## Friday, June 6, 2014

### On Googol and Giggil

On Googol and Giggil

According to the mathematician Edwin Kasner, one day he asked his 6-year-old nephew to make up a name for a very large number. The lad replied by inventing the “googol”, which equals a 1 followed by a hundred zeros:
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
That is, 10100.
He called it a “googol” because of all the 0’s in it. He also named the “googolplex”, which is a 1 followed by a googol zeros: 10googol. I can’t write the googolplex down in non-exponential notation; that would require more zeros than there are elementary particles in the observable universe.
The googol exceeds in magnitude all measures of the known physical cosmos, including size, duration, and number of components. The googol, and even more so the googolplex, just plain blow away the phenomenological universe.
I wonder about that nephew. Many years have passed since that incident; surely he grew up to become an adult himself, perhaps with 6-year-old nephews of his own. I wonder if that fellow did anything significant in math since? After naming the googolplex, surely everything else is anticlimactic…
The googol is a landmark in the Lore of Large Numbers. It’s big enough to be truly LARGE, yet small enough to be easily explained. (Unlike, say, Ackermann’s number.) It’s always there, read to be evoked by any mathematician willing to really stomp on something. For what else are Large Numbers good for?
Far more practical are small numbers. You can hold more of them in a small space; their upkeep is tiny; and they’re useful as finite approximations of infinitesimals. Contemplating these miniature matters, I decided to think small, and thus encountered the Lore of Small Numbers.
Every small number is the reciprocal of a large number, and vice versa; large and small are by definition complementary. And so I wondered what to name the reciprocal of a googol.
1 / googol   =  0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
=       10-100
=       one “googolth”.
“Googolth”? What an awkward name. I propose this name:
1 / googol    =       1 “giggil”.
Even finer is the giggilplex = 1 / googolplex  =  10-googol.
The giggil is tiny, yet it can have a noticeable effect in certain dynamical systems. In a chaotic iteration, errors grow exponentially, and a giggil’s difference in the initial conditions can transform the outcome after 300 doubling times. But even chaos is unperturbed by a giggilplex error for at least three googol doubling times.
The giggil is a model epsilon; it is finite and positive, but smaller than most numbers one meets in practice. Therefore it does quite well for a finite approximation of a calculus infinitesimal. If g = 1 giggil, then
( (x+g)2 - x2 ) /  g             =       2x  + g
Which is the derivative of  x2, to within a giggil.
With the coming of computers, decimal expansions hundreds of places long shall become a commonplace. Therefore the giggil, and even tinier numbers, shall rule the Information Age.