Mailing List Archive: 49091 messages

## [REBOL] Re: On ordinal and cardinal numbers...

### From: joel:neely:fedex at: 10-Jul-2001 3:08

```
Ken Anthony wrote:
> Dave,
>
> Thank you for the geographic analogy...
>

Provided we us it appropriately!  ;-)

In ordinary usage (as in programming) we're more interested
in the "places" than the boundaries *between* the "places".

> Your saying days are discontinuous at midnight? ...
>
> In other words, would July 9th 12:01am belong to July 9th,
> but July 9th 12:00am not?
>
...

> > ... be like having Canada and the US argue over who owns
> > the land under the 49th Parallel.
> > In total there's 0.00 acres there to worry about just like
> >  there's 0.00... seconds to worry about at the `moment of
> > midnight'.
>

The number zero is *neither* positive nor negative.  It is the
boundary *between* positive "territory" and negative "territory"
on the number line.  If we stretch the number line out with a
magnifying glass:

-1         0         1         2         3
...:         :         :         :         :...
...nnnnnnnnnnzpppppppppppppppppppppppppppppp...
...000000000011111111112222222222...

I hope it's clear that the segment marked with "n"s is the
negative territory and the "p"s mark positive territory.  The
z
is always the boundary, and however much we zoom in, it
never is any wider than the smallest thing we can see, so it
has no "width".  On the other hand the segment marked with "0"s
does have width.  It represents the numbers that are at least
zero but less than one.  Written in decimal, those numbers are

0.000000000...   thru   0.999999999...

(where the ellipses mean that we never stop... just like this
thread ;-)  So we can label that "space" with it's lower bound
and call it "space 0".  Similarly the "1"s label "space 1", the
2
s are "space 2", and so on.

This is, of course, not the only way to label spaces.  We could
put the boundaries in the digits and label each segment by the
*nearest* integer, as in

-1         0         1         2         3
...:         :         :         :         :...
...nnnnnnnnnnzpppppppppppppppppppppppppppppp...
...000000000011111111112222222222...

which we normally call "rounding off".  Notice that the same
issue still comes up; there's the old standard argument about
whether 1.5 should round off to 1.0 or 2.0.  In the "good old
days" before computers, the standard rule taught in elementary
school was that one rounds xxx.5 to the nearest *even* number,
so that both 1.5 and 2.5 round to 2.0, while 0.5 rounds to 0.0
and 3.5 rounds to 4.0.  This rule looks rather weird to our
modern
eyes, but had the virtue that it averaged out the
round-off error!  (In general, half of the rounding went up and
half went down, thus cancelling out the accumulated error in a
long calculation with randomly-distributed fractions.)

OBTW, this also show how "standard practice" changes over time!

Speaking of "time", which is how all of this got started, let's
do the same magnification on the time-line (at the scale of
minutes) and think of how we normally write times.

11:58     11:59     12:00     12:01     12:02
...:         :         :         :         :...
...TTTTTTTTTTTTTTTTTTTTmWWWWWWWWWWWWWWWWWWWW...

At the "midnight boundary" between e.g. Tuesday and Wednesday,
(shown by the "T" and "W" labels) the ownership of the *moment*
of midnight could be argued about all night (;-) but I think we
would all agree that the *span* of time which begins at 12:00:00
and includes 12:00:01 through 12:00:59 constitutes the *first*
minute of Wednesday.  During that entire time, a clock that only