Operations on Measurement Units

Date: 08/21/98 at 13:53:06
From: Igor
Subject: Measurement units operations

           Distance
   Rate = -----------
             Time

Why can we divide distance by time, but not add distance to time? In 
other words, why not to introduce a measurement unit (miles+hours) in 
addition to (miles/hour)? Is it possible to teach computers what is 
allowed to sum and what is not?

I know the answer from textbooks, but I am not satisfied.

Date: 08/21/98 at 18:01:20
From: Doctor Rick
Subject: Re: Measurement units operations

Hi, Igor. 

You make an interesting analogy, worthy of some exploration. You're 
saying that we can take the ratio of quantities with two different 
units, and the result is a quantity with units that combine the two: 
meters divided by seconds = meters per second. (We can also multiply to 
make a new unit: a foot-pound is the amount of work you do in lifting a 
1-pound weight 1 foot.) So why can't we do this with addition?

Let's think about why dividing distance by time is useful. If Mike 
walks 4 miles in an hour, his speed is 4 miles/1 hour = 4 miles/hour. 
If Sam walks 8 miles in 2 hours, his speed is 8 miles/2 hours = 
4 miles/hour, also. And the fact that you get the same answer means 
something. If Mike and Sam walked side by side, neither one would pull 
ahead because they are walking at the same speed.

What if we tried to do the same thing with addition? Mike walks 1 mile, 
then he catches the bus and rides for 4 hours. Sam walks 4 miles, then 
rides the bus for 1 hour. They have both gone 5 miles+hours, by your 
idea, but I can't think of anything that this tells us. Mike goes a lot 
farther, Sam takes a lot longer and gets more tired. They just have 
nothing in common.

You know that you can add, say, miles and feet. But first you have to
convert them into the same units: 

   1 mile + 500 feet = 5280 feet + 500 feet = 5780 feet.

Can you see that you can't add miles and feet to get 501 miles+feet? 
It doesn't tell you anything worth knowing, since 500 miles + 1 foot 
would give the same answer. If you can't even add two different length 
measures so they mean anything, surely it's meaningless to add measures 
of different types.

Think about it. There's a deeper question here about why multiplication 
is so different from addition. I'll have to think about that.

- Doctor Rick, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Date: 08/24/98 at 09:16:32
From: Doctor Peterson
Subject: Re: Measurement units operations

Hi, Igor. 

Since I saw Dr. Rick's answer to your question Friday, I've had some 
interesting thoughts I wanted to share with you. It's a great question 
to think about.

The main thing I noticed is that units are inherently multiplicative. 
That is, when you talk about "3 miles" it means "3 times a mile." So 
I'll write 3 miles as:

   3 * mi

to make this clear. That's also why you convert units by multiplying; 
3 times a mile is 3 times 5280 feet.

Now, when you combine units, you are just using the commutative and 
associative properties of multiplication:

   (3 * mph) * (2 * hr) = (3 * 2) * (mph * hr) = 6 * mi

If we try to add quantities with different units, watch what happens:

   (3 * mi) + (2 * hr) = ?

There's nothing you can do with this, just as there's nothing you can 
do to simplify:

   3 * x + 2 * y

if x and y are unrelated variables. Only if they are related, say:

   y = 5 * x

can you do anything with this. Then:

   3 * x + 2 * y = 3 * x + 2 * (5 * x) = 3 * x + 10 * x = 13 * x

Notice that the distributive property, not the commutative property, 
applies in this situation. In fact, if you did try to talk about added 
units like:

   5 * (mi + hr)

it would just expand to:

   5 * mi + 5 * hr

which is not the same as (3 * mi) + (2 * hr). So a unit like (mi + hr) 
is useless. It just doesn't behave like a unit.

Another way to look at this is that when you combine two independent 
units, each of which gives one "dimension" of something, you don't get 
a single new "dimension," but a two-dimensional thing (a vector) that 
can't be measured by any single unit. If you multiply two units 
(independent or not), you get a new one-dimensional entity: an area.

      ^                    ^
      |                    |
   2hr+   + 3mph+2hr       +---+
      |  /                 |   |
      | /                  |6mi|
      |/                   |   |
      +---+-------->       +---+--------->
          3mph

As for teaching computers how to use units, there are several programs 
that do computations with units. I have one called StudyWorks that's 
meant for high school students. It treats units as multipliers, and 
automatically converts them to whatever unit you want. If I type:

   3mph * 2hr =

it responds with:

   9.656*10^3m

and if I change the m to mi, it gives:

   6mi

But if I type:

   3mph + 2hr =

it says "The units in this expression do not match." I presume it 
converts each value to standard metric units and finds that they are 
not equivalent.

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/