Gaps That Do Not Divide Into Hours

Not every gap between two clocks is a whole number of hours. Some are a half, some a quarter, some three quarters. This record contains seventy-two such offsets, and forty-eight of them leave some shared working time.

They are left out of almost every discussion of these arrangements, and the omission is not an accident of rounding. It is a habit of thought, and it costs something specific.

Before a wider rollout, decision-makers should see how the workflow is configured and document who can access each type of information. A useful external reference for the surrounding topic is World Health Organization.

What is in the record

Seventy-two part-hour offsets, between twelve hours behind and twelve hours ahead:

• Forty-eight at a quarter or three quarters of an hour. Of these, thirty-two leave something.
• Twenty-four at a half hour. Of these, sixteen leave something.

Forty-eight of the seventy-two leave shared time, which is two thirds — a slightly better proportion than the whole-hour gaps manage, for the straightforward reason that the part-hour offsets are spread across the same range and a few of them land just inside the edge where a whole hour would have landed just outside.

The law still holds exactly

Everything in the first part of this record applies without modification. Two equal working days lose an hour of overlap for every hour of gap, and that is as true for a gap of three and a half hours as for a gap of three.

A gap of three and a half leaves four and a half shared hours. A gap of seven and three quarters leaves a quarter of an hour. The arithmetic does not care whether the input is round.

The law was never about whole numbers. Whole numbers were about the question, not the answer, and asking the question in whole hours is what makes the rest of it disappear.

Where the two counts come apart

In the whole-hour part of this record, every gap that leaves anything leaves at least a full hour. There is nothing in between nothing and an hour for the arithmetic to land on.

Here there is. A gap of seven and a quarter hours leaves three quarters of an hour of shared time — real, countable, and not an hour. Several offsets in this record leave less than an hour and more than nothing, and they are a category that whole-hour reasoning cannot represent at all.

That column is the whole case for this section. Somebody reasoning in whole hours sees a step from one to zero. What is actually there is a gradual descent through three intermediate values, and an arrangement sitting on one of them is neither of the two things the rounded view offers.

Where a part-hour offset comes from

Two ends whose clocks differ by a fraction of an hour is not an exotic situation. It arises in two ordinary ways.

The clocks themselves differ by a fraction. Some parts of the world keep time at a half-hour or quarter-hour offset from their neighbours. That is a fact about how time is kept and not about any particular place, and it means a gap of four and a half hours is as ordinary a quantity as a gap of four.

The working days differ by a fraction. Far more common. One end starts at nine and the other at half past; one end takes a thirty-minute lunch and the other an hour. The clocks may be a whole number of hours apart and the arrangement is still at a part-hour offset, for the same reason the section on shapes gives: a difference of habit behaves exactly like a difference of clocks.

The second route is the one that catches people, because nobody describes a half-hour difference in start times as an offset. It is simply what time each person happens to begin, and it goes into the arithmetic all the same.

Why they get rounded away

The offsets are not hidden. They are rounded, at the point where the arrangement is first described, by somebody choosing between "four and a half hours apart" and "about four hours apart" in ordinary conversation.

Both of those are reasonable things to say. Only one of them is true, and the rounded version is the one that gets written down, repeated and planned against.

The cost is a half-hour error in one direction or the other, permanently, in the figure everybody uses. It is small enough never to be caught by any single incident and large enough to misplace a daily commitment.

Nobody decides to discard the fraction. It falls off in the sentence that first describes the arrangement, and nothing afterwards ever puts it back.

What follows

The next pages take the half hour and the quarter hour separately, look at what a fractional overlap means in practice, and ask why these offsets get forgotten — which turns out to have a reason, and not a good one.

Next along

• What Half An Hour Of Gap Does
  Twenty-four offsets are half hours and sixteen leave something. The law is unchanged; what changes is that not
• Quarter-Hour Offsets
  Forty-eight offsets sit at a quarter or three quarters of an hour. They are the only place the answer stops be
• An Overlap Of Three And A Quarter Hours
  A fractional answer is not an invitation to round. Rounding down discards time that exists; rounding up spends
• The Forty-Eight That Leave Something
  Forty-eight of the seventy-two part-hour offsets leave shared time: thirty-two at quarter marks and sixteen at