The alphabet for the binary number system consists of the digits 0 and 1,
and a number such as 1011.01 is the binary representation for the number
1 23 0 22 1 21 1 20 0 2 ^1 1 2 ^2. Writing each of these
terms in the familiar decimal system, this number is 8 0 2 1 0 ^1 ⁄ 4
11.25. The binary system is the natural one to use in storing informa-
tion in a computer. Originally, information was displayed by means of a
sequence of lights: when the light was on, the corresponding digit was a
1; off corresponded to 0. Thus, a row of four lights in the order on-off-off-
on corresponded to the binary number 1001 1 23 0 22 0 21 1 2
(^0) , whose decimal value is 8 0 0 1 9. Computers now store informa-
tion magnetically: if a spot is magnetized, the corresponding digit is a 1;
if it is not magnetized, the corresponding digit is a 0.
There is a simple correspondence between an infinite sequence of coin
f lips and binary representations of numbers between 0 and 1. Given a
sequence of f lips, simply replace H with 0 and T with 1, remove the com-
mas, and stick a decimal point to the left of the first digit. The infinite
sequence of coin f lips that alternates heads and tails (H,T,H,T,H,... )
becomes the binary number .01010.... One can also execute this proce-
dure in reverse, going from a binary number between 0 and 1 to an infi-
nite sequence of coin f lips. The search for the ideal random penny thus
morphs into a search for a binary number. The requirement that each
specific sequence of N f lips will occur^1 ⁄ 2 N of the time becomes the re-
quirement that each specific sequence of N binary digits (0s and 1s) will
occur^1 ⁄ 2 N of the time. A number that possesses this property is said to be
normal in base 2. We’ll take a closer look at such numbers shortly.
The Message in Pi
The number pi plays an important role in Carl Sagan’s best-selling novel
Contact,^1 about man’s first encounter with an advanced civilization. One of
the chapters, entitled “The Message in Pi,” outlined the idea that the aliens
had delivered a message for mankind hidden deep in the gazillion digits of
the expansion of pi. Possibly the fact that pi is known to be a transcendental
number prompted Sagan to posit a transcendental message hidden in pi.
The lure of mysticism in pi appeared more recently in the movie Pi and has
doubtless appeared other places of which I am unaware.
It seems awfully unlikely that an alien civilization has so manipulated
the geometry of the universe in which they evolved that the ratio of the
circumference of a circle to its diameter contains a message. In fact, it
seems impossible, considering that the truths, and the parameters, of
plane geometry are independent of wherever it is studied. Not only that,
The Disor ga nized Universe 173