number-theory

Mathematicians call it __he arithmetic of congruences._ You can think of it as clock arithmetic. Temporarily replace the 12 on a clock face with 0. The 12 hours of the clock now read 0, 1, 2, 3, _ up to 11. If the time is eight o__lock, and you add 9 hours, what do you get? Well, you get five o__lock. So in this arithmetic, 8 + 9 = 5; or, as mathematicians say, 8 + 9 _ 5 (mod 12), pronounced __ight plus nine is congruent to five, modulo twelve.

It is shown that the golden ratio plays a prominent role in the dimensions of all objects which exhibit five-fold symmetry. It is also showed that among the irrational numbers, the golden ratio is the most irrational and, as a result, has unique applications in number theory, search algorithms, the minimization of functions, network theory, the atomic structure of certain materials and the growth of biological organisms.