Rules for Significant Figures and Decimal Places

Date: 03/23/99 at 21:29:42
From: Ashley Seither
Subject: Definition and rules

I need to know what significant digits are, and what rules go with 
them.

Date: 03/24/99 at 12:00:26
From: Doctor Peterson
Subject: Re: Definition and rules

When we work with numbers that come from the real world (such as measurements 
from a ruler), the numbers are not exact, but carry some amount of inaccuracy 
with them (because, for example, no ruler is absolutely, perfectly straight). 
There are two main ways we can describe the accuracy of a measurement:

If it is accurate to N *decimal places*, this means that there are N 
digits to the right of the decimal point that you can trust. For 
example, if I measure a length with a ruler marked off with millimeters, then 
the measurement will be accurate to the nearest millimeter. (If I write it in 
meters, to three decimal places: 0.001m.) If I claim to have measured it as 
1.1293m, you know I was guessing about the 3 ten-thousandths, and you would 
round it off to the nearest thousandth: 1.129. If I say it was 1.100 m to three 
decimal places, you know that the two zeroes are not just guesses, but what 
I actually read from the ruler. The ruler will always produce the same number 
of decimal places, since there is a certain minimum size it can measure.

If a number is accurate to N *significant digits* (or figures), this 
means there are N meaningful digits that you can trust. For example, 
in my 1.129m, there are four digits I consider dependable, based on 
how I measured. If I had measured 0.024m with the same ruler, there would 
be only two significant digits. (The zeroes are there only to show the place 
value of the other digits, and are not 'significant'.) The ruler does not 
always produce the same number of significant digits, because it is better at 
measuring larger things. If I tried to measure something smaller than a 
millimeter, it would be useless. It would not give me any significant digits 
at all!

Incidentally, be careful about zeroes in a number. If I told you a road was 
12300m long, according to my car's odometer which shows tenths of a kilometer, 
you would know that only three digits are significant, because I read "12.3." 
The two zeroes, like the zeroes in 0.024, are there only to give the other 
digits the right meaning. But if I used a more accurate instrument, I might 
have read all five digits exactly. You do not know unless I tell you how I 
measured it or how many digits are significant.

Now, what happens to the accuracy of a number when I use it in a 
calculation? Or rather, how does the accuracy of the 'inputs' to a 
calculation affect the accuracy of the 'output'?

When you are adding numbers, you want to look at the number of decimal 
places. For example, if I add 1.2 and 3.45, with different numbers of 
decimal places, I do not know what the hundredths place of 1.2 is, or 
what the thousandths place of 3.45 is. I can put an X for the unknown 
digits and see what happens:

    actual    with X's

      1.2       1.2XX
    + 3.45    + 3.45X
    ------    -------
      4.65      4.6XX

You see, since I do not know all the hundredths I am adding, I really 
have no idea what the hundredths place of the result is (an 'unknown' 
plus 5 is still 'unknown'). So, to be honest, I have to drop the 5 and 
call the answer 4.6 (or else round it up to 4.7), showing that my 
answer is accurate to only one decimal place. (Even the tenths might 
be wrong because of a carry, but it would not be too far off.) So, when 
I add numbers, the result is only accurate to the smallest number of 
decimal places I am adding. In this case, since 1.2 has only one 
decimal place, that is all I can keep in my sum.

On the other hand, if I multiply numbers, what counts is the number of 
significant digits. Suppose I run for 1.45 hours at 6.1 miles per hour. 
Then I have gone 1.45 x 6.1 miles. How accurate is that? Again, I will 
put an X for the unknown places and see what happens:

      actual       with X's

        1.4 5        1.4 5 X
    x     6.1    x     6.1 X
    ---------    -----------
        1 4 5        X X X X
      8 7 0        1 4 5 X
    ---------    8 7 0 X
      8.8 4 5    -----------
                 8.8 X X X X

You can see that the number of significant digits in the result (two) 
is the smaller of the significant digits for the two multiplicands (three 
and two respectively), so I have to write my product as 8.8, rounding it 
to two significant digits and dropping two digits that I worked hard for 
and would otherwise have thought were good. Since 6.1 has only two 
significant digits, I cannot keep more than that in my product.

So those are the rules:

   When you add (or subtract), you keep as many *decimal places* as
   there are in the least accurate number.
   
   When you multiply (or divide), you keep as many *significant
   digits* as there are in the least accurate number.

I should mention that this is only a 'rule of thumb', and it sometimes 
underestimates the precision of an answer. If I had demonstrated 
multiplication with a larger factor in place of the 1.45, you would 
have seen an extra significant digit because of a carry. There are 
more careful rules for measuring the accuracy of a result, when you 
really need to know just how accurate a number is, but significant 
digits work well as a general rule.

In this age of calculators, when you can get many digits in any 
calculation with no trouble, it is important not to keep all those 
digits and thus get a false sense of the precision of your results. 
We do not want to pretend we know seven digits when we really only know 
two or three.

Here are some answers in the Dr. Math archives that may be of interest 
to you:
   
   Rounding to One Digit Accuracy
   http://mathforum.org/dr.math/problems/lori4.16.98.html   
   
   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   

I hope this does not overwhelm you! I wanted to give you not just the 
basic definition, but some background so you could see why it makes 
sense.

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

Date: 03/09/2001 at 08:44:08
From: Solomon JHS Math Club
Subject: Significant Digits Dividing

We reviewed this answer and we still have trouble understanding the rule 
for dividing. For example, the exact answer for 366/2 = 183. Since 2 has 
only one significant digit, does that mean that the most accurate answer 
is 200?

Date: 03/09/2001 at 09:22:32
From: Doctor Peterson
Subject: Re: Significant Digits Dividing

Hi, Solomon.

It depends on whether 2 is a measurement, or a known exact value, as, for 
instance, if you are calculating a radius from a diameter, so you know you 
have to divide by exactly 2. In the latter case, you can think of the 2 as 
having infinitely many significant digits when you apply the rule, since 
EVERY possible digit is known exactly. You have three significant digits 
in 366, so you can keep three in the answer. But if 2 represents, say, 
the number of hours it took to go 366 miles, then the answer should be 
200 mph, since you have only one digit of precision in the time measurement.

We usually talk of significant digits only when there are decimal points 
present to show the assumed precision of the numbers; in fact, we should 
only do this if we know something about the actual measurements, rather 
than just assuming the significant digits from the way a number was written. 
Also, it's best to write numbers in scientific notation if we want to be 
completely clear about accuracy. (Does 23000 have 2 or 5 significant digits? 
If we write it as 2.30 *10^4, we can see that it has 3.) In such a context, 
writing "2" with no decimal place would make it clear that it is an exact 
number.

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