Significant figures (sig figs) represent the precision of a measurement, indicating which digits are reliable. Using them correctly ensures that calculations do not falsely suggest a higher level of precision than the original measurements allow.
Understanding Significant Figures
Identifying significant figures in a given number follows a set of guidelines. All non-zero digits are considered significant. For instance, a measurement of 23.45 grams contains four significant figures.
Zeros positioned between non-zero digits are also significant; these are known as captive zeros. For example, the number 20.03 meters has four significant figures. Conversely, leading zeros, which appear before non-zero digits, are never significant, as they merely indicate the position of the decimal point. A value like 0.0023 liters only contains two significant figures.
Trailing zeros, found at the end of a number, are significant only if the number includes a decimal point. For example, 200. seconds has three significant figures. However, if no decimal point is present, such as in 200 seconds, the trailing zeros are not considered significant, meaning this value only has one significant figure.
Rules for Addition and Subtraction
When performing addition or subtraction with measured values, the rule for significant figures differs from other arithmetic operations. The result of an addition or subtraction must be rounded so that it contains the same number of decimal places as the measurement with the fewest decimal places. This ensures that the final calculated value does not appear more precise than the least precise measurement used in the calculation. The precision of a sum or difference is limited by the component that is least precisely known. If one measurement is known only to the tenths place, adding or subtracting a measurement known to the hundredths place cannot yield a result known to the hundredths place.
Applying the Rules with Examples
Applying the rules for addition and subtraction involves identifying the limiting precision before rounding the final answer. Consider adding 1.23 meters and 4.5 meters. The first number, 1.23, has two decimal places, while the second number, 4.5, has only one decimal place. When these values are added, the initial sum is 5.73 meters. The final answer must be rounded to one decimal place (from 4.5 meters). Therefore, 5.73 meters is rounded to 5.7 meters.
For a subtraction example, imagine calculating the difference between 12.56 grams and 3.2 grams. The value 12.56 grams has two decimal places, while 3.2 grams has one decimal place. The initial result of the subtraction is 9.36 grams. The final answer must be limited to one decimal place. Consequently, 9.36 grams is rounded to 9.4 grams.