Problem 43 - Chemistry The Central Science In Si Units

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, (e) the number of microseconds in a week, (f) the number of pages in this book.

Step-by-Step Solution

Answer

Exact numbers are (b), (d), (e), and (f).

Step 1: Understanding Exact Numbers

An exact number is one that is known with complete certainty. Exact numbers typically result from counting (e.g., 5 apples) or defined quantities (e.g., 1 kilogram = 1000 grams). In contrast, measured quantities, which are subject to some degree of uncertainty (e.g., the mass of an object), are not considered exact.

Step 2: Analyzing Each Option

Let's evaluate each option to determine whether it is an exact number. We'll go through each part and check if it is a counting number or a defined constant. (a) The mass of a 7.5 by 12.5 cm index card is a measured quantity and subject to variation, hence not exact. (b) The number of grams in a kilogram is defined as exactly 1000, thus it is an exact number. (c) The volume of a cup of coffee is typically measured and can vary, so it is not exact. (d) The number of centimeters in a kilometer is defined as exactly 100,000, making it an exact number. (e) The number of microseconds in a week is derived from time measurements and predefined conversions, hence it is also exact. (f) The number of pages in this book is a countable quantity, so it is exact.

Step 3: Conclusion

Having analyzed each option, we conclude that the exact numbers in this exercise are (b) the number of grams in a kilogram, (d) the number of centimeters in a kilometer, (e) the number of microseconds in a week, and (f) the number of pages in this book.

Key Concepts

Measurement Uncertainty

Measurement uncertainty refers to the doubt that exists about the result of any measurement. It is a critical aspect to understand, especially in scientific experiments and real-world applications.
In practical terms, no measurement can be perfectly exact. Every measuring device has limitations, and there is always an element of subjectivity in interpreting readings.
For example, the mass of a 7.5 by 12.5 cm index card can vary depending on factors such as the thickness of the card or atmospheric conditions. This is why different people might get slightly different measurements for the same object.

• Measurement uncertainty is inevitable in all measured values.
• Measurements are often reported with a margin of error or uncertainty range.
• The reliability of a measurement can be improved but never completely zeroed out.

Understanding and accounting for measurement uncertainty is essential in making informed decisions and analyzing results effectively.

Counting Numbers

Counting numbers are those numbers that arise from counting discrete items and are inherently exact. This is because there is no ambiguity when counting whole items.
Consider the number of pages in a book; if you have 300 pages, that's a precise number – there can be no fractions when it comes to counting pages.
Similarly, if you have 12 apples, this is an exact count.

• Counting numbers are whole numbers and do not have decimal or fractional parts.
• They provide certainty because you can definitively say any particular count is accurate.
• Used in various fields such as inventory management, budgeting, and statistics.

Counting numbers differ from measured numbers, as they do not involve estimation and therefore have no uncertainty.

Defined Constants

Defined constants are quantities defined by agreement or legislation. These are exact by their very nature, as they do not change and have no uncertainty associated with them.
For example, the number of grams in a kilogram is 1,000 – this is a defined constant.
Similarly, there are exactly 100 centimeters in a meter and 1,000 meters in a kilometer.

• Defined constants provide standardized measurements across different contexts.
• They enable scientists and engineers to have a common understanding and make reliable calculations.
• Defined constants are integral in conversion between units in mathematics and science.

This permanence is what makes defined constants so powerful and crucial in accurate scientific and engineering calculations.