Significant figures are the meaningful digits in a number, reflecting the accuracy with which a measurement was made. Understanding how to handle these figures prevents calculations from implying a greater level of precision than the original measurements allow. Different mathematical operations, such as multiplication, have specific rules for determining the correct number of significant figures in the result.
Understanding Significant Figures
Identifying the number of significant figures in a given value is a foundational step before performing calculations. All non-zero digits are always considered significant. For example, the number 4,129,45 has six significant figures.
Zeros can be complex. Zeros located between other non-zero digits are significant, such as in 1008, which has four significant figures. Leading zeros, which precede all non-zero digits, are not significant and simply indicate the position of the decimal point, as seen in 0.0025, which has two significant figures. Trailing zeros are significant only if the number contains a decimal point. For instance, 90.7500 has six significant figures, while 100 might have only one significant figure if no decimal point is shown.
The Rule for Multiplication and Division
When performing multiplication or division, the result’s precision is limited by the least precise measurement used in the calculation. The rule states that the answer should have the same number of significant figures as the measurement with the fewest significant figures involved in the calculation.
This rule applies equally to both multiplication and division operations. For example, if one measurement has three significant figures and another has two, the final answer from their multiplication or division must be rounded to two significant figures. This approach maintains consistency with the inherent uncertainty of the least precise input value.
Applying the Rule with Examples
Applying the significant figures rule for multiplication involves identifying the significant figures in each number, performing the calculation, and then rounding the result appropriately. For instance, consider multiplying 2.5 cm by 3.42 cm. The number 2.5 has two significant figures, and 3.42 has three significant figures. The calculator result of 2.5 multiplied by 3.42 is 8.55. According to the rule, the answer must be limited to the fewest number of significant figures from the original numbers, which is two. Therefore, 8.55 is rounded to 8.6 cm², maintaining two significant figures.
Another example involves 10.888 multiplied by 44. The number 10.888 has five significant figures, while 44 has two significant figures. The raw product is 479.072. Since the least number of significant figures is two (from 44), the final answer must be rounded to two significant figures, resulting in 480. When dealing with numbers like 503.29 (five significant figures) multiplied by 6.177 (four significant figures), the product is 3108.82233. The result should be rounded to four significant figures, yielding 3109.
Consider a calculation like 0.012 m multiplied by 100.5 m. The number 0.012 has two significant figures (leading zeros are not significant), and 100.5 has four significant figures. The direct product is 1.206 m². Following the rule, the answer must be limited to two significant figures, so the result is rounded to 1.2 m². These examples illustrate how the rule consistently guides the precision of calculated results, reflecting the precision of the initial measurements.