Question

Which of the following include exact numbers? (a) The speed of light in a vacuum is a physical constant; to six significant figures, it is \(2.99792 \times 10^{8} \mathrm{~m} / \mathrm{s}\) (b) The density of mercury at \(25^{\circ} \mathrm{C}\) is \(13.53 \mathrm{~g} / \mathrm{mL}\) (c) There are \(3600 \mathrm{~s}\) in \(1 \mathrm{~h}\). (d) In 2018 , the United States had 50 states.

Short answer

Options (c) and (d) contain exact numbers.

Step-by-step solution

Understand the concept of exact numbers

Exact numbers are quantities that are counted or defined values, not measured. They have no uncertainty and do not limit the precision of calculations.

Analyze option (a)

Option (a) gives the speed of light in a vacuum: \(2.99792 \times 10^8 \text{ m/s}\). This is a measured value with six significant figures, not an exact number.

Analyze option (b)

Option (b) provides the density of mercury at \(25^{\text{circ}} \text{C}\): \(13.53 \text{ g/mL}\). This is also a measured quantity, not an exact number.

Analyze option (c)

Option (c) states that there are \(3600 \text{ s}\) in \(1 \text{ h}\). This is a defined value and thus an exact number.

Analyze option (d)

Option (d) mentions that in 2018, the United States had 50 states. This is a count and therefore an exact number.

Conclusion

Based on the analysis of each option, the exact numbers are in options (c) and (d).

Key concepts

Significant Figures

Significant figures are essential in scientific measurements as they reflect the precision of a measurement. These figures include all the certain digits in a measurement plus one uncertain or estimated digit. For instance, if you measure a length and find it to be 12.34 cm, the number has four significant figures. This indicates that the measurement is precise up to the hundredths place.

Significant figures help in conveying how accurate and precise a measurement is. Thus, while recording measured values, significant figures must be counted meticulously to avoid overestimating the precision. For example, the speed of light given as \(2.99792 \times 10^{8} \mathrm{m/s}\) contains six significant figures, highlighting its high precision.

Measured Quantities

Measured quantities are obtained through observation or instrumentation. These values always come with a degree of uncertainty because no measurement is perfect. Each measured quantity must be reported with the appropriate number of significant figures to indicate its precision. For example, the density of mercury at \(25^{\circ} \text{C}\) is \(13.53 \mathrm{g/mL}\), indicating that the measurement is precise to the hundredths place.

Measured quantities require the use of units to ensure clarity. The measurement \(13.53 \mathrm{g/mL}\) specifies that for every milliliter of mercury, its mass is 13.53 grams. Both the magnitude and the unit are crucial in representing the measured quantity accurately and completely.

Defined Values

Defined values are quantities known exactly and without uncertainty. These are often the backbone in calculations involving conversions from one unit to another. Defined values are considered to have an infinite number of significant figures because they are precise by definition. For instance, there are \(3600 \text{s}\) in \(1 \text{h}\). This conversion factor is absolute and has no variation.

Another example of a defined value is the number of states in the United States in 2018. There were exactly 50 states, making this a count value and not subject to measurement uncertainties. Defined values enable precise calculations and maintain the accuracy of results.

Precision in Calculations

Precision in calculations refers to how detailed and exact a result is, based on the significant figures of the numbers involved. It's important to preserve precision throughout calculations to ensure the final result is as accurate as possible. The precision of a calculated result is limited by the least precise measurement used in the calculation.

Rounding and Errors
Precision is maintained by following the rules of significant figures during addition, subtraction, multiplication, and division. For instance, when multiplying two numbers, the result should have the same number of significant figures as the factor with the fewest significant figures. Incorrect rounding can lead to errors, so it's crucial to apply the rules consistently.

Avoiding Overprecision
A common mistake is to report a calculated result that appears more precise than the initial measurements warrant. Always be mindful of the precision of your input values and make sure your final answer correctly reflects the same level of precision.