Concrete mathematics : a foundation for computer science

There was a time in
the 70s when ‘mod’
was the fashion.
Maybe the new
mumble function
should be called
‘punk’?
No-l &
‘mumble’.
The remainder, eh?
3.4 ‘MOD’: THE BINARY OPERATION 83
that this would be much less useful. Continuity is not an important aspect
of the mod operation.)
We’ve already seen one special case of mod in disguise, when we wrote x
in terms of its integer and fractional parts, x = 1x1 + {x}. The fractional part
can also be written x mod 1, because we have
x = lxj + x mod 1 .
Notice that parentheses aren’t needed in this formula; we take mod to bind
more tightly than addition or subtraction.
The floor function has been used to define mod, and the ceiling function
hasn’t gotten equal time. We could perhaps use the ceiling to define a mod
analog like
xmumbley = y[x/yl -x;
in our circle analogy this represents the distance the traveler needs to continue,
after going a distance x, to get back to the starting point 0. But of course
we’d need a better name than ‘mumble’. If sufficient applications come along,
an appropriate name will probably suggest itself.
The distributive law is mod’s most important algebraic property: We
have
c(x mod y) = (cx) mod (cy) (3.23)
for all real c, x, and y. (Those who like mod to bind less tightly than multiplication
may remove the parentheses from the right side here, too.) It’s easy
to prove this law from definition (3.21), since
c(x mod y ) = c(x - y [x/y] ) = cx - cy [cx/cy] = cx mod cy ,
if cy # 0; and the zero-modulus cases are trivially true. Our four examples
using f5 and f3 illustrate this law twice, with c = -1. An identity like
(3.23) is reassuring, because it gives us reason to believe that ‘mod’ has not
been defined improperly.
In the remainder of this section, we’ll consider an application in which
‘mod’ turns out to be helpful although it doesn’t play a central role. The
problem arises frequently in a variety of situations: We want to partition
n things into m groups as equally as possible.
Suppose, for example, that we have n short lines of text that we’d like
to arrange in m columns. For aesthetic reasons, we want the columns to be
arranged in decreasing order of length (actually nonincreasing order); and the
lengths should be approximately the same-no two columns should differ by