Question
Where does: 15/2, belong on the Venn diagram that shows Integers, Natural numbers, Rational numbers, and Irrational numbers?
Answers
GPT-4o mini
To determine where \( \frac{15}{2} \) belongs in a Venn diagram that includes Integers, Natural numbers, Rational numbers, and Irrational numbers, let's analyze the number:
1. **Rational Numbers**: A rational number is defined as any number that can be expressed as the quotient \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0\). Since \( \frac{15}{2} \) can be expressed as the quotient of two integers (15 and 2), it is a rational number.
2. **Integers**: Integers include whole numbers and their negatives, such as ... -3, -2, -1, 0, 1, 2, 3, .... Since \( \frac{15}{2} = 7.5 \), it is not an integer.
3. **Natural Numbers**: Natural numbers are the positive integers (1, 2, 3, ...). Since \( 7.5 \) is not a whole number, it is not a natural number.
4. **Irrational Numbers**: Irrational numbers are numbers that cannot be expressed as a simple fraction; they have non-repeating, non-terminating decimal expansions. Since \( \frac{15}{2} \) can be expressed as a fraction, it is not an irrational number.
Putting all of this together, \( \frac{15}{2} \) belongs in the section of the Venn diagram labeled "Rational Numbers". It does not belong in the sections labeled for Integers, Natural Numbers, or Irrational Numbers.
1. **Rational Numbers**: A rational number is defined as any number that can be expressed as the quotient \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0\). Since \( \frac{15}{2} \) can be expressed as the quotient of two integers (15 and 2), it is a rational number.
2. **Integers**: Integers include whole numbers and their negatives, such as ... -3, -2, -1, 0, 1, 2, 3, .... Since \( \frac{15}{2} = 7.5 \), it is not an integer.
3. **Natural Numbers**: Natural numbers are the positive integers (1, 2, 3, ...). Since \( 7.5 \) is not a whole number, it is not a natural number.
4. **Irrational Numbers**: Irrational numbers are numbers that cannot be expressed as a simple fraction; they have non-repeating, non-terminating decimal expansions. Since \( \frac{15}{2} \) can be expressed as a fraction, it is not an irrational number.
Putting all of this together, \( \frac{15}{2} \) belongs in the section of the Venn diagram labeled "Rational Numbers". It does not belong in the sections labeled for Integers, Natural Numbers, or Irrational Numbers.