Question

Indicate which of the following are exact numbers: (a) the mass of a 945-mL can of coffee, \((\mathbf{b})\) the number of students in your chemistry class, \((\mathbf{c})\) the temperature of the surface of the Sun, \((\mathbf{d})\) the mass of a postage stamp, \((\mathbf{e})\) the number of milliliters in a cubic meter of water, (f) the average height of NBA basketball players.

Short answer

The exact numbers are: (b) the number of students in your chemistry class, and (e) the number of milliliters in a cubic meter of water. The remaining quantities have some uncertainty and are not exact numbers.

Step-by-step solution

a) The mass of a 945-mL can of coffee

The mass of a 945-mL can of coffee will vary depending on the density of the coffee and the material used for the can. Since this is a physical property and it may vary, it is not an exact number.

b) The number of students in your chemistry class

The number of students in your chemistry class is a counted quantity and has no uncertainty. Therefore, it is an exact number.

c) The temperature of the surface of the Sun

The temperature of the surface of the Sun is determined through scientific measurements and may vary with time. This number has an associated uncertainty and is not an exact number.

d) The mass of a postage stamp

The mass of a postage stamp would vary depending on the material and design of the stamp. It's a physical property and would have some uncertainty. Therefore, it is not an exact number.

e) The number of milliliters in a cubic meter of water

This is a conversion factor: 1 cubic meter of water is equal to 1,000,000 milliliters. This number is defined and has complete certainty. Therefore, it is an exact number.

f) The average height of NBA basketball players

The average height of NBA basketball players is a quantity that is calculated from the height of all players, which have uncertainties. Moreover, the average height can change as new players join or older players retire. It is not an exact number.

Key concepts

Physical Properties

Physical properties relate to characteristics of a material or substance that can be observed or measured without changing its identity. These might include attributes like mass, volume, and density. For example, the mass of a can of coffee is determined by its physical properties. The material of the coffee and the can itself will affect this measurement. Physical properties can vary with conditions, such as temperature or pressure, and are therefore not considered exact. This variation introduces a level of uncertainty, which means numerical values based on physical properties are usually approximations rather than exact numbers.

Counted Quantity

Counted quantities refer to items or entities that can be counted individually, with each unit being distinctly separate and identifiable from the others. A simple example is counting the number of students in a classroom. Since you can count each student as a whole and individual unit, this quantity does not have any uncertainty.

• Counted quantities are always considered exact numbers.
• They don’t depend on methods or measurements that introduce error.

For this reason, they provide perfect accuracy, which differentiates them from quantities obtained through measurement.

Scientific Measurements

Scientific measurements involve quantifying variables or occurrences that are not easily countable. They require precision instruments and techniques, often followed by estimations of error or uncertainty. For instance, measuring the temperature of the Sun's surface requires scientific measurement tools and calculations.

• These measurements produce approximate values, not exact ones.
• The results depend on the limitations and precision of the tools used.

Due to the inherent uncertainty in such measurements, the calculated numbers for scientific phenomena are subject to variation and interpretation depending on advancing methods and technology.

Conversion Factor

A conversion factor is a defined numerical relationship between two units that express the same quantity. These relationships have no margin for error. For example, the number of milliliters in a cubic meter is exactly 1,000,000.

• Conversion factors are based on fixed mathematical definitions.
• There is complete certainty in these numbers.

Conversion factors serve as bridges to convert one unit to another, ensuring consistency and precision in problem-solving. When using conversion factors, rounding or approximation is not necessary, as these values are inherently exact by virtue of their definition.