Concrete mathematics : a foundation for computer science

70 INTEGER FUNCTIONS
same inequality with floor or with ceiling; but we need to think twice before
deciding which of the two is appropriate.
The difference between. x and 1x1 is called the fractional part of x, and
it arises often enough in applications to deserve its own notation:
{x} = x - lx] . (3.8)
We sometimes call Lx] the integer part of x, since x = 1x1 + {x}. If a real
number x can be written in the form x = n + 8, where n is an integer and
0 < 8 + {y}J.
And since 0 < {x} + {y} < 2, we find that sometimes lx + y] is 1x1 + [y],
otherwise it’s 1x1 + [y] + 1.
3.2 FLOOR/CEILING APPLICATIONS
We’ve now seen the basic tools for handling floors and ceilings. Let’s
put them to use, starting with an easy problem: What’s [lg351? (We use ‘lg’
to denote the base-2 logarithm.) Well, since 25 < 35 6 26, we can take logs
to get 5 < lg35 6 6; so (3.5(c)) tells us that [lg35] = 6.
Note that the number 35 is six bits long when written in radix 2 notation:
35 = (100011)~. Is it always true that [lgnl is the length of n written in
binary? Not quite. We also need six bits to write 32 = (100000)2. So [lgnl
is the wrong answer to the problem. (It fails only when n is a power of 2,
but that’s infinitely many failures.) We can find a correct answer by realizing
that it takes m bits to write each number n such that 2”-’ 6 n < 2m; thus
&(a)) tells us that m - 1 = LlgnJ, so m = 1lgn.J + 1. That is, we need
\lgnJ t 1 bits to express n in binary, for all n > 0. Alternatively, a similar
derivation yields the answer [lg(n t 1 )I; this formula holds for n = 0 as well,
if we’re willing to say that it takes zero bits to write n = 0 in binary.
Let’s look next at expressions with several floors or ceilings. What is
[lxJl? Easy-smce 1x1 is an integer, [lx]] is just 1x1. So is any other expression
with an innermost 1x1 surrounded by any number of floors or ceilings.
Here’s a tougher problem: Prove or disprove the assertion
[JI;TII = lJ;;I, real x 3 0. (3.9)
Equality obviously holds wh.en x is an integer, because x = 1x1. And there’s
equality in the special cases 7c = 3.14159. . . , e = 2.71828. . . , and @ =
(1 +&)/2 = 1.61803..., because we get 1 = 1. Our failure to find a counterexample
suggests that equality holds in general, so let’s try to prove it.
Hmmm. We’d better
not write {x}
for the fractional
part when it could
be confused with
the set containing x
as its only element.
The second case
occurs if and only
if there’s a “carry”
at the position of
the decimal point,
when the fractional
parts {x} and {y}
are added together.
[Of course 7-c, e,
and 4 are the
obvious first real
numbers to try,
aren’t they?)