In scientific work, most numbers are measured quantities and thus are not exact. All measured quantities are limited in significant figures (SF) by the precision of the instrument used to make the measurement. The measurement must be recorded in such a way as to show the degree of precision to which it was made-- no more, no less. Calculations based on the measured quantities can have no more (or no less) precision than the measurements themselves. The answers to the calculations must be recorded to the proper number of significant figures. To do otherwise is misleading and improper.
Determining Which Figures are Significant
Non-zero integers are always significant.
example: 23.4g and 234g both have 3 SFCaptive zeroes, those bounded on both sides by non-zero integers, are always significant.
example: 20.05 has 4 SF; 407 has 3 SFLeading zeros, those not bounded on the left by non-zero integers are never significant. Such zeros just set the decimal point; they always disappear if the number is converted to powers-of-10 notation.
example: 0.04g has 1 SF; 0.00035 has 2 SF. They can be written as 4x10-3 and 3.5x10-4 respectively.Trailing zeros, those bounded only on the left by non-zero integers may or may not be significant.
example: 45.0L has 3 SF; 450L has only 2 SF; 450.L has 3 SF.
Note: To clarify whether a trailing zero is significant, it is preferable to use scientific notation to express the final answer.
example: 450.L can be expressed as 4.50 x 102 or 4.50E02 whereas 450 L would be expressed as 4.5x102.Exact numbers are those not obtained by measurement but by definition or by counting numbers of objects. They are assumed to have an unlimited number of significant figures.
Multiplication and Division Involving Significant Figures
Calculations involving only multiplication and/or division of measured quantities shall have the same number of significant figures as the fewest possessed by any measured quantity in the calculation.
example: 14.0 x 3 = 40, not 42, because one of the multipliers has only one SF.
example: 14.0 x 3.0 = 42, because one of the multipliers has only two SF.
example: 14.0 / 3 = 5, not 4.6, because the denominator has only one SF.
Addition and Subtraction Involving Significant Figures
Calculations where measured quantities are added or subtracted shall correspond to the position of the last significant figure in any of the measured quantities. That is, the final answer is only as precise as the decimal position of the least precise value. The number of significant figures can change during these calculations.
example: 14.16 + 3.2 = 17.36 (this is not the final answer!)
17.4 is the correct answerexample: 46.6 + 5.72 = 52.32 (this is not the final answer!)
52.3 is the correct final answer.
Combined Calculations
In calculations involving addition/subtraction and multiplication/division, significant figure guidelines must be applied before and after each calculation involving addition or subtraction.
example: (3.2 x 4 x 0.035 / 7) + (12 x 0.5) =
0.06 + 6 = 6
Tips for Rounding Off Numbers
A number is rounded off to the desired number of significant figures by dropping one or more digits to the right. If you are rounding to the tens, hundreds place or higher, you must put zeroes in the lesser places (the ones place, for example) to indicate to what place you have rounded. The following guidelines should be observed when rounding off numbers.
- When the first digit dropped is less than 5, the last digit remains unchanged.
- When the first digit dropped is more than or equal to five, the last digit retained is increased by 1.
example: 243 - 240
17.9 - 18example: round 2,457 to the tens place - 2,460 not 246; to the hundreds place - 2,500 not 25.
Conclusion
The precision of the instruments used in collecting data determines the degree to which your results are accurate. Significant figures provide an easy way of indicating that accuracy. Using these guidelines assures that the data resulted from your procedures is not only reproducible, but also allows an observer to understand the degree to which your data is accurate.