Monday, June 15, 2015

Almost Binary

I’m working on a research problem dealing with numbers that are either close to unity or close to zero. So much so, that the mathematical training and practice that I’ve had up until now would not allow me to do this problem. Indeed, in several iterations, I have done pages of derivations that end with a trivial result 1=1 or 0=0. (1=0 means that I have made a very different type of mistake).

So much my time recently has been spent trying to avoid triviality.

As I’ve been working on this problem, I’ve been inventing math, and it blows my mind with a sense of discovery. Then I look it up, and it turns out that it has been discovered in 1760. A shout-out to all of you mathematicians on Wikipedia: Thank you so much!

Notes to self:
2. Make friends with a mathematician

This week it’s averages.
How do you take the average of two numbers, A and B?

1/2(A+B) or (AB)^1/2
Arithmetic mean or geometric mean?

These numbers will be different: am1 and gm1.

But then you can take the arithmetic and geometric mean of am1 and gm1.

½(am1 + gm1) = am2
(am1 x gm1)^1/2 = gm2

and keep on going and going and eventually the answer will converge. That's the arithmetic-geometric mean.

I’ve been playing with this for a while, but I just discovered that Gauss had already been playing with these probably close to 200 years ago.

For me, doing math is meditative, lovely.  I am fully concentrated on the abstract, far from the world of people & feelings & things I have to do.

Here is a partial recording of the part of my brain that is not re-doing 200-yr-old math.

1.     Um. Is this a good use of my time?
2.     How could it be bad for me if it’s so much fun?
3.     Now I understand how to derive things
4.      Omigod I’m sooooo sloooow
5.     I suppose it’s a better use of my time than other things I can be doing
6.     But perhaps not a better use of my time than other-other things I can be doing
7.     In the end I really really want an elegant solution to this problem and I know I’m almost there.
8.     Keep on plugging.

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