10 shows that the number is bigger than^1 / 10 ^2 / 102 , .11... in base 2 shows
that the number is bigger than^1 ⁄ 21 ^1 ⁄ 22 ).
However, in 2001, the base-2 incarnation of Champernowne’s constant
was shown to be normal in base 2. Thus, writing H for every appearance
of 0 and T for every appearance of 1 would be an example of a sequence of
tosses that appear to come from a perfectly random penny.
There are a very few known examples of numbers that are normal in
every base; all the ones that are known are highly artificial.^3 By “highly
artificial,” I mean that you are not going to encounter the number in the
real world. Obviously, we encounter numbers such as 3.089 (the current
price in dollars of a gallon of gasoline in California) and the square root of
2 (when finding the length of the diagonal of a square one foot on a side),
but numbers such as Champernowne’s constant simply don’t show up
when we measure things. Numbers that are normal in every base do not
appear in the real world, but the real line is chock-full of such numbers.
Borel’s normal number theorem^4 states that if you pick a real number at
random (there’s that word again), you are almost certain (in a sense that
can be made mathematically specific) to pick a number that is normal in
every base. In ordinary usage, when a person is asked to pick a number, he
or she will usually pick a number that measures something, such as 5.
When a mathematician describes picking a random real number, he or
she envisions a process somewhat like a lottery, in which all the real num-
bers are put into a hat, the hat is thoroughly shaken, and a number is
picked out by someone wearing a blindfold. If one does that, the number
that is picked out will almost certainly be normal in every base. Again,
“almost certainly” has a highly technical definition, but one can get an
idea of what is meant by realizing that if a real number is picked at random
in the sense described above, it is almost certain that it will not be an inte-
ger. Integers form what is known as a set of Lebesgue measure zero; the
technical statement of Borel’s normal number theorem is that all numbers
except for a set of Lebesgue measure zero are normal in every base.
Transcendentals such as pi seem to be prime candidates for numbers
that are normal in every base. If pi were shown to be such a number, then
Sagan would have been right: the message from the aliens would be en-
coded in the digits of pi, as the encoded message would simply be a se-
quence of digits. However, every message is a sequence of digits, so if you
dig deep enough into pi, you will find the recipe for the ultimate killer
cheesecake, as well as your life story (even the part that hasn’t happened
yet), repeated infinitely often.
Sagan used to talk about how we are all made of star stuff; the explo-
sions of supernovas create the heavier elements that are used to construct
The Disor ga nized Universe 175