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/
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