Significant Figures: Understanding and Application in Calculations

Significant figures are essential in performing precise calculations, whether in scientific research, engineering, or everyday measurements. Understanding how to count and use significant figures correctly ensures that your data and results are both accurate and reliable. This article delves into the rules and applications of significant figures through various examples.

Stage One: Counting Significant Figures

Significant figures (sigfigs) play a crucial role in determining the precision of a measured or calculated value. Here are the fundamental rules for counting sigfigs:

All Non-Zero Digits Are Significant

Any digit that is not zero is considered a significant figure. For example:

12.34 has four sigfigs.

Trailing Zeros

In a number without a decimal point, trailing zeros are significant unless otherwise noted. Examples include:

4500 has two sigfigs.4.500 has four sigfigs.

Leading Zeros

Leading zeros are not considered significant. A more accurate way to represent such numbers is in scientific notation:

0.00405 has three sigfigs (4, 0, 5).

Underlines and Overlines

Exceptional cases where a trailing zero might be significant can be marked with underlines or overlines. These examples can also be better represented in scientific notation:

72overline{0}00 has three sigfigs.3050underline{0}0000000 has five sigfigs.

For instance:

72{0}00 would be better as 7.20 × 104. 3050{0}0000000 would be better as 3.0500 × 1011.

Exact Values

Exact values, such as fundamental constants or conversions, are considered to have infinite sigfigs. For instance:

1 meter is exactly equal to 1000 millimeters, so 1000 mm is treated as infinite sigfigs for conversions between meters and millimeters.

Stage Two: Multiplication and Division

When multiplying or dividing numbers, the result should be rounded to the lowest number of sigfigs. For example:

8.288 mm × 4.00 mm 33.2 mm 0.41 m}{77.215 s} 0.0053 m/s 5.3 × 10-3 m/s

Exact values, like those in conversions, do not limit the sigfigs of the result:

0.0412529 m × 1000 mm}{1 m} 41.2529 mm

Stage Three: Addition and Subtraction

In addition and subtraction, the result should be rounded to the lowest number of decimal places, not significant figures. Examples include:

13.15 kg - 2.2 kg 15.4 kg 0.0915 s - 0.065 s 0.027 s

Stage Four: Squares and Square Roots

Squaring a number or raising it to the second power results in losing one sigfig. Conversely, taking the square root of a number gains one sigfig. Examples are shown below:

(14.08 cm)2 198 cm2 √4.4 s2 2.10 s

Note that not all teachers or professors may require knowledge of these rules. Consult your instructor.

Stage Five: Logarithms and Antilogarithms

Logarithms and natural logarithms of a number are determined by the number's sigfigs. Antilogarithms of a number are determined by the number's decimal places. Examples are:

log10241.9 2.3836 ln88 4.48 101.962 91.6 e4.8818 131.9

Stage Six: Multiple-Step Calculations

In multiple-step calculations, it is generally advised to avoid rounding intermediate values until the final step. However, some instructors recommend keeping one additional sigfig in each intermediate step. As always, consult your instructor.

For example:

9.915 kg - 2.28 kg}{0.82 L} 12.195 kg}{0.82 L} 15 kg/L

In this example, I kept all three decimal places in the numerator in the intermediate step.

Conclusion

Understanding and correctly applying significant figures is crucial for maintaining precision in your calculations. By following the guidelines outlined in this article, you can ensure that your results are reliable and accurate.