mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and apportionment theory, a signpost sequence is a

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

of real numbers, called signposts, used in defining generalized rounding rules. A signpost sequence defines a set of ''signposts'' that mark the boundaries between neighboring whole numbers: a real number less than the signpost is rounded down, while numbers greater than the signpost are rounded up. Signposts allow for a more general concept of rounding than the usual one. For example, the signposts of the rounding rule "always round down" (truncation) are given by the signpost sequence

s_0 = 1, s_1 = 2, s_2 = 3 \dots

Formal definition

Mathematically, a signpost sequence is a ''localized'' sequence'','' meaning the

n

th signpost lies in the

n

th interval with integer endpoints:

s_n \in (n, n+1]

for all

n

. This allows us to define a general rounding function using the

floor function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...

\operatorname(x) = \begin
    \lfloor x \rfloor       & x < s(\lfloor x \rfloor) \\
    \lfloor x \rfloor + 1   & x > s(\lfloor x \rfloor)
\end

Where exact equality can be handled with any tie-breaking rule, most often by rounding to the nearest even.

Applications

In the context of apportionment theory, signpost sequences are used in defining

highest averages method The highest averages, divisor, or divide-and-round methods are a family of Apportionment (politics), apportionment rules, i.e. algorithms for fair division of seats in a legislature between several groups (like Political party, political parti ...

s, a set of algorithms designed to achieve equal representation between different groups.

References

Sequences and series Apportionment methods {{Polisci-stub