To understand why the expression \(\frac{ad + bc}{bd}\) is a rational number given that \(a\), \(b\), \(c\), and \(d\) are integers with \(b \neq 0\) and \(d \neq 0\), we first need to recall the definition of a rational number.
A rational number is defined as any number that can be expressed as the quotient of two integers \( \frac{p}{q} \) where \(p\) and \(q\) are integers and \(q \neq 0\).
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Examining the numerator: The numerator of our expression is \(ad + bc\).
- Since \(a\), \(b\), \(c\), and \(d\) are all integers, the product \(ad\) is also an integer (as the product of two integers).
- Similarly, \(bc\) is an integer.
- The sum of two integers (in this case, \(ad\) and \(bc\)) is also an integer. Thus, \(ad + bc\) is an integer.
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Examining the denominator: The denominator of our expression is \(bd\).
- Since \(b\) and \(d\) are both non-zero integers, their product \(bd\) is also a non-zero integer (the product of two non-zero integers is non-zero).
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Putting it together: Now we can combine these results:
- We have the fraction \(\frac{ad + bc}{bd}\), where the numerator \(ad + bc\) is an integer and the denominator \(bd\) is a non-zero integer.
Since both the numerator and denominator meet the criteria for forming a rational number (the numerator is an integer, and the denominator is a non-zero integer), we conclude that \(\frac{ad + bc}{bd}\) is indeed a rational number.
Thus, \(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\) is a rational number as required.
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