^w00t"

Now "uber" is so big that 1/uber might as well be called zero.

Might be good to use for some experimenting. It doesn't posess certain qualities of infinity, for instance, you could not even theoretically say, the expression 1/x has a limit of Uber as x aproaches zero. Thats wrong for obvious reasons. But Uber has a fixed value, and it can have coefficiants, and you can multiply or add this number to both sides.   

I just thought it was a cool thought. :-)

Last edited by mikau (2005-11-17 14:56:51)


A logarithm is just a misspelled algorithm.

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#5 2005-11-17 17:00:18

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Infinity comparisons

How about (100!)!, that's pretty big, and easy to express.

So w00t could be (100!)! , and Uber could be w00t^w00t

And t00t could be 1/w00t, and Unter could be 1/Uber.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#6 2005-11-17 17:04:24

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: Infinity comparisons

being the reciprocal, shouldn't it be rebu? Thats the inverse of uber. (or is it just uber in a mirror?)


A logarithm is just a misspelled algorithm.

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#7 2005-11-18 04:21:40

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Infinity comparisons

Using that reasoning, half should actually be owt.


Why did the vector cross the road?
It wanted to be normal.

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#8 2005-11-18 06:14:13

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: Infinity comparisons

half of w00t?

Well if half of w00t is owt, twice w00t should be "in".


A logarithm is just a misspelled algorithm.

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#9 2005-11-18 06:15:28

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: Infinity comparisons

oh your talking about two! Stupid me! I gets it. Hey I like it! One owt!


A logarithm is just a misspelled algorithm.

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#10 2005-11-18 09:19:06

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Infinity comparisons

And 10% = net


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#11 2005-11-28 18:16:53

darthradius
Member
Registered: 2005-11-28
Posts: 97

Re: Infinity comparisons

I like to think about...
which is greater?...
the infinite number of real numbers between 1 and 2, or
the infinite number of positive integers?

can we really make a distinction between countably and uncountably infinite sets?


The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
                                                             -Bertrand Russell

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#12 2005-11-28 21:49:26

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Infinity comparisons

So, is there more "expandable" space between 1 and 2, than in all the counting numbers?

Interesting ... it would seem that the Reals offer an extra dimension, but ... infinity is infinity.

That sparked me to think: (let's ignore negatives) are there twice as many whole numbers as even numbers?

It would seem so, but if you then ask is the infinite set of whole numbers twice as big as the infinte set of even numbers, then ...


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#13 2005-11-28 23:03:09

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Infinity comparisons

For every counting number, you can pair it off with an even number by multiplying by 2.

1, 2, 3, 4, 5,   6,   7,   8,   9,  10...
2, 4, 6, 8, 10, 12, 14, 16, 18, 20...

So there are equal amounts of counting and even numbers.

But, some numbers, like 1, 3, 5, 7 etc. are counting numbers but are not even numbers
and all even numbers are counting numbers, so there must be more counting numbers
than even numbers.

YAY! PARADOX!

Let's just all agree that it's a paradox and leave it there before our brains start hurting.


Why did the vector cross the road?
It wanted to be normal.

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