Question

Describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole numbers? Are all whole numbers also rational numbers?

Answer 1

Rational numbers and whole numbers are both subsets of the broader category of numbers, but they have distinct characteristics.

1. **Rational Numbers**: A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) is an integer and \( b \) is a non-zero integer. This means that rational numbers include integers, fractions, and terminating or repeating decimals. Examples of rational numbers include \( \frac{1}{2}, -3, 0.75 \), and \( -\frac{5}{4} \).

2. **Whole Numbers**: Whole numbers are a subset of integers and include all non-negative integers, starting from 0. The set of whole numbers is typically represented as {0, 1, 2, 3, ...}. Whole numbers do not include negative numbers or fractions.

### Relationship between Rational Numbers and Whole Numbers:
- **Are all rational numbers also whole numbers?** No, not all rational numbers are whole numbers. For instance, the rational number \( \frac{1}{2} \) is not a whole number because it is a fraction and not a non-negative integer.

- **Are all whole numbers also rational numbers?** Yes, all whole numbers are indeed rational numbers. Every whole number can be expressed as a fraction by placing it over 1 (for example, the whole number 3 can be written as \( \frac{3}{1} \)). Since this fits the definition of a rational number, whole numbers are included in the set of rational numbers.

In summary, while all whole numbers are rational numbers, not all rational numbers are whole numbers.