isomorphismes

Readers of isomorphismes, you might enjoy powers of two tumblr.

powersoftwo:

2¹⁰⁰ = 1,267,650,600,228,229,401,496,703,205,376 — one nonillion, two hundred sixty-seven octillion, six hundred fifty septillion, six hundred sextillion, two hundred twenty-eight quintillion, two hundred twenty-nine quadrillion, four hundred one trillion, four hundred ninety-six billion, seven hundred three million, two hundred five thousand, three hundred seventy-six (31 digits, 320 characters)

I think I’ve been subscribed since the 30’s. Never a letdown. And of course it’s only going to get more exciting.

This is an amazing fact that comes up in so many applications.  It’s used in the valuation of companies, solution of equations, ……… any time you want to convert an infinite stream into something finite.

f is a proper fraction. (0 < f < 1)

Or, in fancy notation:

Or, in C++:

long big = 9999999999;
single frac = .70;
double total = 0;

for ( i = 0, big, i++)
{
  total += frac^i;
  }

cout << total;                 # in this case, prints 1 / .3 = 10/3
cout << total - 1/(1-frac)     # prints 0 for any value of frac


Isn’t it strange that adding together an infinite number of things can give you a finite answer?  The ancient philosopher Zeno thought that he could disprove reality through the following thought experiment

1. An arrow fired at a tree first covers half the distance to the tree.
2. Then it covers half the remaining distance to the tree.
3. Then it covers half the remaining distance to the tree.
4. Etc….so it only ever covers less than all the distance to the tree!  Because it just keeps adding halves of halves of halves of ….
5. So, since we see it hit the tree, but logically it cannot hit the tree, logically reality must be false!  (Motion is impossible, and we observe motion, so our observations are impossible.)

But calculus proves that:

Take that, Zeno!