What is a Significant Digit?

Date: 12/25/2001 at 21:00:53
From: Yin
Subject: Significant digits

I came across a question asking for an answer with no more than "2 
significant digits." My answer was 236. Should the approximate answer 
be 240 (rounded to the tens) or 2.4 x 10^2? And can you please explain 
what a significant digit is?

Thank you,
Yin

Date: 12/26/2001 at 12:27:08
From: Doctor Rob
Subject: Re: Significant digits

Thanks for writing to Ask Dr. Math, Yin.

The approximate answer could be either 240 or 2.4*10^2, which are
(of course) equal.

Significant digits arise out of inexact measurements. Suppose you
measure the circumference of a circle and find that it is 90 meters 
long. Your measuring device is only accurate enough to tell you that 
the actual circumference is between 85 and 95 meters, so that the 
actual value, rounded to the 10's digit, is 90 meters. Then the 
formula tells you that the circle has a diameter of

   90/Pi = 90/3.14159265... = 28.6478898... meters.

Actually all you can conclude is that the diameter is between

   85/Pi = 27.0563404... meters   and
   95/Pi = 30.2394392... meters.

That means that it is silly to say that the length is 28.6478898...
meters, because this implies a level of accuracy in measurement that 
simply doesn't exist. It would be just as reasonable to say that the 
length is 27.824739248234, or 29.3, or 29, or 30 meters. The best you 
can say is that the diameter is about 30 meters, since both bounding 
numbers round off to that number when rounded to the same number of 
decimal digits. That number of decimal digits (one in this case) is 
the number of significant figures in the answer.

A rough rule of thumb is that the number of significant figures in an 
answer is about the same as the number of significant figures in the 
least accurate of the input data values. In the above problem, there 
was one significant figure in the circumference, so there should be 
about one significant figure in the diameter (and there was).

Feel free to write again if I can help further.

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