pairs (heads-heads, heads-tails, tails-heads, tails-tails) occurs one-quarter
of the time. The following sequence will do that.
H,H,H,T,T,H,T,T, H,H,H,T,T,H,T,T, H,H,H,T,T,H,T,T,...In case it isn’t clear what’s going on here, we repeat the pattern
H,H,H,T,T,H,T,T (heads-heads, heads-tails, tails-heads, tails-tails) end-
lessly. This satisfies two requirements: heads and tails each occur half the
time, and each of the two-f lip possibilities occurs one-quarter of the time.
And yet we still don’t have a random sequence; many easily conceivable
patterns have been left out. Tails, for example, never occurs three times
in a row, as it certainly would given infinitely many f lips, and almost cer-
tainly would, even after only a hundred.
There is a surprisingly deep question contained here: Can you construct
a sequence of f lips that is perfectly in accord with the laws of probability,
in that each specific sequence of N f lips will occur^1 ⁄ 2 N of the time?
Number Systems: The Dictionaries of Quantity
The decimal number system (also known as the base-10 number system)
that we learn in elementary school is similar to a dictionary. Instead
of the letters of the alphabet, the decimal number system uses the char-
acters 0,1,2,3,4,5,6,7,8,9. From these ten characters, it forms all the words
that can be used to describe quantity. It’s an amazingly simple dictionary;
for example, the number 384.07 is actually defined as the sum 3 102 8
101 4 100 0 10 ^1 7 10 ^2 , where the expression 10^2 ^1 / 102. The
quantitative value of the word 384.07 is deducible from the “letters” used
and their positions in the word. I tell my prospective elementary school
teachers that it’s a lot simpler dictionary than the Merriam-Webster,
where one can’t deduce the meaning of the word from the letters that
make it up, and you have to decide in a split second whether duck means
“quacking waterfowl” or “watch out for rapidly approaching object.”
One way to define the real numbers is the set of all decimal representa-
tion of the above form, where we are only allowed to use finitely many
numbers to the left of the decimal point but infinitely many thereafter.
With this convention, 384.07384.0700000.... The rationals are all
those numbers, such as 25.512121212... , that eventually settle down
into a repetitive pattern to the right of the decimal point. A calculator will
show (or you can do it by hand) that .5121212... ^507 / 990.
Instead of the “10” that we use in the decimal system, it is possible to
use any positive integer greater than 1. When “2” takes the place of “10”
in the decimal system, the result is the binary, or base-2, number system.
172 How Math Explains the World