Concrete mathematics : a foundation for computer science

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3
Integer Functions
WHOLE NUMBERS constitute the backbone of discrete mathematics, and we
often need to convert from fractions or arbitrary real numbers to integers. Our
goal in this chapter is to gain familiarity and fluency with such conversions
and to learn some of their remarkable properties.
3.1 FLOORS AND CEILINGS
We start by covering the floor (greatest integer) and ceiling (least
integer) functions, which are defined for all real x as follows:
1x1 = the greatest integer less than or equal to x;
[xl = the least integer greater than or equal to x .
(3.1)
Kenneth E. Iverson introduced this notation, as well as the names “floor” and
“ceiling,” early in the 1960s [161, page 121. He found that typesetters could
handle the symbols by shaving the tops and bottoms off of ’ [’ and ‘I ‘. His
notation has become sufficiently popular that floor and ceiling brackets can
now be used in a technical paper without an explanation of what they mean.
Until recently, people had most often been writing ‘[xl’ for the greatest integer
6 x, without a good equivalent for the least integer function. Some authors
had even tried to use ‘]x[‘-with a predictable lack of success.
Besides variations in notation, there are variations in the functions themselves.
For example, some pocket calculators have an INT function, defined
as 1x1 when x is positive and [xl when x is negative. The designers of
these calculators probably wanted their INT function to satisfy the identity
INT(-x) = -INT(x). But we’ll stick to our floor and ceiling functions,
because they have even nicer properties than this.
One good way to become familiar with the floor and ceiling functions
is to understand their graphs, which form staircase-like patterns above and
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