Significant figures provide an indication of how precisely a number is known. You need to recognize how many digits in a number are significant and how many significant figures should be present in the result of a calculation. Since the treatment of significant figures varies depending on who you are talking to, we will give you our version (the correct version) of how to deal with significant figures.
1. Integers, numbers obtained by counting and defined numbers have an infinite number of significant figures, i.e. they are known with infinite precision:
Number or Number of
Relationship Significant Figures
the integer "5" ∞ (infinite)
3 apples ∞ ( infinite, an exact number)
1000 g in 1 kg ∞ (infinite, defined relationship)
453.6 g in 1.000 lb 4 (measured relationship)
2. All other numbers have a given number of significant figures equal to the number of digits to the right of the first non-zero digit including that digit:
Number Number Significant
Figures
18.2 3
18.20 4
0.0051 2
0.501 3
0.50100 5
501.00x10-3 5
320.000 6
320. 3*
320 2*
300 1*
300. 3*
3.2·10+2 2
* By convention, the decimal is used to indicate whether the zeros in a number greater than one are significant. Note the examples above.
3. When rounding a number that is exactly 5, round to the nearest even integer. This procedure will result in your rounding up about half the time and rounding down the other half. This way you can average out most rounding errors.
3.275, round to 3.28; 3.285, round to 3.28
4. In addition or subtraction the number with the largest uncertainty (smallest degree of precision) determines the uncertainty in the result.
4.010 1.100 0.900
+200.2 -0.3522 +0.376
204.2 0.748 1.276
Note: In the middle example both numbers involved in the calculation have four significant figures, but the answer only has three! In the right example both numbers involved have three significant figures, but the answer has four! The number of significant figures can change doing addition and subtraction.
5. When adding or subtracting numbers that are expressed in scientific notation, first express both numbers to the same power of 10:
3.70x10+3 3.70x10+3
-454 ===> -0.454x10+3
3.25x10+3
6. In a multiplication or division problem the number with the smallest number of significant figures determines the number sig. figs. in the answer. In this example, 2.1 has only two significant figures, so it determines the precision of the answer.
(2.1)x(3.06)/(5.9987) = 1.1
7. A significant figure is gained when taking the logarithm of a number. A significant figure is lost when taking the antilog (raising to an exponent).
ln(3.69) = 1.306 e+1.306 = 3.69
log(0.405) = -0.3925 10-0.3325 = 0.405
8. In complex calculations the number of significant figures is carried through each step of the calculation in the same order that the calculation is performed:
(1.99)x[log(88.3)]/(3.82-3.63) = (1.99)x(1.946)/(0.19) = 20
Note: When
performing complex calculations, a mental track of the correct number of
significant figures should be kept as the calculation is carried out. At least one more than the correct number of
significant figures should, however, be carried through the calculations to
avoid accumulating rounding errors. The
final answer can be expressed to the correct number of sig, figs.