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 are (c) number of inches in a mile, (d) number of ounces in a pound, (e) number of microseconds in a week, and (f) number of pages in this book.

Step-by-step solution

(a) Mass of a paper clip

The mass of a paper clip is determined by measurement and can vary depending on the material and manufacturing process used. Hence, it is not an exact number.

(b) Surface area of a dime

The surface area of a dime is determined by measurement and can also vary slightly due to manufacturing tolerances. Hence, it is not an exact number.

(c) Number of inches in a mile

The number of inches in a mile is defined by convention. There are exactly 63,360 inches in a mile. Hence, it is an exact number.

(d) Number of ounces in a pound

The number of ounces in a pound is also defined by convention. There are exactly 16 ounces in a pound. Hence, it is an exact number.

(e) Number of microseconds in a week

The number of microseconds in a week can be easily calculated using the exact numerical values of the number of seconds in a minute, minutes in an hour, hours in a day, and days in a week. There are exactly 604,800,000,000 microseconds in a week. Hence, it is an exact number.

(f) Number of pages in this book

The number of pages in a book is determined by simple counting, and it should be the same for every copy of the same edition of the book. Hence, it is an exact number.

Key concepts

Measurement

When we talk about measurement, we're discussing the process of determining the quantity, dimension, or extent of something that can be compared against a standard unit. It's essential to understand that measurements are not exact numbers. For example, measuring the mass of a paper clip involves using tools such as a scale. These tools, while accurate to a certain degree, are subject to calibration, precision limits, and human error.
Because of these factors, when we measure attributes like the mass of a paper clip or the surface area of a dime, the results can vary slightly. Hence, they are approximate rather than exact. Consider combining measurement with precision, where precision indicates how consistent repeated measurements are. However, it's crucial to recognize that precision doesn't equate to accuracy. While precision means results are close to one another, accuracy refers to how close a measurement is to its true value.
Whenever you measure, always remember that these values are best estimates, not exact figures.

Convention

Conventions in mathematics and science provide us with standardized numerical values that don't change. They serve as a framework that helps society use terms and measurements consistently.
For instance, the number of inches in a mile is a convention. We all agree that there are exactly 63,360 inches in a mile. Similarly, the number of ounces in a pound is another convention. With 16 ounces in a pound, this value does not change, regardless of where or how it's applied. This consistency allows us to use these numbers reliably across various applications, making them exact numbers.
Conventions arise from a collective agreement or a standardization process that gives us a mutual understanding and base to reference from. It's crucial in science and mathematics to ensure everyone is working within the same parameters.

Counting

Counting is the process of identifying the total number of items in a collection by adding them up one by one. This process results in what we call exact numbers. Whenever you count something, like the number of pages in a book, you get an exact figure.
This is because each item in the count, such as a page, is discrete and indivisible in context. For instance, if a book says it has 300 pages, you trust that count to be precise. This is distinct from measurement, where results can vary slightly. Counting offers us the assurance of precision because we're not estimating but rather tallying distinct items.
Simple counting leads to whole numbers and helps provide certainty and specificity, important aspects in various applications ranging from daily tasks to complex mathematical computations.

Numerical Values

Numerical values refer to the number representation we use in various contexts to express measurements, counts, magnitudes, and more. It's important to distinguish between exact and non-exact numerical values.
Exact numerical values are derived from conventions or counting, like the known quantity of 16 ounces in a pound or counting the number of books on a shelf. These figures stay the same no matter who calculates them. In contrast, numerical values gained through measurement are often estimates. They give us a close approximation rather than an exact figure because of potential variations in how measurements are taken and the precision of the instruments used.
Understanding the distinction between these two helps in science, mathematics, and everyday life, providing clarity in communication, calculations, and decision-making.