Question
In the proof that a/b+c/d, where a, b, c, and d are integers and b and d' are non-zero, explain why ad+bc/bd is a rational number.
Answers
GPT-4o mini
To show that the expression \(\frac{ad + bc}{bd}\) is a rational number, let's analyze each part of the expression with the conditions given.
1. **Understanding the components**:
- \(a\), \(b\), \(c\), and \(d\) are integers.
- \(b\) and \(d\) are non-zero integers.
2. **Examining \(ad\)**:
- Since \(a\) and \(d\) are both integers, the product \(ad\) (which is the result of multiplying two integers) is also an integer.
3. **Examining \(bc\)**:
- Similarly, since \(b\) and \(c\) are integers, the product \(bc\) is an integer as well.
4. **Combining \(ad\) and \(bc\)**:
- The sum \(ad + bc\) involves adding two integers (i.e., \(ad\) and \(bc\)). The sum of two integers is also an integer.
5. **Examining the denominator \(bd\)**:
- Since \(b\) and \(d\) are both non-zero integers, their product \(bd\) is also a non-zero integer.
6. **Final Expression**:
- We now consider the fraction \(\frac{ad + bc}{bd}\).
- The numerator \(ad + bc\) is an integer.
- The denominator \(bd\) is a non-zero integer.
7. **Conclusion**:
Since the fraction \(\frac{ad + bc}{bd}\) has an integer in the numerator and a non-zero integer in the denominator, it meets the definition of a rational number. Therefore, \(\frac{ad + bc}{bd}\) is indeed a rational number.
In summary, because both \(ad + bc\) is an integer and \(bd\) is a non-zero integer, the expression \(\frac{ad + bc}{bd}\) is a rational number as required.
1. **Understanding the components**:
- \(a\), \(b\), \(c\), and \(d\) are integers.
- \(b\) and \(d\) are non-zero integers.
2. **Examining \(ad\)**:
- Since \(a\) and \(d\) are both integers, the product \(ad\) (which is the result of multiplying two integers) is also an integer.
3. **Examining \(bc\)**:
- Similarly, since \(b\) and \(c\) are integers, the product \(bc\) is an integer as well.
4. **Combining \(ad\) and \(bc\)**:
- The sum \(ad + bc\) involves adding two integers (i.e., \(ad\) and \(bc\)). The sum of two integers is also an integer.
5. **Examining the denominator \(bd\)**:
- Since \(b\) and \(d\) are both non-zero integers, their product \(bd\) is also a non-zero integer.
6. **Final Expression**:
- We now consider the fraction \(\frac{ad + bc}{bd}\).
- The numerator \(ad + bc\) is an integer.
- The denominator \(bd\) is a non-zero integer.
7. **Conclusion**:
Since the fraction \(\frac{ad + bc}{bd}\) has an integer in the numerator and a non-zero integer in the denominator, it meets the definition of a rational number. Therefore, \(\frac{ad + bc}{bd}\) is indeed a rational number.
In summary, because both \(ad + bc\) is an integer and \(bd\) is a non-zero integer, the expression \(\frac{ad + bc}{bd}\) is a rational number as required.