Decimal Places and Significant Figures

Date: 02/05/99 at 04:04:13
From: zakia ali
Subject: Number skills

If a length of 5.738km is rounded to 5.7km, then the accuracy is 
either "to .....d.p." or "to .....s.f."
   
I am stuck on this problem. Could you help me?

Date: 02/05/99 at 12:46:29
From: Doctor Peterson
Subject: Re: Number skills

Welcome to the Doctor's office, Zakia.

I'm happy to help, and I'll give you a little more than you asked for 
- an explanation of why "d.p." and "s.f." matter.

"Accurate to N decimal places" means the number of digits to the right 
of the decimal point that you can trust is N. For example, if I 
measure a length with a ruler that shows millimeters, the measurement 
will be accurate to one millimeter, or 3 decimal places if I write it 
in meters (0.001 m). If I claimed to have measured it as 1.1293 m, 
you'd know I was guessing about the 3, and would round it off to the 
nearest thousandth: 1.129. The ruler will always produce the same 
number of decimal places.

So your number, 5.7 km, is presumed to be accurate to one decimal 
place. 

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

In your number there are two significant figures, 5 and 7.

The number of decimal places matters when you are adding the numbers. 
For example, if I add 1.2 and 3.45, with different numbers of decimal 
places, I don't 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 don't know all the hundredths I'm adding, I have no 
idea what the hundredths place of the result is, so I have to drop the 
5, and call the answer 4.6, accurate to only one decimal place. (Even 
the tenths might be wrong because of a carry, but it won't be too far 
off.) When I add numbers, the result is only accurate to the smallest 
number of decimal places I'm adding. In this case, since 1.2 has only 
2 decimal places, that's all I can keep in my answer.

On the other hand, if I multiply numbers, what counts is the number of 
significant figures. Suppose I run for 1.45 hours at 6.1 miles per 
hour. Then I've gone 1.45 * 6.1 miles. How accurate is that? Again I'll 
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 figures in the result (2) 
is the smaller of the significant figures for the two multiplicands (3 
and 2), so I have to write my answer as 8.8, rounding it to two 
significant figures and dropping two digits that I worked hard for and 
would otherwise have thought were good. Since 6.1 has only two 
significant figures, I can't keep more than that in my answer.

So that's why decimal places and significant figures are both useful. 
I hope that gives you a better feel for this concept. It's especially 
important in the age of calculators, which give us a false sense of 
accuracy in cases like this!

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