Exact Numbers and Conversion Statements

Date: 07/25/2005 at 21:26:28
From: Jason
Subject: Exact Numbers and Conversion Statements

I am currently taking chemistry, and something that keeps coming up is 
exact numbers, which are, by definition, numbers obtained by counting 
(as opposed to measuring) as in 8 apples.  Also, an exact number could 
be part of a definition, like a conversion statement such as 1 inch = 
2.54 cm. 

Is there a general rule for determining whether or not a conversion
statement is exact?  I believe that fairly recently, the inch was 
redefined to be EXACTLY 2.54 cm.  Are there others that have been 
defined as exact?  And are conversion statements from metric to 
English or vice versa the only conversion statements that might be 
inexact? 

I would really appreciate your help here, since knowing the exactness 
of a conversion factor plays a huge role in dimensional analysis and 
rounding an answer to the correct number of significant digits 
(because an exact number has no limits on significant digits, but an 
inexact number does).

And on a more general note, what other types of numbers are classified
as exact (e.g., infinite, infinite repeating, fractions, pi)? I know a 
lot of times it has to do with the context.

An example of my dilemma is in the following problem:

What is the mass of a troy ounce of gold in grams? 
1 troy ounce = 20 pennyweight (exact)
1 pennyweight = 24 grains (exact)
1 grain = 0.0648 gram (NOT exact)

I used dimensional analysis, assuming that the "one troy ounce" I was 
starting with was inexact (because it is a measurement), so I assumed 
the answer should have only ONE significant digit, but the answer key 
said the answer should have 3 significant digits because of the 
0.0648. 

I am really lost now, and I don't know how to distinguish between a 
measurement that is inexact (which I thought a measurement always is) 
and one that is exact (which I thought they never were, until this 
problem).  The only explanation I can think of is that the "one troy 
ounce" wasn't actually measured, they just wanted us to use it for the 
sake of converting.  PLEASE HELP!!!!!!!!!!!!!!!  Thanks.

Date: 07/25/2005 at 22:37:23
From: Doctor Peterson
Subject: Re: Exact Numbers and Conversion Statements

Hi, Jason.

The general rule is that a number is exact if you are told it is 
exact. There is nothing more general that can be said.

Any formula you are given should make it clear whether the constants
in it are exact.  Generally, within a system (such as the 20 and 24 in
your example), all numbers are exact integers; between systems, such
as your 0.0648, the numbers are usually approximate, and that should
be indicated by using an "approximately equal" sign, or saying "to
three significant digits", or something like that.  But since it is so
unusual for a conversion factor between systems to be exact, you can
take it the other way and, in that setting, assume a number is inexact
unless it is explicitly stated, as it will be for 2.54.

You asked about "infinite, infinite repeating, fractions, pi".  You
didn't really mean "infinite numbers", but rather "infinite decimals".
If a decimal is given to you indicating how it repeats, as with a bar
over it, then that is exact because you are being told every digit in
the number.  If you are just given some decimal places and are not 
told how the rest of it behaves, then obviously it can't be considered
exact.  That is true of pi, which as an irrational number can't be
expressed exactly.  As for fractions, they are exact unless you are
told that the fraction is approximate.  For example, the 5/9 or 9/5 in
conversion between Fahrenheit and Celsius is an exact fraction.  This
means that in such cases you don't need to consider the number of
significant digits.  Other fractions, such as 22/7 for pi, are
approximations; to use this with significant digits, you would have to
determine HOW accurate it is, by writing it as a decimal and comparing
to a good decimal approximation to pi.  A decimal approximation makes 
a lot more sense in such a context.

Now, in your question about "the mass of a troy ounce in grams", there
is no number given!  There is no measurement here!  It is "the mass of
an exact troy ounce", not "a mass measured as 1.0 troy ounces".  That
means you can't take "1" as having a specific number of significant
digits; so you have to take it as exact.  So the only number in your
calculation that has a specific precision is the last conversion
factor, and only that limits the precision of your answer.

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


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