Significant Digits in Measurement

Date: 08/13/2001 at 13:38:01
From: Tricia 
Subject: Significant digits in measurement

I understand that there are rules to determine the significant digits 
in measurement; however, I do not comprehend the basic concept behind 
the use of significant digits. It seems to me that all digits are 
significant, especially zeros for place value. An example of my 
confusion would be: How can 3000 have only one significant digit (3)?
or How can 0.0050 have only two significant digits (50)? 

Thank you for your help!

Date: 08/13/2001 at 15:34:17
From: Doctor Peterson
Subject: Re: Significant digits in measurement

Hi, Tricia.

The basic idea is that all digits that are not there ONLY for place 
value purposes are significant. 

Suppose I built a building that rose 10 stories high, but the first 
two stories were just columns holding the rest up above a highway. The 
first two "stories" would be important, certainly; but they wouldn't 
really count as part of the building, because they only hold it in 
place, without actually containing anything useful. The building has 
only eight "significant stories"! Zeros used only for place value at 
the right of a number are mere scaffolding holding it "above" the 
decimal point, and don't really contain any information.

It's a lot easier to talk about significant digits when you write 
numbers in scientific notation, which is designed to neatly separate 
significant digits from the size of the number, by ensuring that no 
extra zeros are required in order to write it. 

If we write 3000 as 3 * 10^3, we can see we have only one significant 
digit. If I write it instead as 3.000 * 10^3, it has four, because I 
am explicitly telling you all four digits. When I write it as 3000, 
you can't tell whether the 0's are there because I know they are 
correct, or just because I had to put them there to write the number 
(placeholders). You really can't say just from looking at the number 
what I meant.

Similarly, if we write 5.0 * 10^-3, we clearly have two significant 
digits. In this case, that is just as clear from 0.0050, because you 
know the zeros before the 5 are there only as placeholders. If those 
digits were non-zero, I would have had to write their correct values, 
whereas if the zeros in 3000 were really non-zero, I have the option 
of writing them or rounding them off.

Scientific notation eliminates the need to write any digits as mere 
placeholders, so all the digits you see are significant (unless you 
write unnecessary zeros on the left, which would be silly).

Here are several answers in our archives that are relevant to this 
question:

   Significant Figures and Scientific Notation
   http://mathforum.org/library/drmath/view/56291.html   

   Rules for Significant Figures and Decimal Places
   http://mathforum.org/library/drmath/view/58335.html   

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

I searched the archives (http://mathforum.org/mathgrepform.html   ) for 
"significant digits" to find them.

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