To determine why the expression [tex]\(\frac{a d + b c}{b d}\)[/tex] is a rational number, let's analyze it step-by-step:
1. Given:
- [tex]\(a, b, c,\)[/tex] and [tex]\(d\)[/tex] are integers.
- [tex]\(b \neq 0\)[/tex] and [tex]\(d \neq 0\)[/tex].
2. Expression Breakdown:
- The expression we need to consider is [tex]\(\frac{a}{b} + \frac{c}{d}\)[/tex].
- Common denominator for [tex]\(\frac{a}{b}\)[/tex] and [tex]\(\frac{c}{d}\)[/tex] would be [tex]\(bd\)[/tex].
- Therefore, we rewrite the addition as: [tex]\(\frac{a}{b} + \frac{c}{d} = \frac{a d}{b d} + \frac{b c}{b d}\)[/tex].
3. Combining the Fractions:
- Combining the fractions gives: [tex]\(\frac{a d + b c}{bd}\)[/tex].
4. Analyzing the Numerator and Denominator:
- The numerator [tex]\(ad + bc\)[/tex] is a sum of products of integers.
- According to the closure property of integers, the sum of products of integers ([tex]\(a, b, c, d\)[/tex]) is an integer. Therefore, [tex]\(ad + bc\)[/tex] is an integer.
- The denominator [tex]\(bd\)[/tex] is a product of two non-zero integers, which means [tex]\(bd\)[/tex] is also an integer and not equal to zero.
5. Conclusion:
- Since [tex]\(ad + bc\)[/tex] and [tex]\(bd\)[/tex] are both integers, and [tex]\(bd \neq 0\)[/tex], the expression [tex]\(\frac{a d + b c}{b d}\)[/tex] is a quotient of two integers.
- In mathematics, a number that can be expressed as the quotient of two integers (where the denominator is not zero) is defined as a rational number.
So, the expression [tex]\(\frac{a d + b c}{b d}\)[/tex] is a rational number because, by the Closure Property, [tex]\(ad + bc\)[/tex] and [tex]\(bd\)[/tex] are both integers, and thus [tex]\(\frac{ad + bc}{bd}\)[/tex] is a quotient of two integers, which fits the definition of a rational number.
