Question

Indicate which of the following are exact numbers: (a) the mass of a 7.5 - by \(12.5-\mathrm{cm}\) index card, \((\mathbf{b})\) the number of grams in a kilogram, \((\mathbf{c})\) the volume of a cup of Seattle's Best coffee, (d) the number of centimeters in a kilometer, \((\mathbf{e})\) the number of microseconds in a week, \((\mathbf{f})\) the number of pages in this book.

Short answer

The exact numbers from the given list are: (b) the number of grams in a kilogram, (d) the number of centimeters in a kilometer, and (e) the number of microseconds in a week. The other quantities are not exact numbers due to factors like material, size, and uncertainty.

Step-by-step solution

(a) Mass of a 7.5 x 12.5 cm index card

The mass of a 7.5 x 12.5 cm index card cannot be an exact number as it depends on the material of the card and can vary between different cards of the same size. So, this is not an exact number.

(b) Number of grams in a kilogram

By definition, there are exactly 1000 grams in a kilogram. Thus, this is an exact number.

(c) Volume of a cup of Seattle's Best coffee

The volume of a cup of Seattle's Best coffee is not an exact number because it depends on the size of the cup used and other factors such as how much coffee is poured. Therefore, this is not an exact number.

(d) Number of centimeters in a kilometer

By definition, there are exactly 100,000 centimeters in a kilometer (1 kilometer = 1000 meters, 1 meter = 100 centimeters). Thus, this is an exact number.

(e) Number of microseconds in a week

A week has a fixed duration of 7 days, where each day has 24 hours, each hour has 60 minutes, each minute has 60 seconds, and each second has 1,000,000 microseconds. So, we can calculate the exact number of microseconds in a week as follows: \(7 \times 24 \times 60 \times 60 \times 1,000,000 = 604,800,000,000\) microseconds. This is an exact number.

(f) Number of pages in this book

The number of pages in a book is an exact number, as it refers to a specific, countable quantity. However, since we don't have the information about the book in question, we can't provide the actual numerical value. Nonetheless, this would be considered an exact number.

Key concepts

Measurement

Measurement is the process of determining the quantity, size, or extent of something. In scientific contexts, measurement is key, as it ensures that we can compare results, repeat experiments, and share findings reliably across the globe. It typically involves an instrument and always needs a unit of reference, like meters or kilograms.

Inquiring whether a measurement can be exact involves understanding its inherent limitations. For example, while you can measure the mass of an index card, this mass will vary based on the card's material and thickness. Therefore, unlike simple counting, which can be exact, measurements that depend on physical properties are typically not exact. Measurement can be broken down to:

• Quantitative observation - Amplitude of a property, like temperature or mass.
• Instrumental precision - Precision can vary based on the quality and calibration of tools.
• Environmental factors - Conditions like air pressure can influence measurement ends.

Understanding these elements helps us grasp why some values can be precise, while others cannot.

Unit Conversion

Unit conversion is the process of changing a measurement from one unit to another. This is an essential skill in science, helping scientists and students understand different interpretations of data. For example, converting a measurement in grams to kilograms involves recognizing that 1000 grams equals one kilogram.

With certain unit conversions, values are exact due to standardized definitions. Among the examples from the exercise:

• The conversion from grams to kilograms is exact (1000 grams = 1 kilogram).
• Similarly, the conversion from centimeters to kilometers involves strict definitions (1 kilometer = 100,000 centimeters).

With exact numbers in unit conversions, you can be certain of their precision, as they derive from fixed relationships in measurement systems Precision in unit conversions allows scientists to make calculations knowing they won't introduce measurement errors. It's crucial to ensure arithmetic steps adhere to these exact values without rounding or estimating.

Precision in Chemistry

Precision in chemistry is about how consistently results can be obtained with repeated measurements. It signifies the reproducibility of a measurement and is vital for ensuring reliable scientific results. Unlike general counting, precision in chemistry doesn't guarantee accuracy, but it helps indicate how well experimental results can be replicated.

Exact numbers, such as the defined number of units in a conversion, don't impact the precision of a measurement, but they provide a foundation upon which precise experiments can build. Chemicals measured with precise tools yield better-replicated results. However, knowing the number of days in a week, which equals an exact value of microseconds, aids in precision when dealing with time-based chemical reactions.

Achieving precision in chemistry involves:

• Properly calibrated equipment to avoid systematic errors.
• Consistent measurement techniques, reducing random errors.
• Recognizing the need for significant figures in reporting results, which maintain raw data's integrity.

Precision ensures that experiments and discoveries deliver credible, consistent results regardless of conditions or repeated tests.