Answer Key for Lesson 1.1 Rational and Irrational Numbers

First, identify the types of values in the given set. Begin by distinguishing between those that can be expressed as a fraction of two integers and those that cannot. The first category includes values like 1/2, 5, and 0.75. These can be written as ratios, which is their defining characteristic. The second group consists of values like √2, π, and e, which cannot be precisely represented as a fraction of integers.

Next, verify the decimal expansions of these values. Numbers in the first category have either terminating or repeating decimals, making them easier to work with in practical situations. On the other hand, values in the second group have non-terminating, non-repeating decimals that continue infinitely without a clear pattern.

For further clarification, test specific examples. Take the value 3/4. This is a simple fraction, and its decimal form is 0.75, a terminating decimal. Now consider √3, which approximates to 1.732… This is an example of a value that cannot be expressed exactly as a fraction and has a never-ending decimal expansion.

These distinctions are fundamental for solving problems involving these values. Understanding their properties allows you to determine the appropriate methods for working with them in various mathematical contexts.