Physics 105,106
INSTRUCTIONS ON SIGNIFICANT FIGURES
AND ROUNDING OFF
1
How to tell whether figures are significant
Many numbers, such as 2, 1/2, 3, 1/3, 100 etc., are exact numbers and there is no approximation associated
with them. usually the context of a problem will make clear which numbers should be considered as exact; they
should then be used as such and should not be considered as limiting the number of significant digits that will be
retained in the calculation. For example, if the number 1/2 appears it does not mean that only one significant digit
will be retained in the result. On the other hand, many numbers are only approximations to some ‘true’ value, and
the degree of accuracy of the approximation
√ will then depend on how many significant digits are available in this
approximate value. Such numbers as π and 2 are exact numbers when expressed this way, but when given in digital
form, such as 3.141593 and 1.414214, they are approximate numbers. Here they are given to seven significant figure
accuracy; to four figure accuracy they would be 3.142 and 1.414.
A significant figure is any of the digits 0, 1, 2, 3, . . . 9. Note that 0 (zero) is also a significant digit EXCEPT
when it is used only to fix the decimal point or to fill the places of unknown or discarded digits. Thus, in the number
0.000395, the number of significant digits is only three; they are the digits 3, 9, and 5, the zeros serve only to indicate
the decimal place and hence they are not significant. Similarly in a number like 564000 there is nothing shown to
indicate whether the zeros are significant or not, and they are therefore assumed to be nonsignificant. That is, the
number is assumed to have only three significant digits. One way to remove any ambiguity about whether trailing
zeros are or are not significant digits is to write the number in exponential form as 5.640 × 105 , 5.6400 × 105 , or
5.64000 × 105 , the number of significant figures being specifically indicated by the number of digits (zeros counting)
included in the factor to the left of the × sign. Also to write a number as 200. means that the zeros preceding the
decimal point should be considered to be significant.
To further clarify the use of zero as a significant digit in problems in Physics 105 and 106, let us state the following
as our guide.
Lead Zeros: In a number such as 0.03426 the leading zeros serve only to fix the decimal point and as such will
never be considered as significant digits.
Trailing Zeros: If there is only one trailing zero such as in 30, 120, 1450, etc. we will consider it to be a significant
digit; particularly if the numbers are small (i.e. not more than three digits), or if the other numbers in the
problem have a corresponding number (or a greater number) of significant digits. To not consider a single
trailing zero as significant tends to ‘demote’ 0 (zero) as a digit – it no longer has a full status equal to that of
the digits 1, 2, 3, . . . 9 – and this creates a bias in our number system that can be a vexing problem at times.
On the other hand, if there are two or more trailing zeros, such as 14500 or 145000, then the common practice
is that they are not considered significant digits. However, for our purposes in Physics 105 and 106, it is better
in most cases to not drop all trailing zeros as nonsignificant, but to judge how many are to be considered as
significant on the basis of other data in the problem. For example, if we have the number 6000 appearing in a
problem along with another number such as 0.529, then we will consider 6000 to have three significant figures
also. (Any particularly obvious ambiguities will be handled by using the exponential form for the number.)
Trailing Zeros in Decimal Fractions: Trailing zeros written in a decimal fraction such as 0.23900 or 0.4000 will
always be considered as significant digits. They are not serving in any way to indicate the decimal place, and
the very fact that they are written at all indicates that they are significant.
2
How Many Figures Do You Keep?
As a general rule the accuracy of a final result can not be greater than the accuracy of the least accurate datum
used in computing that result. Hence the number of significant digits retained in a result should usually be the
same as that of the data item with the smallest number of significant digits. This rule holds if the computation is
Significant Figures and Rounding
2
dominated by multiplication and division processes; however when additions and subtractions become particularly
important in the computation process, then additional considerations sometimes apply. For example, suppose you
have the computation
R = A − (B × C)
where
then
A = 1245, B = 14, and C = 2.3 .
R = 1245 − (14 × 2.3) = 1245 − 32.2
R = 1212.8 ' 1213 .
You might say that because B and C both have only two significant figures that the final result R should only have
two also, i.e., R = 1200. But this is not correct. Notice that even though the product of B and C is really only
accurate to two figures (i.e. 32), when it is subtracted from 1245 the last significant digit of 32 corresponds to the
same decimal position as the last significant digit of 1245, hence this digit in the result R is still significant and R
has four significant figure accuracy (1213) as a result. One must be careful to watch for this sort of situation in doing
problems – it is not a rare occurrence.
3
Rounding Off
To retain full accuracy in a problem it is essential that round-off errors be kept to a minimum. This means
that one should round off only once, when the final result is reached. Do not round off at intermediate steps in the
computation.
To avoid accumulation of roundoff errors it helps to carry more significant figures in the computation than are
given in the original data. This way one can retain full accuracy of the data until the final result is reached. If two
figures are given in the original data, consider the data accurate to three, or four, figures and carry this accuracy
through the computation to the end, then round off to the two figures originally given. Similarly, given three figure
accuracy, carry four or five figures in the computations, etc. With electronic calculators these extra digits require
little extra effort and final accuracy of results is assured.
In rounding numbers the rule commonly followed is that if the part being dropped off is less than 5 then one
rounds down to the lower number, and if the part being dropped is equal to or greater than 5 one rounds up to the
next higher number. This rule introduces a systematic bias in that one rounds up more frequently than rounding
down. To equalize things, a better rule is to say that if the part being dropped is exactly equal to 5 (or 50 or 500,
etc.) then you round to the nearest even number. Since this results in rounding down about as often as up when
the part being dropped is exactly 5 (or 50, or 500, etc.) the net effect is to minimize any systematic bias in rounding
errors. To illustrate this rule,
14.9650 would be rounded to 14.96,
and
if only four figures are desired.
13.2750 would be rounded to 13.28,