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

Sunday, March 13, 2016

True power of correctness

Talked about my disruptive approach to mathematical discovery, where was more into brainstorming ideas where meant many were useless or wrong, but would put up public which helped me have them critiqued. So different am sure from what most math people do, but am NOT a mathematician, where like to emphasize, as frees me to do things different.

Still some people can ignore even the simplest disclosures and try to hold you to rules they think apply. And with disruption of established ways? Yeah that word "disruption" can mean some will react badly, and the insulting names game can seem appealing to them, but to me is childish. I call it a playground mentality. Which to me emphasizes is talking a behavior and not labeling a person or group of people.

And so long ago seems now, when was actively arguing out math ideas can remember distinctly when something remarkable happened. And yes had times when was convinced some idea was correct only to have it fall apart in shambles, worst after MONTHS. But at some point I flipped on my own ideas as realized the math does not care.

If you have a correct mathematical approach it does not care. It does not know you exist. The math has no opinion on you. It has no feelings whatsoever it is just correct. It will work.

At some point arguing with others, which just don't bother now, realized an odd feeling as the slings and arrows of words kind of made me flinch but was like a shield was there. Eventually it's like this odd feeling as you realize nothing is getting through. The math deflects all attempts against it. Blocks all attempts at reason to find fault, as there is none.

And the words ceased to have meaning.

The math does not care. It protects without thought. Defends without effort.

I was shielded by correctness not because the math defended ME but because I relied upon it.

It is fun then, as you attack it. Try to take down your own ideas. Get creative at approaches to debunk it.

The true researcher should be the most virulent in testing against own ideas. Best ones don't care what you think. Flawed approaches can reflect your need or mental limits, or those of others if you rely upon ideas handed to you when wrong.

There is no point in defending correct mathematics, as trust me, if it IS correct it is unassailable. Perfectly shielded. I try to think it defends me but must fight my emotion.

The math does not care.

James Harris

Thursday, March 10, 2016

Disruptive approach to simpler math

One of the things I realized over a decade ago with my approach to mathematical discovery was my leverage of the web to check ideas. So I'd brainstorm for ideas, put them up public, and get rapidly critiqued, usually meaning the idea failed. But sometimes would rapidly innovate to something different, robust and perfect.

Like my way to count prime numbers which is SO CLOSE to what was known but is more compact because of innovative approach.

Will give the sieve form as is smallest to show, and actually is MUCH prettier with the summation sign but not so sure on how to do that with Blogger or if I can. Interested readers can just write it out to see it that way.

With integers ONLY, where pj is the jth prime:

P(x,n) = x - 1 - sum for j=1 to n of {P(x/pj,j-1) - (j-1)}

The P(x,n) function will count prime numbers up to and including x, if n equals the count of primes up to sqrt(x), and if as you iterate you never let n be greater than the count of primes up to and including sqrt(x), if the function receives an n greater than that value it just needs to reset it to that count.

That's mine. I found it.

For example P(100,4) = 25.

(Wish to pursue more? I use labels on most posts, and beneath this one should see the prime counting label. Click on it for more related posts!)

My way of doing research was very disruptive as I wasn't so excited about mathematical errors, as much as idea generation, and tossing of erroneous approaches rapidly. Where I used the web for feedback to help figure out when things were wrong. And now finally with the web can distribute perfect ideas which withstood all challenges.

As mathematics is awesome in a fascinating way: correct ideas can't be broken, cannot fail, cannot lose against challenge. So why not hit them with everything you've got?

The math doesn't care.

James Harris