Question

When multiplying or dividing measured quantities, what determines the number of significant figures in the result?

Short answer

The number of significant figures in the result of multiplying or dividing measured quantities is determined by the quantity with the least number of significant figures.

Step-by-step solution

Understanding Significant Figures in Multiplication and Division

When multiplying or dividing measured quantities, the number of significant figures in the result is determined by the measured quantity with the least number of significant figures. Unlike addition and subtraction, where the precision is based on the decimal place of the least precise number, multiplication and division focus on the total number of significant figures.

Identify the Number of Significant Figures in Each Quantity

Before performing the multiplication or division, identify the total number of significant figures in each of the measured quantities involved in the calculation. Remember that all non-zero digits are significant, as well as any zeros in between them or after them if there is a decimal point.

Perform the Calculation

Multiply or divide the quantities as you would normally do.

Adjust the Result to the Correct Number of Significant Figures

Round the result to have the same number of significant figures as the measured quantity with the fewest significant figures used in the calculation. Use standard rounding rules; if the digit to be removed is less than 5, simply drop it, if it's 5 or more, increase the last remaining digit by one.

Key concepts

Measured Quantities

Accuracy and precision are paramount when dealing with measured quantities. These refer to the numbers that are obtained through measuring instruments and have a limitation on their precision based on the tool used. For example, if you measure a length with a ruler that has millimeter divisions, the quantity you measure will only be precise to the nearest millimeter. It's crucial to recognize that the number of significant figures in a measured quantity reflects the precision of the measurement.

The number of significant figures provides insight into the reliability of a measurement. If a scale provides a weight as 20.0 grams, the presence of two decimal places indicates a higher precision than a scale that provides a weight as 20 grams. It is this indication of precision that becomes influential when these quantities are used in mathematical operations like multiplication and division.

Precision in Calculations

In calculations involving measured quantities, precision must be preserved. This is where the concept of significant figures becomes a cornerstone. Precision in the context of calculations does not mean just getting a result with many digits after the decimal point. Rather, it is about how much those digits are worth trusting. When we perform operations such as multiplication or division, we can only be as precise as our least precise measurement.

Therefore, when calculating with measured quantities, each step must consider the precision of the measurements involved. Let's take the multiplication of two measured quantities, if one has three significant figures and the other has four, the result should be reported with only three significant figures – in line with the least precise measurement. This ensures that the reported answer does not imply a greater level of accuracy than the measurements themselves warrant.

Rounding Rules

The rounding of measured quantities according to significant figures is guided by established rounding rules. When the result of a multiplication or division is to be rounded, we look at the digit immediately to the right of the last significant figure we want to keep. There are two simple rules to follow:

• If that digit is less than 5, we leave the last significant figure as is and remove all the digits to the right.
• If the digit is 5 or greater, we increase the last significant figure by one, also removing all digits to the right.

It's critical to apply these rules correctly to avoid overstating the precision of a result. These rules also help to maintain the integrity of the data and to convey the accuracy of measurements effectively. By understanding and employing these rules, one can ensure that the outcomes of calculations reflect the true precision of the involved measurements.