Question

describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole numbers? Are all whole numbers? Give examples to support your answer.

Answer 1

Rational numbers and whole numbers are two different categories of numbers in mathematics, each with its own properties and definitions.

### Rational Numbers:
- A rational number is any number that can be expressed as the quotient or fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). This includes positive and negative numbers, as well as zero.
- Examples of rational numbers include:
- \( \frac{1}{2} \) (a positive rational number)
- \( -3 \) (which can be expressed as \( \frac{-3}{1} \))
- \( 0 \) (which can be expressed as \( \frac{0}{1} \))

### Whole Numbers:
- Whole numbers are a subset of the number system that includes all non-negative integers, starting from zero. The set of whole numbers is \( \{ 0, 1, 2, 3, 4, \ldots \} \).
- Examples of whole numbers include:
- \( 0 \)
- \( 1 \)
- \( 25 \)
- \( 100 \)

### Relationship:
1. **Not All Rational Numbers Are Whole Numbers**:
- Rational numbers include fractions, integers, and numbers that can be expressed as decimals (e.g., \( 0.5 \)). Not all rational numbers fit the criteria of whole numbers.
- For example, \( \frac{1}{2} \) is a rational number but not a whole number. Similarly, negative numbers like \( -1 \) or non-integer rational numbers like \( \frac{3}{4} \) also do not belong to the set of whole numbers.

2. **All Whole Numbers Are Rational Numbers**:
- Whole numbers are a specific type of rational number. Any whole number \( n \) can be written as a fraction \( \frac{n}{1} \), which fits the definition of a rational number.
- For instance, \( 3 \) can be expressed as \( \frac{3}{1} \), thus it is a rational number.

### Conclusion:
In summary, while all whole numbers are indeed rational numbers, not all rational numbers are whole numbers. The whole numbers form a subset within the broader set of rational numbers.