What are Significant Figures?
Significant figures are a reflection of the measurement’s uncertainty. The digits in a measured quantity, including all digits known exactly and one digit whose quantity is uncertain.
Recording measurement provides information about both its magnitude and uncertainty. For example, if we weigh a sample on a balance and record mass as 1.2637g, assume that all digits, except the last, are known precisely. We presume that the last digit has an uncertainty of at least ±1, giving an absolute uncertainty of at least ±0.0001 g or a relative uncertainty of at least,
The number of significant figures is equal to the number of digits in the measurement, except that a zero (0) used to fix the decimal point location is not considered significant. The definition can be ambiguous. For example, how many significant figures are in the number 100? If measured to the nearest hundred, then there is one significant figure. If measured nearest ten, then two significant figures are included. To avoid ambiguity, we use scientific notation. 1×102 has one significant figure, whereas 1.0 ×102 has two significant figures.
For measurement using logarithms, such as pH, the number of significant figures equals the number of digits to the right of the decimal, including all zeros. Digits to the left of the decimal are not included as a significant figure only indicate the power of 10. A pH of 2.45, therefore, contains two significant figures.
The exact number, like the stoichiometric coefficient in chemical formula or reaction, and unit conversion factors, have an infinite number of significant figures. A mole of CaCl2, for example, contains exactly two moles of chloride and one mole of calcium. In the equality
1000 mL = 1 L
Both numbers have an infinite number of significant figures.
Significant figures are also important because they guide us in reporting the results of an analysis. When using measurement in a calculation, the result of that calculation can never be more specific than that measurement uncernitity.
As a general rule, mathematical operations involving addition and subtraction are carried out to the last digit, which is significant for all numbers included in the calculation. The sum of 135.621, 0.33, and 21.2163 is 157.17 since the last significant digit for all three numbers is in the hundredth place.
When multiplying and dividing, the general rule is that the answer contains the same number of significant figures as that number in the calculation having the fewest significant figures. Thus,
It is important to remember that these rules are generalizations. What is conserved is not the number of significant figures but absolute uncertainty when adding or subtracting and relative uncertainty when multiplying or dividing. For example, the following calculation reports the answer to the correct number of significant figures, even though it violates the general rules outlined earlier.
For example, following the list of significant numbers and indicating which zero is significant,
0.168, 70.9, 400.0, 0.0850
0.168 | three significant figures |
70.9 | three significant figures; zero is significant |
400.0 | Four significant figures, all zero, are significant |
0.0850 | Three significant figures; only the last zero significant |
