• Source: Signpost sequence

In mathematics and apportionment theory, a signpost sequence is a sequence 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
…


{\displaystyle s_{0}=1,s_{1}=2,s_{2}=3\dots }





Formal definition


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



n


{\displaystyle n}

th signpost lies in the



n


{\displaystyle n}

th interval with integer endpoints:




s

n


∈
(
n
,
n
+
1
]


{\displaystyle s_{n}\in (n,n+1]}

for all



n


{\displaystyle n}

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




round
⁡
(
x
)
=


{



⌊
x
⌋


x
<
s
(
⌊
x
⌋
)




⌊
x
⌋
+
1


x
>
s
(
⌊
x
⌋
)








{\displaystyle \operatorname {round} (x)={\begin{cases}\lfloor x\rfloor &xs(\lfloor x\rfloor )\end{cases}}}


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




References

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