Significant Figures
Scientists imply the level of precision in measurements by how they report the number. Unlike in mathematics where 25 and 25.0 are identical, a measurement of 25 cm in science means something drastically different than a measurement of 25.0 cm.
Scientific measurements are reported to one digit more than what is known with certainty. A reported value of 25 cm implies that the actual value is somewhere between 24 cm and 26 cm, approximately. In contrast, a reported value of 25.0 cm implies that the actual value is somewhere between 24.9 cm and 25.1 cm, approximately.
Counting Significant Figures
To know which digits in a number are "significant", first let's go over some terminology. Non-zero digits are 1, 2, 3, 4, 5, 6, 7, 8, and 9 (any digit other than zero). Leading zeros are those that come before all the non-zero digits (but not necessarily before the decimal point). Trailing zeros are those that come after all the non-zero digits (but not necessarily after the decimal point).
Rule 1: Non-zero digits are always significant. For example, 25 has two significant figures.
Rule 2: Leading zeros are never significant. For example 0.025 has two significant figures.
Rule 3: Trailing zeros are significant if a decimal point is shown in the number, but may or may not be significant if no decimal point is shown. By convention, it is assumed that trailing zeros without a decimal point are not significant. For example, 250.0 has four significant figures, but 2500 only has two definitive significant figures. In these cases, it is best to write the number in scientific notation to avoid ambiguity.
Rule 4: Zeroes that are between non-zero digits are always significant.
Rounding
You will often need to round calculated numbers to the proper number of significant figures. For example, 25.18 rounded to one significant figure is 30, rounded to two significant figures is 25, and rounded to three significant figures is 25.2.
How would you round a number like 99.99 to three significant figures? You could either write 100. or 1.00×102.
Multiplication and Division of Measurements
The number of significant figures in the product or quotient of two or more measurements cannot be greater than that of the measurement with the fewest significant figures. For example: 25.0 × 1.0 = 25
Addition and Subtraction of Measurements
The number of digits after the decimal point in the sum or difference of two or more measurements cannot be greater than that of the measurement with the fewest digits after its decimal point. For example: 25.0 + 1 = 26
Logarithms and Antilogs
The number of digits after the decimal point of log(x) should be equal to the number of significant figures of x. For example: log (3.5×105) = 5.54
For an inverse log of x (10x), the number of significant figures in the answer should be equal to the number of digits after the decimal point in x. For example: 10–3.421 = 3.79×10–4
Multi-step Calculations
When doing a series of mathematical operations, you should not round at each individual step. You should keep track of the number of significant figures produced at each step, then round at the end.
For example: (30.62 ÷ 2.0) + 0.4 = 16
To arrive at this answer, you would first analyze each step...
1) 30.62 ÷ 2.0 = 15.31. If rounded, the result would have only have two significant figures, and thus no digits after the decimal point.
2) Take the unrounded value, 15.31, and add 0.4. 15.31 + 0.4 = 15.71. From step 1, we know the result should have no digits after the decimal point, and so we round to the ones place, 15.71 ≈ 16.
You may be wondering, "Why is 16 a better answer than 15.71? Isn't the number more accurate with more digits?". But remember, the original values (30.62, 2.0, and 0.4) each have some degree of error in them (since nearly everything in science is a measurement). If we report the answer as 15.71, we imply that we know it's between 15.70 and 15.72, and that is simply not true. The accumulated error puts the ones place in doubt, so by reporting the answer as 16 we imply that it's roughly between 15 and 17.