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The approach:
d = 30 ft ==> d / ft = 30
is sometimes called 'quantity calculus'. This is the recommended
method of ISO, IBWM, IUPAC, and others.
Dr. Roy Jensen
(==========)-----------------------------------------¤
Lecturer, Chemistry
W5-19, University of Alberta
780.248.1808
On Sat, 02 May 2015 14:09:54 -0700, you wrote:
This can in some sense be considered an extension of the_______________________________________________
"backwards bicycle" discussion, making it more practical
and more directly relevant to learning and teaching physics.
Suppose we measure the floor of a certain hallway using
a yardstick, and find that it is ten yards long. There
are good reasons to formalize this as
L = 10 yd [1]
where L is the length, 10 is a pure number, and yd is
a unit of measurement.
We can invoke the rule
1 yd = 3 ft [2]
We don't need to remeasure the hallway using a one-foot
ruler; we can use a physical and mathematical model of
the universe, plus a little bit of algebra, to conclude
that
L = 30 ft [3]
In passing we note that equation [2] is not a rigorous
mathematical fact; it depends on a number of provisos,
notably the assumption that the floor is flat. These
assumptions are often reasonably accurate and are often
left unstated, but sometimes they break down and cause
problems.
========================
We should recognize that there is another approach,
another way of formalizing things. One could write:
y = distance measured in yards [4]
and in the present example
y = 10 [5]
where the RHS is a pure number, without units, without
even dimensions.
Using this approach, the rule is
? f = 3 y ? [6]
where
f = distance measured in feet [7]
and the model tells us
f = 30 [8]
The contrast between equation [6] and equation [2]
could not be more extreme. Asking students to switch
from one approach to the other is like asking them
to switch from a regular bicycle to a backwards
bicycle. There will be lots of crashes.
The temptation is to treat the "f = 3 y" approach
as a misconception. However, we must be careful
not to blame the students for this. It is quite
likely that they have been taught to do things this
way. I've seen the "f = 3 y" approach used by people
who have too little mathematical sophistication
*and* by people who have too much:
*) Those who don't know how to think in algebraic
terms find the process for getting from equation
[2] to equation [3] to be very mysterious. They
think that "math" is synonymous with "arithmetic".
They can perform arithmetic on numbers, but they
cannot manipulate anything more abstract than
that.
*) I've also seen the "f = 3 y" approach used --
and taught! -- by famous mathematicians /1, 2/.
I reckon this is what comes of knowing too much
math and not enough physics.
========
Constructive suggestion: Although there are cases
where it might /seem/ necessary to use the "f = 3 y"
approach, there are ways of avoiding this quagmire.
In particular, suppose we are using an old-school
computer language that can only deal with numbers
(i.e. plain old dimensionless numbers). We can
write L__yd = 10 and document the convention that
the double underscore means both "measured in units
of" and "divided by".
This allows us to maintain the conceptual distinction
between the abstract dimensionful quantity L and
the dimensionless variables L__ft and L__yd. This
allows us to write yd__ft = 3 as it should be. If
desired, we can adopt the additional convention that
SI units are implicit, so that
in = 0.0254 exactly,
ft = 12*in,
yd = 3*ft,
L = 10*yd, and
L__ft = L/ft.
Nowadays another option is to use a computer algebra
system. That allows us to write L=10*yd where yd
remains an algebraic abstraction, without any
numerical value (SI or otherwise).
As for the width of the hallway, we can represent it
as W__yd = 2, no problem.
In contrast, expressions such as "f=30" or "y=10" or
"nbits=15" are considered bad practice. They tell
you the units of measurement, but not what is being
measured. For example, if the length in yards is
y=10, how do we represent the width?
Bottom line: It is tempting to ask "how many feet" or
"how many yards" and give dimensionless answers to such
questions ... but I suggest we explicitly train students
to overcome this temptation. They ought to standardize
on the non-backwards bicycle, rather than trying to
switch from backwards to non-backwards, again and again.
References for how /not/ to do things:
/1/ Steven Strogatz _The Joy of X_
/2/ Khan Academy, flagship offering: algebra lessons
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