Concrete mathematics : a foundation for computer science

If the USA ever
goes metric, our
speed limit signs
will go from 55
mi/hr to 89 km/hr.
Or maybe the high.
way people will be
generous and let us
go 90.
The “shift down”
rule changes n
to f(n/@) and
the “shift up”
rule changes n
to f (n+) , where
f(x) = Lx + @‘J
6.6 FIBONACCI NUMBERS 287
(This identity is true by inspection when n = 0 or n = 1, and by induction
when n > 1; we can also prove it directly by plugging in (6.123).) The ratio
F,+,/F, is very close to 4, which it alternately overshoots and undershoots.
By coincidence, @ is also very nearly the number of kilometers in a mile.
(The exact number is 1.609344, since 1 inch is exactly 2.54 centimeters.)
This gives us a handy way to convert mentally between kilometers and miles,
because a distance of F,+l kilometers is (very nearly) a distance of F, miles.
Suppose we want to convert a non-Fibonacci number from kilometers
to miles; what is 30 km, American style? Easy: We just use the Fibonacci
number system and mentally convert 30 to its Fibonacci representation 21 +
8 + 1 by the greedy approach explained earlier. Now we can shift each number
down one notch, getting 13 + 5 + 1. (The former '1' was Fz, since k, > 0 in
(6.113); the new ‘1’ is Fl.) Shifting down divides by 4, more or less. Hence
19 miles is our estimate. (That’s pretty close; the correct answer is about
18.64 miles.) Similarly, to go from miles to kilometers we can shift up a
notch; 30 miles is approximately 34 + 13 + 2 = 49 kilometers. (That’s not
quite as close; the correct number is about 48.28.)
It turns out that this “shift down” rule gives the correctly rounded number
of miles per n kilometers for all n < 100, except in the cases n = 4, 12,
62, 75, 91, and 96, when it is off by less than 2/3 mile. And the “shift up”
rule gives either the correctly rounded number of kilometers for n miles, or
1 km too mariy, for all n < 126. (The only really embarrassing case is n = 4,
where the individual rounding errors for n = 3 + 1 both go the same direction
instead of cancelling each other out.)
6.7 CONTINUANTS
Fibonacci numbers have important connections to the Stern-Brocot
tree that we studied in Chapter 4, and they have important generalizations to
a sequence of polynomials that Euler studied extensively. These polynomials
are called continuants, because they are the key to the study of continued
fractions like
00 +
al + -
a2 +
a3 +
1
1
a4 +
1
1
1
1
a5 + ___ 1
a6 + - a7
(6.126)