To explain why [tex]\(\frac{a}{b}+\frac{c}{d}\)[/tex], where [tex]\(a, b, c,\)[/tex] and [tex]\(d\)[/tex] are integers and [tex]\(b\)[/tex] and [tex]\(d\)[/tex] are non-zero, is a rational number, let's follow these steps:
1. Definition of Rational Numbers:
A rational number is any number that can be expressed as the quotient of two integers, i.e., [tex]\(\frac{p}{q}\)[/tex] where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \ne 0\)[/tex].
2. Addition of Fractions:
To add two fractions, we need a common denominator. The common denominator for [tex]\(\frac{a}{b}\)[/tex] and [tex]\(\frac{c}{d}\)[/tex] is [tex]\(bd\)[/tex]. Thus, we rewrite the fractions accordingly:
[tex]\[
\frac{a}{b} = \frac{a \cdot d}{b \cdot d} = \frac{ad}{bd}
\][/tex]
[tex]\[
\frac{c}{d} = \frac{c \cdot b}{d \cdot b} = \frac{cb}{bd}
\][/tex]
3. Combining the Fractions:
Now, we add the rewritten fractions:
[tex]\[
\frac{ad}{bd} + \frac{cb}{bd} = \frac{ad + bc}{bd}
\][/tex]
4. Closure Property:
According to the Closure Property of integers, if [tex]\(a\)[/tex] and [tex]\(d\)[/tex] are integers, then [tex]\(ad\)[/tex] is also an integer; similarly, if [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are integers, then [tex]\(bc\)[/tex] is also an integer. Therefore, both [tex]\(ad + bc\)[/tex] and [tex]\(bd\)[/tex] are integers obtained by addition and multiplication of integers.
5. Form of a Rational Number:
Since [tex]\(ad + bc\)[/tex] and [tex]\(bd\)[/tex] are both integers, and [tex]\(bd \ne 0\)[/tex] (since [tex]\(b \ne 0\)[/tex] and [tex]\(d \ne 0\)[/tex]), the fraction [tex]\(\frac{ad + bc}{bd}\)[/tex] is the quotient of two integers where the denominator is non-zero. By definition, this means [tex]\(\frac{ad + bc}{bd}\)[/tex] is a rational number.
Therefore, [tex]\(\frac{a}{b} + \frac{c}{d}\)[/tex] is indeed a rational number because it can be expressed as the quotient of two integers. This concludes the proof that the sum of the fractions [tex]\(\frac{a}{b}\)[/tex] and [tex]\(\frac{c}{d}\)[/tex] is a rational number.
