Out of range

Numbers are getting out of their range. That is, numbers are wandering into the next higher range. The ranges are defined on my Hamlet page. What numbers not getting out of range means is that budgets can be billions of dollars but not googols. Here are some examples of why numbers don't get out of their ranges, one for each range boundary:

1. A committee of 1,500 can't get the job done.

2. If you sue for $30 trillion, you will never collect.

3. A googol or more years is preposterous - that's more years than there are elementary particles.

4. Disk drives are getting too big - the numbers of their possible states are inconceivable.

5. Logicians try too hard to define huge ordinal numbers and hence finite numbers. They can never exceed W, defined as the largest number that we will ever be able to define (notation due to Rudy Rucker).

Recently, however, numbers are getting out of their range. I mentioned earlier that my home disk drive should have a literary number of possible values, as the sequence of characters can be or could be a literary work. But my current disk drive is 80GB, putting the total number of configurations in the inconceivable range instead. Now I hear that 9/11 families are suing rich Saudi families for $1 trillion. This is 100 trillion pennies and so comes close to being a scientific (a quadrillion or above) number instead of a statistical number (between a thousand and a quadrillion). Can they ever get that much?

There's a good reason why in our common life we almost never have numbers much greater than one quadrillion: 1,000,000,000,000,000. The total worth of the world is probably about $10-50 trillion, or in the early quadrillions of pennies. Objects that we meet in our lives, such as cars, buildings, hula hoops, gold necklaces, space shuttles and so forth, have to have a net worth less than that of the entire world, and to measure worth, we need at least a penny, which is coming close to being worthless nowadays. The worth of all these objects must therefore be less than a quadrillion of units we can perceive. This is why we see trillion a lot in our everyday lives, but not a quadrillion.

Numbers rarely get out of their ranges. I will never be able to use all of my disk drive, and the 9/11 families will never be able to collect all that money.

PRIMES is in P

That is grammatically correct. The problem called PRIMES is that of asking of a number whether it is a prime number or not; that is, whether it is divisible only by itself and 1. If I say 2773 = 47 x 59, you can check it out by multiplying those two numbers. To see whether 2773 is prime or not, we need to check all the numbers below 2773 to see if they divide 2773. There are tricks, such as only checking up to the square root of 2773, and only checking prime or odd numbers, but it still is a tedious task for large numbers. Fortunately, there is Fermat's little theorem which states that if you raise, say, 3 to the 2542 power and keep only the remainder upon division by 2543, then if 2543 is prime, you will get 1. But many composite numbers such as 561 will give 1 as well, so this is not quite a good test for primality. But if you bang around for numbers and raise them to the one-less power and get 1 every time, that is evidence, but not conclusive, that the number is prime. It was an open question as to whether it was easy to prove it is prime.

I heard in the news recently that the problem was proven to be easy; that is, "in P". This means the time it takes to solve it is proportional to some power or polynomial on the number of digits in the number. Checking each divisor out is exponentially complicated - the time it takes explodes exponentially with the number of digits in the number. The Indian mathematicians Manindra Aginwal, Nitin Saxena, and Neeraj Kayal found a much easier way of doing it, and have printed it in a paper that is available by going to http://www.cse.iitk.ac.in/news/primality.html. You apply their algorithm to a number p, and in a fairly short time the computer says that it is either prime or composite. If it says it is prime, that settles it. If it says it is composite, then you have absolutely no idea of what the factors are. That is a much harder problem - so hard that secret codes and encryption are based on it.

So one by one these problems are being settled. The big one, that of P = NP, is still a long way from being solved. Will I see it in my lifetime?

Imaginique

I went to a show at Busch Gardens last night to celebrate my birthday. We chose a weekday to do it because we figured the crowds would be less. But when we got there there were long lines waiting to get in. The gates opened up 30 minutes before the start time of 8:30 pm to an enormous outdoor theater. There was room for everyone. As far as the acts themselves - it was all in fantastic color and showed some differences from other events I have been to, especially concerts and circuses. Clowns called on audience members to mimic them. Then the show started, featuring a female singer and driving music, and three women high on trapezes. Then came these tube creatures who lead to two men setting on a plank on a ball. Then came this flying shirtless man using a white dangling cloth as a swing, followed by the "work of three" - three strong looking shirtless men doing all kinds of acrobatics including standing on each other in a 3-tower. Despite the crowds, we had no trouble getting home. The show was presented on one of these balmy summer nights. It was somewhat warm, but occasionally a breeze would come. It was a really nice night and a memorable one - the night we did something unusual and traveled an hour to see an imaginative show - Imaginique. Hope to see more of this in the theme parks in the future.

Sports Society

We live in a sports society. I found this in the newspaper this morning. It was an editorial describing a hypothetical sale of baseball caps. These caps have pictures of birds on them with the name of the bird below it; nothing else. The birds on these caps are oriole, blue jay, goldfinch, and cardinal. The article commented that the goldfinch hat will sell much less than the other three, because it is not the name of a major league baseball team. That says something about our society. We are too hung up on sports. When we talk about Indians (either from India or American), there is a good chance we mean the ones from Cleveland. Mariners are not on ships but are on ballfields, quite often in Seattle. Reds are not communists or lefties; they are from Cincinnati. The Land of the Brave is Atlanta, Georgia. Vikings came from Minnesota, not Scandanavia. The only place which has patriots in this country is New England. And where does jazz come from? Certainly not from New Orleans. From Utah! Now, listen to all that Utah jazz. Cool, man.

It's not just baseball and football. This is a country where the death of Dale Earnhardt is more important than the death of Richard Nixon. It's all over the place. Maybe this is just a way of getting levity in a world full of bad news. But I rather either watch the news or go out running or swimming then watch sports on television. And maybe you will see me wearing a goldfinch hat some day.