Question

Indicate which of the following are exact numbers: (a) the mass of a paper clip, (b) the surface area of a dime, (c) the number of inches in a mile, (d) the number of ounces in a pound, (e) the number of microseconds in a week, (f) the number of pages in this book.

Short answer

The exact numbers among the given quantities are: (c) the number of inches in a mile, (d) the number of ounces in a pound, (e) the number of microseconds in a week, and (f) the number of pages in this book.

Step-by-step solution

(a) The mass of a paper clip

The mass of a paper clip is not an exact number, as it can vary depending on the size, material, and manufacturing of the paper clip. There could be variations and uncertainties in the mass of different paper clips.

(b) The surface area of a dime

The surface area of a dime is not an exact number either, as it may vary based on wear and tear, manufacturing differences, and the like. There could be minor variations in the surface area of a dime.

(c) The number of inches in a mile

The number of inches in a mile is an exact number, determined by definition: 1 mile is exactly equal to 63,360 inches. This is a relationship set by definition and does not have any uncertainty.

(d) The number of ounces in a pound

The number of ounces in a pound is an exact number because it is defined as exactly 16 ounces in one pound. There is no uncertainty in this value, as it is established by definition.

(e) The number of microseconds in a week

The number of microseconds in a week is also an exact number because it is determined by counting and conversions. There are 1,000,000 microseconds in a second, 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 7 days in a week. This calculation (\(1{,}000{,}000 \times 60 \times 60 \times 24 \times 7\)) results in an exact number of microseconds.

(f) The number of pages in this book

The number of pages in the given book is an exact number because it refers to a countable quantity. The number of pages does not have any uncertainty and remains constant unless new pages are added or removed.

Key concepts

Measurement Uncertainty

Measurement uncertainty is a fundamental concept in science that reflects the lack of exactness in any measurement. No physical measurement can be made with absolute certainty, and hence there is always a degree of uncertainty associated with any measurement.

This concept is particularly important when considering measurements like the mass of a paper clip or the surface area of a dime, as mentioned in the exercise. Variations in manufacturing, wear and tear, or inconsistencies in measurement tools can contribute to measurement uncertainty. For instance, paper clips may have different masses due to slight differences in material or size, indicating that the mass of a paper clip is not an exact number and is subject to variation.

Understanding measurement uncertainty helps in making more accurate and reliable scientific observations, which is vital in both experiments and practical applications. Scientists and engineers often express this uncertainty as a range, or as a percentage, to indicate the potential error in their measurements.

Units of Measurement

Units of measurement are standardized quantities used to express and compare the properties of objects. Common units include meters for length, kilograms for mass, and seconds for time. The exercise demonstrates the importance of units by dealing with inches, ounces, and microseconds.

Standardization of Units

Without standardization, communication and replication of results in science would be nearly impossible. For example, the number of inches in a mile or ounces in a pound are fixed by definition. Bodies like the International System of Units (SI) maintain standard units to ensure coherence in measurements across the globe.

Units of measurement not only establish consistency but also allow for precise communication across different fields of study and industries. The certainty in these units is crucial for building, manufacturing, health, and many other areas where precise quantifications are necessary.

Metric Conversions

Metric conversions involve changing a measurement from one unit to another within the metric system or between different measurement systems. The metric system is a decimal-based system of measurement used worldwide, leading to simpler conversion processes.

Role in Scientific Calculations

Conversions are essential in scenarios where different systems of measurement are used, such as converting inches to centimeters or pounds to kilograms. In the exercise, we see the conversion of time units from microseconds to weeks to arrive at an exact number. Understanding how to perform these conversions accurately is a key skill in science and everyday life. The exactness in the number of microseconds in a week stems from the fixed conversion factors within the metric system, emphasizing the importance of precise calculations when dealing with metric conversions.

Scientific Definitions

Scientific definitions are precise, universally accepted terms that describe specific concepts or quantities in science. Unlike everyday language, where a word might have several meanings, scientific definitions aim to reduce ambiguity and ensure clear understanding.

In the context of the exercise, exact numbers like the number of inches in a mile or ounces in a pound are determined by scientific definitions. These defined relationships are not guesses or estimates; they are exact values that have been agreed upon to facilitate accurate communication and computation in science and various other fields.

The Importance of Definitions

Without these exact definitions, measurements could become inconsistent and unreliable. Defining quantities allows scientists to create standards for measuring and comparing physical properties across different domains and applications, ensuring better precision and reproducibility in their work.