HP Customer Support's explanation of how unit reduction works.
(Q_and_a)
Item: 149 by _red@hpcvbbs.cv.hp.com [Mark Mabee]
Subj: Calculating with Units
Date: 22 Oct 1992
The HP 48 Owner's Manual states that "Units are automatically converted
and combined during the calculation." I find that when I divide a
distance by a speed [that] I do not get an answer expressed as a time,
but rather as a compound unit. For example: 600_mi divided by 50_mph
will not produce 12_h, but 12_mi/mph.
Is this a failing of the HP 48's design, am I doing something wrong, or
is my HP 48 broken?
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Resp: 3 of 3 by sanker@hpcvbbs.cv.hp.com [Greg Sanker]
Date: 23 Oct 1992
I forwarded this thread to Customer Support, what follows is their
response.
--
Putting aside such judgement words as "failing", "wrong", and "broken"
for a moment, let's try and understand what is happening. For example,
compare the following two calculations and their results:
600_mi 50_mph / ===> 12_mi/mph
600_mi 50_mi/h / ===> 12_h
How are these two calculations different?
The significant difference is that the simple unit mph is used in the
first example and the compound unit mi/h is used in the second. (Note
that mph is "simple" by virtue of the fact that it is a single unit name
with no applied operators and mi/h is "compound" because it is a mixture
of simple units and operators.)
In the second calculation, the result is intuitive, and (most likely)
the one expected by most users.
Rule #1: If a result would contain multiple occurrences of a simple
unit, those occurrences are combined into one occurrence.
Thus 12_mi/(mi/h) is automatically reduced to 12_h, 5_m/s/s becomes
5_m/s^2, and so on.
In the first calculation, the HP 48 follows the same rule that we just
quoted. The result does not contain multiple occurrences of a simple
unit, so no cancellation is performed. The confounding point is that
the unit mph is implicitly compound -- that is, it can be factored into
other, simpler units. In this case, mph can be factored into simpler
units of length and inverse time. What is worse, the unit name itself
implies that these units are miles and inverse hours. So, why don't the
mi units cancel?
Rule #2: Implicitly compound units are not automatically factored so
that Rule #1 can be applied.
Well, why not? Isn't this what unit management is all about:
simplifying units?
The answer is probably best given by example. Suppose that you are a
solar engineer and want to measure the radiant power striking the
surface of your solar cell. You measure the power in watts of radiant
power per square meter of solar cell, or W/m^2. The HP 48 is very
helpful here in that it understands 1_W/m^2 and allows you to perform
calculations with this unit. For example, total power of 15_W on a
5_m^2 solar cell would yield 3_W/m^2.
Very useful.
But, if rule #2 were not in force and implicitly compound units were
dismantled for automatic simplification, watts, whose base SI units are
kg*m^2/s^3, divided by square meters, would reduce to kg/s^3.
Not very useful.
Thus, rule #2 is not arbitrary, but necessary in order to retain the
sense of the calculation.
Understanding these two rules helps you to organize your calculations.
If you want automatic simplification, use compound units that contain
simple, readily cancelable units. I.e, use mi/h rather than mph if you
want time in hours as a result.
If you want to coerce unit strings into other, equivalent, unit strings,
you can use CONVERT, UFACT, and UBASE. Custom unit menus are also
extremely handy for this.
-- HP Calculator Support