Significant Digits and Zero

Date: 05/19/99 at 16:47:37
From: Gale Carroll
Subject: Significant digits

What is the best way to explain significant digits? When are zeros 
significant and when are they not significant? What is the meaning of 
significant in regard to this concept? Certainly the zeros in the 
number 5,000 are important, but apparently not significant. How about 
508? How about 23.0? How about 0.05? How about 46.50?

Date: 05/20/99 at 09:21:12
From: Doctor Peterson
Subject: Re: Significant digits

Hi, Gale. I'm always happy to help a teacher.

A digit is significant if it contributes to the value of the number. 
Since zeroes on the left (if they are to the left of the decimal 
point) can be dropped without affecting the number, they are 
insignificant. Zeroes on the right (if they are to the right of the 
decimal point) are considered significant because they tell you that 
that digit is not some non-zero value, as do zeroes between digits. 
Zeroes between the decimal point and non-zero digits, however, are 
present only to keep the other digits in the right place, and are not 
significant - they're certainly important to the number, but are 
there only as a scaffolding, not as part of the building. 

In the case of a number like 5000, the zeroes are really ambiguous, 
since they might be there for either reason; when we pay attention to 
significant digits, we usually write numbers in scientific notation so 
that we don't have to write any zeroes in that position, and can avoid 
the ambiguity. On the other hand, if you wrote 5000.0, we would assume 
all the digits were significant, since decimal places are included 
that aren't just required to show place value. If you write 5000., it 
is generally taken to mean that all the zeroes are significant.

In your examples, the significant digits are

    5000      (1 - though the zeroes COULD be significant)
    =

     508      (3 - everything between first and last nonzero digits)
     ===

      23.0    (3 - zero could have been omitted, must have a purpose)
      == =

       0.05   (1 - zero couldn't be dropped, has no significance)
          =

In general, significant digits are:

    000xx0xx000       Between first and last non-zero digits
       =====

    000xx0xx000.000   Between first non-zero digit and last decimal
       ============

    000xx0x.x0xx000   Between first non-zero digit and last decimal
       ==== =======

    000.000xx0xx000   Between first non-zero digit and last decimal
           ========    

Here's an interesting site with several links to explanations of 
significant digits:

    Significant Figures and Rounding Rules, Christopher Mulliss
    http://www.angelfire.com/oh/cmulliss/   

Other answers in our archives dealing with this topic include

   Significant Digits
    http://mathforum.org/dr.math/problems/casey9.7.98.html   

   Significant Figures and Scientific Notation
    http://mathforum.org/dr.math/problems/smith5.21.97.html   


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