## [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 shows minutes will read 12:00, leading to the conclusion that (for times, at least) the convention is that the first minute of an hour (as a duration, not a boundary) is labeled 00. -jn- --------------------------------------------------------------- There are two types of science: physics and stamp collecting! -- Sir Arthur Eddington joel-dot-neely-at-fedex-dot-com