Significant figures are extremely important when we perform measurements and make calculations based on our measurements. An exact measurement does not exist because there is always some uncertainty present in a measurement dependent upon the degree of precision of our measuring device. When we record significant figures, we are recording the numbers we know to be reliable based on our device. For example, a measurement of 7.25 g is said to have 3 significant figures. A measurement of 19.30 g is said to have 4 significant figures. In each measurement, we know for certain all of the digits except for the last. In 7.25 g, we are certain the object has a mass of at least 7.2 g, but because our balance is only reliable to centigrams, we cannot have more significant figures than three. In the case of 19.30 g, we have 4 significant figures resulting from our measurement made on the same balance.
As with most concepts in science, we have developed rules to help determine the number of significant figures in a measured quantity.
All non-zero digits in a measurement are significant.
1. 141 m has 3 significant figures
2. 1, 560 g has 3 significant figures
Zeroes between nonzero digits are significant.
1. 1002 g has 4 significant figures
2. 305 cm has 3 significant figures
Zeros that are used as placeholders are not significant (Leading Zeroes)
1. 0.007 g has 1 significant figure
2. 0.0113 km has 3 significant figures
3. 0.30 N has 2 significant figures (zeroes to right of decimal point are significant)
Zeroes at the end of a number, but not to the right of a decimal point are not necessarily significant (Trailing Zeroes)
1. 1800 km may be 2, 3 or 4 significant figures
2. 30 kg may be 1 or 2 significant figures
***To clear up any ambiguity in the last rule, writing the number is scientific notation allows us to clearly identify the number of significant figures.***
1. 1.8 x 103 km (2 significant figures)
2. 1.80 x 102 km (3 significant figures)
In Addition and Subtraction, the result is always rounded to the same number of digits as the measurement with the fewest decimal places.
1. 6.80 cm (3 significant figures) + 4.5 cm + 5.432 cm (4 significant figures) = 16.732 cm, which should be rounded to 16.7 cm (3 significant figures) to match the measurement with the fewest decimal places (4.5 cm).
2. 3 g – 1.4 g – 0.255 g = 1.345 g, which should be rounded to 1 g to match the measurement with the fewest decimal places (3 g).
In Multiplication and Division, the result will be rounded to the same number of significant figures as the component with the least number of significant figures.
1. 2.4 m (2 significant figures) × 3.45 m (3 significant figures) = 4.278 m2, which should be rounded to 4.3 m2 to match the measurement with the least significant figures (2.4 m).
2. 7.94 g / 355.2 mL = 0.02234731 g / mL, which should be rounded to 0.0223 g / mL to match the measurement with the least significant figures (7.94 g).