Final answer:
The correct placement of numbers within sets and subsets of rational numbers includes integers, positive and negative fractions, all of which can be expressed as ratios with a non-zero denominator. To perform operations like addition, a common denominator must be found, and the hierarchy of sets shows the relationship between different types of numbers.
Step-by-step explanation:
Understanding Rational Numbers and Their Sets
The correct placement of numbers in the sets and subsets of rational numbers relies on a clear understanding of number properties and operations. When discussing rational numbers, we are talking about numbers that can be written as the ratio of two integers, where the denominator is not zero. This includes integers, fractions, and both positive and negative numbers. The construction of a common denominator is essential for adding fractions with different denominators. For instance, to add ½ and ⅓, we find a common denominator (which is the least common multiple of the denominators), in this case, 6. We can rewrite the fractions as ⅔ and ⅖, respectively, and then add them directly to get ⅙.
To understand the hierarchy, we can visualize a set diagram. Integers form a subset of rational numbers since they can be expressed as a ratio of themselves to 1 (e.g., 3 as ⅓), while fractions like ½ and ⅓ are also in this set as they are ratios by definition. All of these are part of the larger set of rational numbers.
Understanding the sets and subsets of numbers is critical in differentiating between different types of numbers, and when adding them, finding a common denominator simplifies the process significantly. Utilizing step-by-step explanations, examples, and graphical representations can assist students in mastering these concepts.