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Thursday, June 01, 2006

I HaVe ThE POWER!!

Well, does the title remind you of the classic “He-Man” Cartoon series?? Well I’m not posting about cartoons but this has to do with “Power.” But that’ll come later.

First of all, I wanted to post my intro too but was waiting for Twish to mail me the pics from the camp. So that’s me saying: ”I have the Power!!”. You want a formal Intro? Here you go:

Rasagy aka RashTheGr81 at KVPY Camp

Name: Rasagy Sharma (a.k.a. ‘Rash’ The Gr81)

Residing in: Delhi

Interests: Dramatics (Done loads of TV and Radio Shows), PlayingSoccer (Some of you who played with me know that – It’s a second religion for me!), of course anything to do with Computers (I just luv Computers!) and last but not the least, making Friends (and continuing the friendship too!)

Enough about me. So after reading Bravura’s post, I was sitting in front of my computer wondering about what went wrong there. Got some ideas, but couldn’t join them all up. And then a short chat with Twish only raised my ideas to higher powers… Yes. It was powers about which I have been thinking all night. Well, most of such weird looking problems arise from the fact that what we have been taught about powers, multiplication etc. is too basic (coz we were too young to understand complex nos. at that time). And even now, we haven’t updated our information about such elementary things, though we do know complex nos. etc. And thus arise such ques.

Well here are some examples:

1. 10 = 20

Then won’t 1 = 2 ?? (Say we have 2x=22, then we say that x = 2, don’t we??!!)

2. To prove: 3 = 4
Proof:
Let a+b = c
4a-3a + 4b-3b = 4c-3c
:.4a+4b-4c=3a+3b-3c
:. 4*(a+b-c)=3*(a+b-c)
:. 3 = 4

Maybe most of you can find what’s wrong up there (I think you all Should find that out easily!). But ever wondered why even your computer’s scientific calculator can’t compute (-8)1/3 ? Even though the answer is an obvious -2 ? (It says invalid input for the function: Got some clue??). Though its fine that it can't compute (-8)1/2.

Another nice problem (which Twish told me during the camp) is the following:

We know that

4*4 = 4 + 4 + 4 + 4

:. 4*4= 4 + 4 +… 4 times.

:. x*x = x + x + … x times.

Differentiating,

2*x = 1 + 1 + 1 … x times.

:. 2*x = x

Hence 2 = 1!!!

All such problems arise because we forget our assumptions – the definitions of the functions – the domains of the functions and we tend to generalize a formula we learnt in junior classes here.

Now what I was thinking was this:

x = 11/3

Then x3 = 1. And we get the solutions for x. So as this n increases, the no. of solutions for x also increase (which is n itself).

But if n-> infinity , we get only one value for x = 1. (Actually here we can’t replace n by infinity as n-> infinity but n is not equal to infinity.)

Also,

2 0 = 1
1 0 = 1
0 0 = ?

Well infact 0 0 an indeterminate form. So we can treat it as an exception.

But what if n is a complex no.?? According to Twish, it’ll lead to infinite solutions for x. I reckon that’s what messes everything in Bravura’s problem. So can some body explain to me what’ll happen with complex powers? (Do we have any physical interpretation other than just saying Z = 2 i finding Z??)

Sorry for writing such a Llllllooooonnnngggg post, but I think it'll compensate for my absence in the coming few days when I'll go out of station. So I hope to find many posts by then (posts by members other than Twish's Dead Ends - come on others plz post!)

And before ending, a bit of cheerleading… (yeah I’m at least good at that!) U ppl are doing a gr8 job posting stuff, but do visit the blog regularly (Too busy studying huh?) And do give more ideas as to what more to add in the blog..

So SwItCh On YoUr FuSeD BuLbS!!

~RashTheGr81~

6 comments:

TwIsTeR said...

Nice discussion !

I agree that basics are the loopholes that I've utilized in making all these false proofs !!!

But sometimes it gets very tricky to catch them. The gud part is that once to CATCH them ! U get a much deeper understanding of the subject !!!

PPL DONT FORGET TO READ BY DEAD ENDS PART 2 !!!

Twish

bravura said...

I agree with Rash totally. I think somebody should post some material on powers. especially complex powers so that all of us get some clue as to what is happening.

sparekh said...

hey: how do you get infinite solutions for raising something to the power i?
the problem swetabh posted:
e^2πi = cos2π + isin2π
= 1
but e^2nπi = the same thing.
so does the set of general soln.s appear from these trigonometric functions actully?
spidy

sparekh said...

about the 2x = x thingy,
it works out fine for:
ax = x+x+x+.. a times
bx = x+x+x+.. b times
but when it comes to
x.x = x+x+x+... x times the problem arises.
i think we must consider one x constant and one of them a variable. sort of partially differentiate the expression. pls tell me why.
great post rash.
spidy

zinx said...

Hey Rash, the flaw in the logic is simple. look,when we say
4*4=4+4+4+4, but the problem is
x*x isn't = x+x+x....x times bcoz x is a real no.
for ex- let x=3^(1/2) that is irrational. how can you write
x+x+....3^(1/2) "times". this addition law holds good only for natural nos. another ex-
(-3)*(-3) isn't = (-3)+(-3)+(-3).
thus y=x+x+......x times is a discontinuos function (for natural nos.only) and hence can't be differentiated.

Arun Chaganty said...

I'd like to say something about the complex powers bit, even if the thread is a bit dead (like shrodinger's cat).

My personal take on it is that when you raise something to the power of a complex number, you endup getting something in the reduced form of:

(blah)^real*(blah)^i*imaginary
and blah = e^ln (blah').
so we get e^i(theta) form. and can continue. I don't see where we get infitum of solutions here.

But on the other hand, if we take logrithims of negative and complex numbers, we get a set of solutions. Here I believe we should just take a small set, which gives the primary solutions, like we do for trignometric inverses.

You are very right about the fact that we haven't bothered to expand our horizons, especially with complex numbers. For example sin(complex), or evaluating arctrignometric (complex) (or even arccos(-2)). These are all activities we should do. Ok then that's my take