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Sunday, August 06, 2006

ok. this thought has been bothering me for somedays. but at last i've shoved off my letharguc spirit and lighted my bulb.
so lets talk about the biggest subject of all: infintãs.
lets observe the set of square numbers. now how many square numbers are there? For each square number would have a positive square root. so there should be as many squares as the number of positive roots (we're talking about perfect squares mind you). And how many positive square roots do we find? As many natural numbers obviously. Thus there are as many square numbers as there are natural numbers.
But then, all natural numbers are not squares, so there should be more of natural numbers. And so there are, in finite set. But then, one could argue that when you observe the set of natural numbers and the set of square numbers, till infinte numbers, there should be as many squares as there are N numbers. So as you approach infinty the ratio of square numbers (S) : natural numbers (N) should become 1. ?
Lets see. For n=1 (observing the set of N) S=1, N=1, S/N =1
For n=2 S=1, N= 2, S/N =1/2
For n=3 S=1, N=3, S/N=1/3 (I'm just writing this to show you the trend)
For n=4 S=2, N=4, S/N=1/2

thus you'll see, for n=m^2 S/N is always = m/m^2 = 1/m
as n→ ∞, m→ ∞, so S/N → 0.
So infact when we're going up the number ladder, the S/N ratio is actually decreasing. So are we moving away from infinity or are we coming closer to it.
I think the crux lies in the realisation that we should stop thinking of infinity as a big number.
Notice that initially, the S/N ratio decreases, then when it encounters a square number it suddenly increases. (I would like to give a graph of it, but i dont know how to; i am just glad enough i remembered how to post)
Look at these numbers:
for n=m^2, S/N ratio has surged up a little. It was lowest at n=(m^2)-1. The previous peak was at n= (m-1)^2
So for n = (m-1)^2 ; S/N = 1/(m-1)
n= (m^2) -1 ; S/N = 1/(m+1)
n = m^2 ; S/N = 1/m
now obviously 1/(m-1) > 1/m > 1/(m+1)
now imagine the n corresponding to 1/(m+1) [say n2] be ∞ {just suppose it!}
now the n corresponding to 1/m [say n3] would be ∞ +1.
at ∞ the S/N ratio would become 0. But at ∞+1 S/N would leap up to 1 (i'm not supporting this mind you; i'm just trying to explain things). and then because we cant understand or differentiate which ∞ is which, we can see them separately by 1) mathematics and 2) logical analysis. But this doesnt make much sense does it? And I am stumped by the mysterious infint├ús. But I hope I've set your minds whirring, so i'll be expecting posts.
Stay tuned for some more mind-boggling stuff on infinity.


Arun Chaganty said...

Hey. I've posted some graphs.
One thing I'd like to point out is that your basic function is
S/N = [sqrt(N)]/N...[] is greatest int fn.

When N -> inf, we can absolutely ignore the little fractional part in comparision with N, and it becomes:
S/N -> 1/N
which goes to zero.

This post really was interesting. But I'f I'm right, whenever you can put a one - one relation between two unbounded sets, then they have same cardinality. I have some trouble with this concept.

If you consider the cardinalities of even and natural sets, the one-one relation is f(n) : N -> E, and e = 2n
Now if we take the same E/N ratio as you have here, we would get 2. Yet they are said to have the same cardinality (aleph null). Your example is much better, because it gives zero as the lt. I haven't understood the concept in general.

sparekh said...

hey arun, you just started on my next topic! This one just set the stge for going into cardinality and countable and uncountable infinities. thanks for the graphs.

Arun Chaganty said...

Any time. I'm really looking forward for the post. (because I've a lot of doubts in the field).

sparekh said...
This comment has been removed by a blog administrator.
sparekh said...

my function S/N was [no. of squares : no. of natural numbers] going by the same , E/N would yield- n/2n = 1/2. so i'm not sure what you meant.

Arun Chaganty said...

sorry, I meant 1/2 only, just a little typo.

sparekh said...

oh ok