Significant Digits and Irrational Numbers

Date: 09/24/2005 at 22:20:25
From: Alex
Subject: How do you determine the signifigant digits of irrationals

Dear Dr. Math,

How can you determine the signifigant digits of a non-terminating or 
irrational number like pi?

I know the rules for finding signifigant digits of an answer, but 
what if your answer doesn't terminate?  How do you express the 
signifigant digits?

Date: 09/24/2005 at 23:05:46
From: Doctor Peterson
Subject: Re: How do you determine the signifigant digits of irrationals

Hi, Alex.

Any exact number (not a measurement, but a known number such as pi or
sqrt(2)) is considered to have infinitely many significant digits, in
itself, since there is no limit to the number of digits we COULD know. 
However, in a specific calculation, we can only use some approximation
to it, which means using some chosen number of significant digits,
which will restrict our accuracy.  The proper thing to do, then, is to
be sure to use at least enough sigdigs in your calculation that it
will not affect the accuracy of the result.  For example, if you are
finding the area of a circle using a radius given as 1.23 meters, you
would want to use at least three digits of pi, say 3.14 or 3.142. 
Most likely, these days, you would just use the full accuracy of the
constant built into your calculator, and not have to worry!
Effectively, then, you have infinitely many sigdigs and can ignore it.

You mention your answer.  You always determine the number of sigdigs
there not from what the answer looks like itself, but from the sigdigs
in the given numbers, and then round your answer appropriately.  So it
doesn't matter whether the answer you calculate terminates or not--you
MAKE it terminate at the appropriate place!

See this page for other aspects:

  Significant Figures/Digits   
http://mathforum.org/library/drmath/sets/select/dm_sig_digits.html 

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


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