Question

When adding or subtracting measured quantities, what determines the number of significant figures in the result?

Short answer

The number of significant figures in the result of an addition or subtraction is determined by the measurement with the fewest decimal places.

Step-by-step solution

Understand the Concept of Significant Figures

Significant figures in a measured quantity represent all the certain digits plus the last uncertain digit. In measurements, significant figures are determined by the precision of the measuring instrument and reflect the reliability of the measurement.

Rules for Significant Figures in Addition and Subtraction

When adding or subtracting measured quantities, the number of decimal places in the result should be equal to the smallest number of decimal places in any of the quantities being combined. Unlike multiplication and division, where the number of significant figures is based on the measurement with the least number of significant figures, addition and subtraction are based on the least precise measurement (i.e., the one with the fewest decimal places).

Determine Significant Figures in the Result

After performing the addition or subtraction, you should round the result to the same number of decimal places as the measurement with the fewest decimal places. This ensures that the result is not unjustifiably precise.

Key concepts

Measurement Precision

Understanding measurement precision plays a critical role in science and engineering, as it defines the reliability of measurements. Precision relates to how close multiple measurements are to each other and to the level of detail that can be confidently discerned from a single measurement. For example, if you use a ruler that is marked in millimeters, your precision is to the nearest millimeter.

When measurements are recorded, precision is indicated by the number of significant figures given. These figures include all the numbers that are known reliably, plus the first uncertain number. It's essential for students to recognize that more significant figures in a measurement imply higher precision, which affects how we perform calculations, especially in addition and subtraction.

Significant Figures Rules

The rules for significant figures are designed to ensure that the reported results of calculations are as accurate as the starting measurements allow. Here's a simple way to grasp the basic rules:

• All non-zero digits are considered significant.
• Zeros between non-zero digits are also significant.
• Leading zeros (those before any non-zero digits) are not significant.
• Trailing zeros in a number without a decimal point are ambiguous and may or may not be significant based on additional information.
• In a number with a decimal point, trailing zeros (those after a decimal point and after a non-zero digit) are significant.

When calculations involved are either addition or subtraction, the rule is straightforward: the number of decimal places in the result should not be greater than the number of decimal places in the least precise measurement being used in the calculation.

Rounding Measured Quantities

Rounding is the process of reducing the digits in a number to the desired level of precision, based on significant figures rules. After performing operations like addition or subtraction, it is often necessary to round the result to ensure it reflects the precision of the input data.

Here's how to apply it:

• Identify the least precise measurement in the data set based on the fewest number of decimal places.
• Round the final result of your calculation to match this decimal place precision.
• If the digit immediately to the right of the last significant figure is 5 or greater, round up. If it's less than 5, round down.

Applying the concept of rounding after performing addition or subtraction allows us to provide results that are as reliable as our least precise piece of data, maintaining integrity in the measurement process and preventing overestimation of accuracy.