Question

Which of the following is the most accurate measurement? (a) \(9 \mathrm{~m}\) (b) \(9.0 \mathrm{~m}\) (c) \(9.00 \mathrm{~m}\) (d) \(9.000 \mathrm{~m}\)

Short answer

(d) 9.000 m is the most accurate measurement.

Step-by-step solution

Understand Significance of Decimal Places

The accuracy of a measurement is often indicated by the number of decimal places or significant digits in which it is expressed. More decimal places suggests a more precise measurement.

Compare Given Measurements

Compare the number of decimal places in each measurement: (a) has no decimal places, (b) has one, (c) has two, and (d) has three.

Identify the Most Accurate Measurement

Since the number of decimal places indicates the precision and assuming each is measured correctly, the measurement with the most decimal places is the most accurate. Here, (d) has the highest number of decimal places.

Key concepts

Significant Digits

Significant digits, also known as significant figures, are essential in understanding the accuracy and precision of a measurement. They include all the numbers that are known with certainty, plus one last digit, which is a little uncertain or estimated. To grasp this concept, consider using a ruler. If the ruler marks millimeters, and we measure a length that falls directly on a millimeter line, we are certain of it up to the millimeter. But if it falls somewhere between two lines, we estimate the last fraction of a millimeter. That's where our significant digits come into play.

For example, if we have a measurement of 9.00 meters, we have three significant digits: the '9', and both of the zeros after the decimal. These zeros are significant because they were deliberately measured or estimated by the person giving the measurement. To improve understanding, one can practice identifying significant digits by going over more measurements and learning the rules that dictate when zeros are significant. An increased familiarity with significant digits will boost a student's confidence in assessing the precision of different numbers.

Decimal Places

Decimal places directly relate to the precision of a measurement by illustrating how finely a measurement is subdivided. They are counted from the decimal point to the last non-zero digit. For instance, in the measurement 9.00 meters, there are two decimal places.

To understand decimal places better, it's like zooming in on the scale of a ruler: the more decimal places, the closer the zoom and the finer the detail. Thus, you can measure more precisely. It is vital for students to practice counting decimal places for a variety of numbers to develop a sharper eye for detail and precision. By comparing numbers like 9 m, 9.0 m, 9.00 m, and 9.000 m, one can see that as the number of decimal places increases, so does the level of detail and, by extension, the precision of the measurement.

Precision in Measurements

Precision in measurements refers to the consistency and repeatability of a set of measurements. It denotes how close the measurements are to each other, not necessarily how close they are to the actual or true value. A precise measurement is one where there is little spread among the repeated measures. It's like throwing darts and consistently hitting the same spot, although not necessarily the bullseye.

In the context of our problem, measurement (d) 9.000 m indicates a high level of precision due to its three decimal places. The additional zeros show that the measurement has been taken with a tool that can measure to the thousandth of a meter, and the measurer has noted down the extra detail observed. To improve comprehension of precision, one could engage in practical exercises, such as using measuring instruments with different scales, like a standard ruler versus a micrometer, to appreciate how the tools' precision affects the reported measurements.