Wednesday, March 16, 2016

Why bother with sqrt(3) approximation

Turns out a simple method for calculating the square root of three is to use the following.

sqrt(3) approximately equals xn+1/yn+1, where:

xn+1 = 362xn + 627yn

yn+1 = 209xn + 362yn

and x0 = 1, and y0 = 0;

Which is just for fun, as have noted every time I put up one of these that of course you can just use a calculator. But it is interesting to me, if I just came across that myself, maybe could make me a bit curious. If you try simplest case x1 = 362 and y1 = 209, so 362/209 approximates sqrt(3). And can see how well it works there.

(362/209)2 approximately equals: 3.0000228

Where clipped a bit as no reason to show all of what pc calculator gave.

Iterate once, and you get, x2 = 262087, y2 = 151316

And: (262087/151316)2 approximately equals 3.00000000004367

Where arbitrarily clipped again. And that method goes to infinity, so you can iterate as much as you like.

Will show one more.

Iterate again: x3 = 189750626 and y3 = 109552575

 (189750626/109552575)2 approximately equals 3.0000000000000000833

And 189750626/109552575 approximately equals 1.732050807568877 showing only digits that match with sqrt(3).

For a mathematician? Probably a trivial thing. Is easily explained how it works, and goes back to number theory that is considered rather old. Though I used my own tool to find it, so for me that part is cool. Does my approach work further? Nope. What I did works for sqrt(2) and sqrt(3) and nothing else. Though I do wonder if I were more clever if could figure out how to get something from it to work further. Tantalized by the clues.

Am confident that if you program it into a computer you can quickly exceed its capacity to check the result. But still, is just the sqrt(3).

So why bother with it.

Not even a question really. For me? I like playing with numbers. For some maybe there is some mystery, like why does it work? How does it work?

I think that's cool.

Not every one wants to be a math expert and look at only advanced mathematics that pushes the limits of even the greatest minds on the planet to understand.

I know I don't. Sometimes I just like to play with numbers, just for fun. Sharing that? Why not?

So it's easy mathematics. Guess what. I like easy.

James Harris
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