Why Not Meter to the Meter Power?

Date: 08/10/2001 at 23:46:19
From: John Engel
Subject: In an exponent, why not only a unit?

Dear Sir,

This question maybe belongs to physics. I wonder why it is excluded 
that we can encounter a quantity with dimensions such as m^s or m^m
(meter to the power of meter).

Thank you.

Date: 08/11/2001 at 01:12:52
From: Doctor Achilles
Subject: Re: In an exponent, why not only a unit?

Hi John,

Thanks for writing to Dr. Math.

That question used to bother me when I was taking physics too. I can't 
give you a proof for all time that m^s and m^m are completely 
nonsensical units, but the reason they are not used is simply that no 
one has ever (to my knowledge) made any sense of them. In other words, 
they may mean something, but no one has figured it out.

Let's just start with a meter. That's a useful unit for physics 
because distance plays an important part in physics. If you take a 
meter and square it, you get a meter long and a meter wide, and area 
is a useful and understandable concept. You can cube a meter and get 
a space, which is also useful and understandable. In some advanced 
math/physics, you can raise a meter to the 4th power and get a 
hyperspacial region, which is useful if you are studying physics in 
four or more dimensions.

And on you can go up through m^5, m^11, m^54234, etc.  For as many 
dimensions as you're willing to work with, you can define a unit 
that measures how much hyperspace something occupies in that many 
dimensions. Granted, it isn't easy to understand how 5- or 
11-dimensional space might work (and I don't think anyone really has 
much desire to do math in 54234 dimensions), but we can at least in 
principle understand what it _means_.

What happens when you take a meter and extend it to a meter's number 
of dimensions? What does that even mean? Well, a meter has an 
(uncountably) infinite number of points in it, so maybe it's just an 
(uncountably) infinite number of dimensions. But then how would the 
unit m^m differ from the unit m^ft (a foot also has an (uncountably) 
infinite number of points in it)? How would m^m differ from m^s (a 
second has an (uncountably) infinite number of instants in it).

When I think about what you do to a meter when you raise it to a 
number, and then I try to apply the same process using meters or 
seconds instead of numbers, I can't even begin to understand what it 
might mean.

Hope this helps. Please write back if you have any other questions or 
if you'd like to talk about this some more.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/