Significant Figures; RoundOff Errors; Conversion Factor Method
Particularly in the first term of CHEM 1000, there is a considerable emphasis on working
problems and doing calculations; they form the basis of the quizzes and examinations. Many
students from high school have encountered the following frustration. You work a problem and
get an answer of 45.2; your friend's answer is 45; another student got 45.1936. Sometimes
students and even teachers say "It doesn't really matter; so long as you understand it . . . "
Nonsense! It does matter! If the problem is based on measured experimental data, there is only
one
right answer. Otherwise, all we are doing is sweeping the mess under the carpet and trying to
pretend it doesn't exist. Intellectually, that is very unsatisfying. In the little illustration above, one
answer is correct and two are wrong. If these appeared on one of our quizzes or exams, two of
the students would receive a deduction of ½ or 1 mark for "wrong sig. figs." One of our
objectives in CHEM 1000 is to educate experimental scientists to handle data properly. Put out a
little effort right now and it will pay big dividends for the rest of your scientific life. Very soon it
will become second nature to use significant figures correctly.
In practice, the correct handling of sig. figs. can be taken care of by a few simple roughand
ready Rules
.
1.
Put all the data into the calculation using all
the digits you've got. Train yourself to do
this even when it is a rough "first attempt".
2.
When you multiply and/or divide numbers, it is the number with the fewest sig. figs.
which controls the answer.
3.
When you add and/or subtract numbers, it is the number with the fewest decimal places
which controls the answer.
Number of Significant Figures
Let us first be clear on the number of sig. figs. A few examples will suffice (number of sig. figs.
in brackets). 4.168 (4). 416.8 (4). 2.0000 (5). 2×10
6
(1). If it was accurate to 3 sig. figs. you
would write it as 2.00×10
6
; the power of 10 just shifts the decimal place but does not affect the
accuracy. 0.000359 (3); it is the same as 3.59×10
4
with a shift of the decimal point. "The tunnel
is 200 m long." Bad! Ambiguous! It isn't clear if the tunnel is 2×10
2
(1), 2.0×10
2
(2) or 2.00×10
2
(3) m long. It may be necessary to use ×10
1
; e.g. 60 to 1 sig. fig. is expressed by 6×10
1
.
RoundOff Errors
One of the advantages of using sig. figs. properly is that you avoid roundoff errors. Everyone is
familiar with the procedure for rounding off: if the last digit is 0 to 4, don't change the previous
digit; if it is 5 to 9, raise the previous digit by 1. For example, if the number of sig. figs. is (2),
67.3 or 67.0 round off to 67, but 67.5 or 67.9 round off to 68.
The following little problem shows how indiscriminate rounding off can introduce errors.
Consider taking exactly ½ of a number several times in succession. (a) 97×½ = 48.5 which
rounds off to 49 (rounded off to maintain the 2 sig. figs. quoted for 97); (b) 49×½ =24.5 which
rounds off to 25; (c) 25×½ = 12.5 which rounds off to 13; etc. But anyone knows that all we have
done here is to work out 97×½×½×½ = 97/8 = 12.125 or 12 (to 2 sig. figs.). Something is clearly
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 Fall '09
 Hempstead
 pH, Decimal, International System of Units, PA., sig. figs.

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