Asked by hgfcbhygf
In the proof that ab+cd ��� ��� + ��� ��� , where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd ��� ��� + ��� ��� ��� ��� is a rational number.(1 point) Responses By the Closure Property, ad+bcbd ��� ��� + ��� ��� ��� ��� reduces to a whole number, which is a subset of rational numbers. By the Closure Property, the fraction with numerator A d plus b c and denominator b d reduces to a whole number, which is a subset of rational numbers. By the Closure Property, ad+bc ��� ��� + ��� ��� and bd ��� ��� are both quotients of integers, and so ad+bcbd ��� ��� + ��� ��� ��� ��� is a quotient of two integers. By the Closure Property, A d plus b c and b d are both quotients of integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers. By the Closure Property, ad+bc ��� ��� + ��� ��� and bd ��� ��� are both integers, and so ad+bcbd ��� ��� + ��� ��� ��� ��� is a quotient of two integers. By the Closure Property, A d plus b c and b d are both integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers. By the Closure Property, a quotient of imaginary numbers is a rational number. By the Closure Property, a quotient of imaginary numbers is a rational number.
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GPT-4o mini
To explain why \( \frac{ad + bc}{bd} \) is a rational number when \( a, b, c, \) and \( d \) are integers and \( b \) and \( d \) are non-zero, we can break down the reasoning as follows:
1. **Identify the Components**: In the expression \( \frac{ad + bc}{bd} \), we have:
- \( ad \): This is the product of two integers (since \( a \) and \( d \) are both integers), and thus \( ad \) is an integer.
- \( bc \): This is also the product of two integers (since \( b \) and \( c \) are integers), and thus \( bc \) is an integer.
2. **Sum of Integers**: The numerator of our fraction is \( ad + bc \). Since both \( ad \) and \( bc \) are integers, and the sum of two integers is also an integer, \( ad + bc \) is an integer.
3. **Denominator**: The denominator \( bd \) is the product of two non-zero integers \( b \) and \( d \). Since both \( b \) and \( d \) are non-zero, \( bd \) is also a non-zero integer.
4. **Quotient of Integers**: The fraction \( \frac{ad + bc}{bd} \) is then a quotient of two integers, where the numerator \( ad + bc \) is an integer and the denominator \( bd \) is a non-zero integer.
5. **Definition of Rational Numbers**: By definition, a number is rational if it can be expressed as the quotient of two integers where the denominator is not zero. Therefore, \( \frac{ad + bc}{bd} \) fits this definition because both the numerator and denominator are integers, and the denominator is non-zero.
Thus, \( \frac{ad + bc}{bd} \) is a rational number.
In summary, the correct explanation aligns with the principles of integer arithmetic and the definition of rational numbers, confirming that \( \frac{ad + bc}{bd} \) is indeed a rational number.
1. **Identify the Components**: In the expression \( \frac{ad + bc}{bd} \), we have:
- \( ad \): This is the product of two integers (since \( a \) and \( d \) are both integers), and thus \( ad \) is an integer.
- \( bc \): This is also the product of two integers (since \( b \) and \( c \) are integers), and thus \( bc \) is an integer.
2. **Sum of Integers**: The numerator of our fraction is \( ad + bc \). Since both \( ad \) and \( bc \) are integers, and the sum of two integers is also an integer, \( ad + bc \) is an integer.
3. **Denominator**: The denominator \( bd \) is the product of two non-zero integers \( b \) and \( d \). Since both \( b \) and \( d \) are non-zero, \( bd \) is also a non-zero integer.
4. **Quotient of Integers**: The fraction \( \frac{ad + bc}{bd} \) is then a quotient of two integers, where the numerator \( ad + bc \) is an integer and the denominator \( bd \) is a non-zero integer.
5. **Definition of Rational Numbers**: By definition, a number is rational if it can be expressed as the quotient of two integers where the denominator is not zero. Therefore, \( \frac{ad + bc}{bd} \) fits this definition because both the numerator and denominator are integers, and the denominator is non-zero.
Thus, \( \frac{ad + bc}{bd} \) is a rational number.
In summary, the correct explanation aligns with the principles of integer arithmetic and the definition of rational numbers, confirming that \( \frac{ad + bc}{bd} \) is indeed a rational number.
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