Significant figures represent the meaningful digits in a measurement, reflecting its precision. In scientific and mathematical calculations, using significant figures ensures that the result accurately reflects the certainty of the input values. This practice prevents answers from appearing more precise than the measurements from which they were derived. Understanding how to apply significant figures is especially important in operations like multiplication, where precision can easily be misrepresented. This article will specifically explore the rule for multiplying significant figures and its practical application.
Identifying Significant Figures
Determining significant figures is a foundational skill for accurate scientific calculations. All non-zero digits are considered significant. For instance, the number 45.67 has four significant figures, as each digit contributes to the measurement’s precision.
Zeros positioned between two non-zero digits are also significant. In a number like 2005, the zeros are “captive” and count as significant figures, resulting in four. Leading zeros, which appear before the first non-zero digit, are not significant; they merely indicate the position of the decimal point. For example, 0.0034 has only two significant figures, as the initial zeros do not convey precision. Trailing zeros, those at the end of a number, are significant only if the number contains a decimal point. So, 120.0 has four significant figures, but 120 without a decimal point has only two.
The Rule for Multiplying Significant Figures
When performing multiplication with measured values, the precision of the result is limited by the least precise measurement involved in the calculation. The rule for multiplying significant figures dictates that the final answer should be rounded to have the same number of significant figures as the measurement with the fewest significant figures. This principle ensures that the calculated product does not suggest a higher level of accuracy than the original data supports.
This rule applies equally to division operations, as both multiplication and division propagate uncertainty in a similar manner. By adhering to this rule, scientists and mathematicians maintain consistency in reporting the precision of their results. It prevents the misleading implication that a calculator’s output, which often displays many digits, possesses inherent accuracy beyond the limitations of the initial measurements.
Step-by-Step Examples
To illustrate the multiplication rule, consider two examples. Suppose we want to calculate the area of a rectangular object with a measured length of 12.3 cm and a width of 4.5 cm. The length, 12.3 cm, has three significant figures, while the width, 4.5 cm, has two significant figures.
Multiplying these values yields an unrounded result of 12.3 cm 4.5 cm = 55.35 cm². The final answer must be limited to two significant figures, matching the least precise measurement (4.5 cm). Therefore, 55.35 cm² is rounded to 55 cm².
As another example, calculate the mass of a substance using its volume and density: 25.0 mL 1.055 g/mL. Here, 25.0 mL has three significant figures (the trailing zero is significant because of the decimal point), and 1.055 g/mL has four significant figures. The unrounded product is 25.0 1.055 = 26.375 g. Following the rule, the answer must be rounded to three significant figures, matching 25.0 mL’s precision. Consequently, 26.375 g rounds to 26.4 g.