Concrete mathematics : a foundation for computer science

Home of the
Toledo Mudhens.
(Or, by pessimists,
half-closed.)
3.2 FLOOR/CEILING APPLICATIONS 73
Level 5. Given an explicit set X, find an interesting property P(x) of its
elements. Now we’re in the scary domain of pure research, where students
might think that total chaos reigns. This is real mathematics. Authors of
textbooks rarely dare to ask level 5 questions.
End of digression. But let’s convert our last question from level 3 to
level 4: What is a necessary and sufficient condition that [JLT;Jl = [fil?
We have observed that equality holds when x = 3.142 but not when x = 1.618;
further experimentation shows that it fails also when x is between 9 and 10.
Oho. Yes. We see that bad cases occur whenever m2 < x < m2 + 1, since this
gives m on the left and m + 1 on the right. In all other cases where J;; is
defined, namely when x = 0 or m2 + 1 6 x 6 (m + 1 )2, we get equality. The
following statement is therefore necessary and sufficient for equality: Either
x is an integer or m isn’t.
For our next problem let’s consider a handy new notation, suggested
by C. A. R. Hoare and Lyle Ramshaw, for intervals of the real line: [01. 61
denotes the set of real numbers x such that OL < x 6 (3. This set is called
a closed interval because it contains both endpoints o( and (3. The interval
containing neither endpoint, denoted by (01. , (3), consists of all x such that
(x < x < (3; this is called an open interval. And the intervals [a.. (3) and
(a. . (31, which contain just one endpoint, are defined similarly and called
half- open.
How many integers are contained in such intervals? The half-open intervals
are easier, so we start with them. In fact half-open intervals are almost
always nicer than open or closed intervals. For example, they’re additive-we
can combine the half-open intervals [K. . (3) and [(3 . . y) to form the half-open
interval [a. . y). This wouldn’t work with open intervals because the point (3
would be excluded, and it could cause problems with closed intervals because
(3 would be included twice.
Back to our problem. The answer is easy if 01 and (3 are integers: Then
[(x..(3) containsthe (?-olintegers 01, o~+l, . . . . S-1, assuming that 016 6.
Similarly ( 0~. . (31 contains (3 - 01 integers in such a case. But our problem is
harder, because 01 and (3 are arbitrary reals. We can convert it to the easier
problem, though, since
when n is an integer, according to (3.7). The intervals on the right have
integer endpoints and contain the same number of integers as those on the left,
which have real endpoints. So the interval [oL.. b) contains exactly [rjl - 1~1
integers, and (0~. . (31 contains [(3] - La]. This is a case where we actually
want to introduce floor or ceiling brackets, instead of getting rid of them.