Accuracy in Measurement

Date: 01/09/2007 at 17:03:49
From: Sam
Subject: Accurate Measurement

Dear Doctor Math, 

I read one of the questions in the forum, where this person was saying
that if pi is irrational, than either the diameter or circumference
must be irrational.  In the reply, it said that it has to be 
irrational because nothing can be measured exactly. 

I don't accept that.  How can one of the circumference or diameter be
irrational since c/d = pi?  How can we know pi if we are dividing by
an irrational number, which can be the circumference or diameter?

Also, my friend said it was impossible to measure something all the
way to an atom exactness.  I don't see why we can't measure exactly. 
How can the well shaped world be based on estimations, not exacts?  

Please help me with this paradox! 

- Sam

Date: 01/09/2007 at 23:15:57
From: Doctor Peterson
Subject: Re: Accurate Measurement

Hi, Sam.

You're referring to

  Accuracy in Measurement
    http://mathforum.org/library/drmath/view/54660.html 

There are some tricky ideas involved here, and I think you've 
misunderstood one or two of them--which is not at all surprising!  In 
particular, I don't think either Dr. Jeremiah or I said that "it has 
to be irrational because nothing can be measured exactly".  Rather, 
because we can't measure anything exactly, we can't say based on 
measurement whether a length in the real world is rational or not.

Math deals with an ideal world in which lines and planes have no
thickness, and there are no atoms.  We know that if, in that imaginary
world, a circle has a rational radius, then its circumference must be
irrational, because their ratio is pi, which is an irrational number.
Pi is not determined by measuring actual circles in the real world,
but by doing calculations (well, actually proving theorems) based on
ideal considerations; and it can be shown that the resulting number is
irrational.

Because the real world is made of atoms, both the circle we are trying
to measure, and the ruler we are trying to use, have uncertain edges-- 
bumpy, or fuzzy, if you will.  Where, exactly, do we measure the 
radius to?  Furthermore, we ourselves are finite; we can't focus our
eyes, even with lenses, "all the way down" to see things that are
infinitely small.  No matter how far we turned up the magnification,
there would be details smaller than we could see; so we couldn't be
sure the edge we were measuring lined up EXACTLY with the mark on the
ruler (even if that mark didn't have a finite size and therefore look
thick under the microscope!).

Luckily, the world isn't based on our measurements--our measurements
are based on the world!  It is what it is, and we just measure as
accurately as we can, and use those numbers.  Just a few decimal 
places of accuracy are enough for any measurement that is going to
make a difference.  We take those measurements, then model the world
based on simplifying assumptions (such as that surfaces are really
flat and lines really are straight--pretending that our world is the
ideal world of Euclidean geometry) and do calculations based on that.
The fact that our results will be only approximations (usually pretty
good approximations) doesn't hurt, because the real-world measurements
we compare them to are approximations anyway.

It is worth being aware of all this, however: because we can't measure
everything perfectly, we can never FULLY understand our world.  That's
why, for example, we can't predict the weather very well: even a very
small error in our measurement, say, of the temperature and velocity
of air in a certain part of a jungle, might have a significant effect
on the weather elsewhere a few days later, and make the difference
between a hurricane hitting a city or missing it!  We need to be
humble in our use of science, realizing that what the world actually
is is far beyond us--all we can do is study what we can, and be amazed 
that we can figure out so much of it.

If you have any further questions, feel free to write back.


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

Date: 01/10/2007 at 16:40:09
From: Sam
Subject: Thank you (Accurate Measurement)

Thanks for your reply.  It really helped me get over this paradox, and
made me feel much more assured towards math.  Thanks!!!