Thursday, September 15, 2005

Square Roots

I hope most of you are familiar with the process of long division of finding square roots. Have you ever wondered why the method works? If yes, probably you would have figured out the answer to that query, and you need not waste your precious time reading this post further.

However, if you haven't pondered over this matter, then its nice for me, as I have some people to read this post. Well, I learnt this long division process way back in my 5th grade, and at that time such a thought did not occur to me as to why the method worked. That's no big deal, very few people at that age are expected to think in that way.

I never inquired about this matter till my 10th grade, but sometime after my 10th grade, I happened to come across some book on "Theory of Equations" and in that book I studied about numerical solutions of equations. Then the thought occurred to me, as to why not apply the methods on a quadratic equation. The process almost matched the long division process of finding square roots, although it was more general enough to handle equations of any degree. Even then I was not satisfied and I thought that there must be a much simpler explanation for the square root method, and after some thought I grasped the idea behind that long division process.

Just to keep you curious, I will not explain the idea in this post, but will rather let you ponder over this simple problem. If you have got any ideas to offer please comment on this post. If not, please wait for a future entry in the blog.

What I find strange is that the idea, although simple in execution and proof, is not explained anywhere in any textbook whatsoever, and I stumbled upon it only after getting to "Theory of Equations". Contrary to this, the rationale behind the Euclid's algorithm of finding HCF (highest common factor) of two numbers is explained at length in many textbooks on number theory. It is based on two simple facts that:
  1. if b divides a, then HCF (a, b) = b
  2. if b does not divide a, and leaves remainder r, then HCF (a, b) = HCF (b, r)
So applying the division after a finite number of times you will get a zero remainder and the last divisor will be the required HCF.

The tragedy with books is that this simple rationale behind the Euclid's algorithm is explained at such a later stage that by the time you understand it your quest for knowing it dies. Moreover, number theory texts emphasize using this technique to find numbers m & n such that:

m*a + n*b = HCF (a, b)

rather than using it to find the HCF.

Hope this slight digression does not detract you from square roots. Please give it a thought to find the rationale behind the long division process. (Just to encourage: the idea is very simple and surely you don't need "Theory of Equations").


Anonymous Ramanujam said...

Books dont explain anything about what you missed.
I think it is because, these can only be explained in person and they dont know how to put these down in writing.
I did not follow your logic in print, so I say this,

12:50 PM  
Anonymous Anonymous said...

I agree with the previous comment. I think you had good intention of explaining the facts, when you wrote the article. But some how I got lost, may be I don't have much knowledge about the subject or you need to explain it in better way.

11:40 PM  

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