Question
Are all rational numbers whole numbers? OR are all whole numbers rational numbers?
Use the numbers -3 and 7 to explain your answer.(3 points)
Use the numbers -3 and 7 to explain your answer.(3 points)
Answers
GPT-4o mini
To determine the relationship between rational numbers and whole numbers, we need to understand what each term means:
1. **Rational Numbers**: A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). This includes integers, fractions, and finite or repeating decimals.
2. **Whole Numbers**: Whole numbers are all non-negative integers, which include 0, 1, 2, 3, and so on.
Now, let's analyze the numbers -3 and 7 in relation to these definitions:
- **Number 7**:
- 7 is a whole number (it is non-negative and an integer).
- 7 can also be expressed as a rational number: \( \frac{7}{1} \), where both 7 and 1 are integers and the denominator is not zero. Therefore, 7 is a rational number.
- **Number -3**:
- -3 is not a whole number because whole numbers do not include negative integers.
- However, -3 is a rational number, as it can be expressed as \( \frac{-3}{1} \).
From these points, we can conclude that:
- **Not all rational numbers are whole numbers** (as -3 is a rational number but not a whole number).
- **All whole numbers are rational numbers** (as 7 is both a whole number and a rational number).
In summary, while all whole numbers (like 7) are indeed rational numbers, not all rational numbers (like -3) are whole numbers.
1. **Rational Numbers**: A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). This includes integers, fractions, and finite or repeating decimals.
2. **Whole Numbers**: Whole numbers are all non-negative integers, which include 0, 1, 2, 3, and so on.
Now, let's analyze the numbers -3 and 7 in relation to these definitions:
- **Number 7**:
- 7 is a whole number (it is non-negative and an integer).
- 7 can also be expressed as a rational number: \( \frac{7}{1} \), where both 7 and 1 are integers and the denominator is not zero. Therefore, 7 is a rational number.
- **Number -3**:
- -3 is not a whole number because whole numbers do not include negative integers.
- However, -3 is a rational number, as it can be expressed as \( \frac{-3}{1} \).
From these points, we can conclude that:
- **Not all rational numbers are whole numbers** (as -3 is a rational number but not a whole number).
- **All whole numbers are rational numbers** (as 7 is both a whole number and a rational number).
In summary, while all whole numbers (like 7) are indeed rational numbers, not all rational numbers (like -3) are whole numbers.