Rounding to Specific Number of Significant Figures

Date: 05/30/2003 at 21:59:20
From: Amber
Subject: How to round off to specific significant figures

I am not sure how to round off properly if the question is, for 
example, asking me to round off to five significant figures... what 
do I do?

I am confused as to what digits are considered significant figures.

Date: 05/30/2003 at 23:37:14
From: Doctor Peterson
Subject: Re: How to round off to specific significant figures

Hi, Amber.

See the Dr. Math archives:

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

  Significant Digits and Zero
  http://mathforum.org/library/drmath/view/57160.html 

The basic idea is that all digits starting with the first (leftmost) 
non-zero digit are significant (that is, their value counts). In the 
case of numbers like 200, with zeros between the last non-zero digit 
and the (implied) decimal point, it is not clear from the way the 
number is written whether they are significant; you would have to be 
told. That is why significant digits should really be counted only 
when a number is written in scientific notation, where there can be 
only one digit to the left of the decimal point anyway.

Let's take an example: I'll round 102.0304 to four significant digits.

We count starting at the first non-zero digit: 1,0,2,0,3,0,4. So our 
example has seven significant digits.

The only digits in a number like this (with a decimal point in the 
middle) that could be insignificant would be zeros at the beginning, 
like this:  00102.0304. The first two zeros don't contribute anything 
to the value of the number; it would still have seven significant 
digits. Zeros at the end, like 102.030400, would be significant, 
because they tell us that whoever made the measurement did read zeros 
in those digits (if he isn't lying).

We want to round to four; so we chop off the last three (which are 
the LEAST significant digits, since they add only a fraction to the 
value of the number). Now we have

  102.0304
       xxx
  102.0

This has four significant digits. (Even the last zero is significant, 
because it tells us that the number of tenths is zero.)

All we have to do is to make sure we got the nearest number of this 
form (stopping at the tenths) to our original number. 102.0304 is 
between 102.0 and 102.1, and because 3 is less than 5, it is less than 
halfway from one to the other. That makes it closer to 102.0, so 
we don't have to round up.

So that's the answer: 102.0.

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

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