Significant Non-Zero Digits

Date: 11/27/2001 at 21:03:03
From: Sarah Marcum
Subject: Significant digits

We were learning about significant digits in my Algebra II class 
today, and our teacher asked this question:  

How many significant digits are there in a number with no non-zero 
digits?  

Example:  00.000  Are there any?

We had a pretty good discussion about this, but didn't come up with a 
definite answer. Thanks for your help.

Date: 11/27/2001 at 23:41:10
From: Doctor Peterson
Subject: Re: Significant digits

Hi, Sarah.

My first impression is that there are no significant digits, and I 
think a consideration of the meaning of significant digits will 
confirm that.

The concept of significant digits is really just a simple rule-of-
thumb representation of relative precision. A number with, say, three 
significant digits, such as 1.23, represents the range 1.225 to 1.235, 
with a relative error of 0.005/1.23 = 0.004. A number with four 
significant digits would have about a tenth of that relative error. We 
can define "number of significant digits" more precisely as the 
negative common logarithm of this relative error, in this case 2.39:

    SD = -log(error/value)

       = -log(0.005/1.23) = -log(0.004) = 2.39

This is less than 3 because our number is smaller than the average 
three-digit number; for 5.00 we would get -log(0.005/5.00) = 3 
exactly.

If the number itself is zero, then you can't talk about relative error 
at all, since you can't divide by zero. Therefore the concept of 
significant digits is meaningless in this case. If we apply my 
definition, we get -log(0.0005/0) = -infinity, not 0. So I guess I'm 
glad I said "no significant digits" rather than "zero," because the 
proper answer is that the number of significant digits in this case is 
UNDEFINED!


I want to clarify one thing in case it confuses you, though it is not 
central to what I said, and may be more for your teacher's interest.

The log-of-relative-error (LRE) I discussed is not actually a 
definition of significant digits, but a more exact measure of 
precision for which the number of significant digits is used as an 
estimate. You can actually define the number of significant digits 
this way:

    SD = floor(log(value/next_digit))

What I mean by this is that if you divide the number (say, 1.23) by a 
number formed by putting a 1 in the first decimal place NOT written in 
the number (in this case, 0.001), take the common logarithm, and then 
take the "floor" of this log (the greatest integer that does not 
exceed it), you get the number of significant digits:

    SD = floor(log(1.23/0.001))
       = floor(log(1230))
       = floor(3.0899)
       = 3

This is related to the LRE by this formula:

    SD = floor(LRE + log(5))

This is true because we can rewrite the definition of LRE as

    LRE = -log(error/value)
        = log(value/(5*next_digit))

This corrects for the fact I pointed out, that the LRE is low for 
numbers whose first digit is less than 5. It doesn't affect my 
conclusion about zero.

I hope this doesn't just confuse you more!

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