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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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