Rounding a number to nearest even

[Lie]

On Apr 12, 3:44 am, hdante <hda... at gmail.com> wrote:
(snip)
> > In this table, we consider that a number is rounded down when the
>
>  But then, the "Round up" table gives inconsistent results if, by the
> same argument, we consider 2.0 -> 2 rounding up. (you get 12 round ups
> and 8 round downs just by "rethinking" the argument). So, "rounding
> up" is, at the same time, better and worse than rounding to nearest
> even.
>

It's not round up, why? In the usual sense -- when not comparing
against round-half-even -- the number range we're talking is from x to
lim((x+1)-y)[y -> 0 from the positive side], e.g. 1 to nearly 2 (thus
the integer 2 itself is not included in the number range we're talking
since it belongs to the next set), then we choose a uniformly
distributed samples, i.e. 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8,
and 1.9. From these samples, we chose which are rounded down and which
are rounded up, it happens that 1.0 is rounded down, while 1.5 is
rounded up.

IF for the sake of argument, you choose the number range to be lim(x +
y)[y -> 0 from the positive side] to x + 1, e.g. barely above 1 to 2,
then the number sample you use is 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7,
1.8, 1.9, and 2.0, in this second number range, there are 5 round
downs (1.1, 1.2, 1.3, 1.4, 1.5) and 5 round ups (1.6, 1.7, 1.8, 1.9,
2.0), but how logical is it to choose this number range (barely above
1 to 2) against choosing the more natural range (1 to nearly 2): in
the correctly chosen number range (1 to nearly 2) all numbers in the
form of 1.x (where x is any positive number and 0) is contained within
it and there is nothing else in it, but in the second number range
(barely above 1 to 2) the number 1.0 is not included while the number
2.0 is contained in it, clearly not a clean separation of numbers in
the form of y.x where y is pre-determined and x is variable from other
possible values of y.

In short, choosing that x.0 is rounded down and x.5 is rounded up is
arbitrary but not without a reason.