Significant Figures

Date: 02/25/99 at 04:14:53
From: Em
Subject: Significant Figures

I am having trouble with significant figures. I do not understand why:

a) 62.3 multiplied by 5.7 = 360, but 62.30 multiplied by 5.70 = 355. 

The question says to express your answer with an appropriate number of 
significant figures. Please help me, as I do not understand why we get 
360 and 355.

Date: 02/25/99 at 12:15:12
From: Doctor Peterson
Subject: Re: Significant Figures

The idea here is that if one of the numbers you are multiplying is only 
accurate to two significant digits, you can only trust two significant 
digits of the result, so you round to that accuracy.

When the numbers being multiplied are given as 62.30 and 5.70, there 
are 4 and 3 significant digits respectively, so you can keep 3 digits 
in your answer, 355. But when you are only given 62.3 and 5.7, you 
should only keep 2 significant digits, so you round it up to 360.

Here's one way to see why this is. (I'll use a different example, and 
explain why below.)

The multiplication of 12.30 by 5.70 looks like this:

          1 2.3 0
        *   5.7 0
        ---------
          0 0 0 0
        8 6 1 0
      6 1 5 0
      -----------
      7 0.1 1 0 0

If we don't know the last digit of each number, but represent each 
unknown digit by X, your multiplication looks like this:

          1 2.3 X
        *   5.7 X
        ---------
          X X X X
        8 6 1 X
      6 1 5 X
    -------------
      7 0.X X X X

I've written X wherever I don't know what a digit is, because I'm 
multiplying or adding an unknown digit. The X's show that I can't trust 
the last digits, and should round off to 70. If you look closely, 
you'll see that the significant digits in the answer come from the 
significant digits of the  5.7, the number with the fewest significant 
digits. So that's the rule: you keep as many significant digits in the 
product as there are in the factor with the fewest significant digits.

Now here's your original problem:

          6 2.3 X
        *   5.7 X
        ---------
          X X X X
      4 3 6 1 X
    3 1 1 5 X
    -------------
    3 5 5.X X X X

You'll notice that it looks as if we have more valid digits than the 
rule says! That's because the first digits of both numbers are 
relatively large, so that you get an extra digit. The rule is an 
approximation, and is a little on the conservative side, assuming that 
it's better to keep too few digits than to trust too many in some 
cases. We could probably modify the rule slightly to take the extra 
digit into account, but the simple rule has been found to be good 
enough. 

The important thing, of course, is that whether you call the answer 
355 or 360, you are rounding off several more digits from the actual 
product, 355.11, that are really meaningless. 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 get a false 
sense of the precision of your results. We don't want to pretend we 
know seven digits when we really only know 2 or 3.

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