Ch.1 - Matter, Measurement & Problem Solving

Problem 79a

Chapter 1, Problem 79a

Which numbers are exact (and therefore have an unlimited number of significant figures)? a. π = 3.14 c. EPA gas mileage rating of 26 miles per gallon d. 1 gross = 144

Verified step by step guidance

Understand the concept of exact numbers: Exact numbers are those that are counted or defined, not measured, and therefore have an unlimited number of significant figures.

Identify the nature of π: π is a mathematical constant that represents the ratio of a circle's circumference to its diameter.

Recognize that π is an irrational number: It cannot be expressed as a simple fraction and has an infinite, non-repeating decimal expansion.

Determine if π is exact: Although π is a constant, it is not considered an exact number in the context of significant figures because it is often approximated in calculations (e.g., 3.14, 22/7).

Conclude that π is not an exact number: In the context of significant figures, π is not exact because it is typically approximated, and thus does not have an unlimited number of significant figures.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exact Numbers

Exact numbers are values that are counted or defined rather than measured. They have an infinite number of significant figures because they are not subject to measurement uncertainty. Examples include the number of students in a classroom or defined constants like 100 centimeters in a meter.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. This includes all non-zero digits, any zeros between significant digits, and trailing zeros in the decimal portion. Understanding significant figures is crucial for accurately reporting measurements and calculations in chemistry.

Measurement Uncertainty

Measurement uncertainty refers to the doubt that exists about the result of a measurement. It arises from limitations in measurement tools and techniques, affecting the number of significant figures that can be reported. Recognizing uncertainty helps in determining how precise a measurement is and how it should be communicated.

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