the question arises as to how one would translate the message in pi. The
digits used in expressing pi are a function of the base of the number, and
there would have to be a dictionary that translates blocks of digits into
characters in the language in which the message is presented. For exam-
ple, ASCII is the code that translates blocks of eight binary digits stored
in a computer into printable characters; the number 01000001 (whose
decimal representation is 65) corresponds to the character A. In one
sense, however, Sagan was right: mathematicians believe that pi contains
not only whatever message the aliens encoded, but every message, re-
peated infinitely often!
A normal number in base 10 is one in which, on average, each decimal
digit, such as 4, appears^1 / 10 of the time—but each pair of successive deci-
mal digits, such as 47, appears^1 / 100 of the time (there are 100 such pairs,
from 00 to 99), each triple of successive decimal digits, such as 471, ap-
pears^1 /1,000 of the time, and so on. This is the mathematical equivalent of
the “ideal random penny” for which we searched a little while ago, except
that instead of an ideal random penny with two sides whose tosses would
generate a normal number in base 2, we would imagine a perfectly bal-
anced roulette wheel with 10 numbers, 0 through 9. It is possible to for-
mulate an equivalent definition of normality for any number base. For
example, a number that is normal in base 4 is one such that each of the 4N
possible N-digit sequences occurs^1 ⁄ 4 N of the time.
Are there any normal numbers, in any base, that we have actually found?
Obviously, we cannot f lip pennies (perfect or not) forever. There are a few
that are known. David Champernowne, who was a classmate of Alan Tur-
ing (whose proof of the unsolvability of the halting problem appears
in chapter 7), constructed one in 1935. This number, known as Cham-
pernowne’s constant, is normal in base 10. The number is
.123456789101112131415... , which is formed simply by stringing to-
gether the decimal representations of the integers in ascending order.
When I saw this result, I jumped to the conclusion that Champernowne’s
constant^2 was also normal in other bases—after all, it’s an idea, rather
than a specific number, and I naively assumed that whatever proof
method worked to show that it was normal in base 10 would work for
other bases as well. If there’s a Guinness Book of Records for Most Conclu-
sions Erroneously Jumped To, I’m probably entitled to an honorable men-
tion. If you look at Champernowne’s constant in base 10, it’s between .1
and .2—but in base 2, it’s a different number. In binary notation, 1 1,
2 10, 3 11, 4 100, 5 101, 6 110, 7111, so Champernowne’s con-
stant in base 2 starts off .11011100101110111.... Any number whose base
2 representation starts off .11... is bigger than^3 ⁄ 4 (just as .12... in base
174 How Math Explains the World