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Tuesday, October 24, 2006

Boring Numbers?

Hey. It's been a long time since I've posted anything, so I'll make a candid proof I saw on the net somewhere.

This deals with a really fundamental theory:

There is no such thing as a uninteresting number.

As we deal with the earth-shaking proposition, one must have a deep fundamental knowledge of sets. Let us limit our theory to the infinite set of natural numbers.

On with the proof:
Let N be the set of all natural numbers, of which X represents the set of uninteresting number. First let us order this set X in numerical order. Let us now analyse the first number of this set. Our understanding of what makes a number interesting is so fragile that to produce such an example is a great step forward. Thus this number being the FIRST uninteresting number is of vital importance, bringing our interest to it. Thus it can not belong to the set, and isn't the first element. Let X' be the revised set omitting this fraud. Thus the second term of set X is the first of set X'. We can apply the same reasoning here, and easily omit this element, and continue to do so for all other numbers of the set, making it a null set. Thus it is easily proved that there is no uninteresting number.

Lovely proof isn't it? But there does exist a very important logical consideration. If one considers ONLY the negative numbers, our present ordering is befuddled, but do not panic. Suppose we order them in REVERSE order, letting greatest come first. Thus we can proceed by the same steps as above.

Now, we come to a very important juncture. If we combine these two sets, and look at ALL integers, we're stuck. For sure the negativity is an important difference between '+' and '-' numbers, and we can't ignore it. A set of Z has no beginning or end, so we CAN'T order it, and use this lovely proof for it. So it seems that the interesting-ness of certain '+' or '-' numbers makes other numbers loose there interesting-ness. But I thought interesting-ness was an intrisic property of a number. I.E. that a number can be called interesting because of a property that it exhibits, i.e. primality (it being prime) or it's square-ness/cube-ness/n-power-ness, or a pattern exhibited by its digits (like 12321).
Appealing to this logic, we could either conclude two things: That there exist no uninteresting numbers, or that the interesting-ness of a number does infact depend extrisicly as well as intrisicaly, and that it depends on a negative counterpart (whether that is the negative of the same number or not can not be concluded though).
As much as I'd like to take the latter, for it's more complicated, and that it IS earth-shaking in its statement, I'll abuse Occam's Razor and say that the former is obviously true.
Occam's Razor is: All explanations are simple, and the more complex one is more probably wrong. Note the keyword: Probably. It's generally quoted with overexagerrations like:
When you see a reflection:
1) Light is getting reflected
2) There is an alternate reality, which can be entered through the window, and which is inhabited by people who look EXACTLY like you, and happen to do EXACTLY what you are in front of the mirror.
Sometimes I think that life would be a lot more fun if it weren't so obviously 1.
That's all for now. I hope to be back this weekend with a post on the brain.
Signing off and Id Mubarak,
Arun

6 comments:

Anonymous said...

Heyy Arun.

Interestring proof. I'd like to present an idea myself. But could you please tell me what is the Theorem of Mathematical Induction for once ?
( Reasons shall follow )

TWISH

RaSh said...

WoW!
It was really nice to read this... and at first it did sound a bit funny!

But yeah, we still can't give an example of an uninteresting number, even if we are considering the set of both +ve and -ve nos., can we??

Arun Chaganty said...

@Rash: Umm, wasn't that what I had devoted the last 2 or so para's for. I don't think anyone has ever thought so much for such a proof, but actually, this is a rather concrete proof of the same, and I think that it is actually valid.

@Twish: The principle of mathematical induction basically is a logical proof of extention. If you can show that:
1) The statement is true for the base case, i.e. 0 generally,
2) That if it is true for n, n-1, n-2, it IMPLIES truth for n+1.

Then, having proven the base case, case for 2 is true, and thus case for 3, and so on. It's like a recursive proof. Both statements are essential. If you don't have the second, you don't have a valid recursion, but if you don't have the first, you don't have a valid base case (I'm assuming that you still remember recursion from your programming days). Come to think of it, I've looked at almost everything through the eyes of a programmer these days.

Twishmay said...

Thanx 4 that! & now chekout my new post...

TWISH

Arun Chaganty said...

Sure have. But I have to say that my proof is in fact true (produce a counter example why don't you), and very accurate. So is your 1st proof. The 2nd one is kinda loony.

Twishmay said...

Hmm

Chek the comment for the second one. If poss, tell me a chat id .. YAHOO or GMAIL.

The 2nd 1 is infact better than the first one.


Im not sick.
Im my senses bro ;-)

TWISH