Re: apples and oranges

[Doug Craigen <dcc@cyberspc.mb.ca>]

Ed Eckel wrote:
> Kind folks:
>
> In a recent discussion on mathematically representing the
> relationship between mass and acceleration for a given
> force, the following issue arose.  We have been taught from
> early grades that we can't add apples and oranges.   And we
> have been taught that multiplication is a shorthand for
> repetative addition.  How come we can multiply apples and
> oranges and get something reasonable?  That is, why is it
> that mass times acceleration yields something useful when
> addition of mass and acceleration does not?

Actually, adding apples and oranges can give quite a reasonable and
tasty punch. 

As far as repetition goes though, what exactly would be the meaning of 6
orange repetitions?

> I am at a bit of a loss on this one and would appreciate
> some suggestions on how to deal with this (interesting)
> issue constructively.

We cannot add or subtract quantities with different units but we can
multiply or divide them.  I think your basic question is why this is so,
not why the particular result of multiplying mass and acceleration
produces something useful.  In fact, acceleration itself is such a
quantity - m/s^2.

Just work it through a bit and I think it becomes plain enough - what do
you get if you add 5 kg and 7 meters?  Is it 12 kg+m?  If so, what do
you get if you add 5000 grams and 7 meters?  Is it 5007 g+m?  These are
the same thing, and yet is there any sensible way converting 5007 g+m to
12 kg+m?  The 12 kg+m might have come from 1 kg plus 11 m, which would
be 1011 g+m.  So both 5007 g+m and 1011 g+m can be converted to 12 kg+m
.... and a little consideration will lead you to suspect that any number
could be converted to any number.  The fact is, you don't get a
self-consistent way of dealing with the result unless you know the
numbers that went into making it up.

The unit of kg*m on the other hand is perfectly self-consistent -
regardless of whether we have any physical use for it or not.  5 kg and
7 meters gives 35 kg*m.  5000 g and 7 meters gives 35000 g*m, which can
be divided by 1000 g/kg to get 35 kg*m.  The same factor that converts
the constituent units can consistently convert the resultant unit. 
Likewise, 1 kg times 11 m gives 11 kg*m, and 1000 g * 11 m gives 11000
kg*m, which gives back 11 kg*m when dividing by 1000 g/kg.  So we have
something that at least can be dealt with in a consistent manner.

A great study in what you can do by mixing and matching units (which
fundamentally means you can get the answer to be any number you desire)
can be found in pyramid numerology.  I'm hazy on the details now, but I
was quite amused looking through some of this a few years ago.  How is
it that a single object with only 2 independent parameters to its
dimensions can be claimed to encode so many different numbers?  The
trickery possible from what I wrote above is a good starting point to
answering that question.

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Doug Craigen          "Technology with purpose"
                       http://www.dctech.com