Question
In the proof that the sum of rational numbers a/b and c/d where a,b,c and d are integers
Answers
GPT-4o mini
To prove that the sum of two rational numbers is a rational number, consider two rational numbers expressed in the form of fractions:
\[
\frac{a}{b} \quad \text{and} \quad \frac{c}{d}
\]
where \( a, b, c, \) and \( d \) are integers, and \( b \) and \( d \) are non-zero (since denominator cannot be zero).
We want to find the sum:
\[
\frac{a}{b} + \frac{c}{d}
\]
To add these two fractions, we need a common denominator. The common denominator of \( b \) and \( d \) is \( bd \). We express each fraction with this common denominator:
\[
\frac{a}{b} = \frac{a \cdot d}{b \cdot d} = \frac{ad}{bd}
\]
\[
\frac{c}{d} = \frac{c \cdot b}{d \cdot b} = \frac{cb}{bd}
\]
Now we can add the two fractions:
\[
\frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{cb}{bd} = \frac{ad + cb}{bd}
\]
The numerator \( ad + cb \) is an integer (since it is the sum of the products of integers), and the denominator \( bd \) is a non-zero integer (since \( b \neq 0 \) and \( d \neq 0 \)).
Thus, we can conclude that:
\[
\frac{ad + cb}{bd}
\]
is a rational number.
Since the sum of two rational numbers is in the form of a fraction where both the numerator and the denominator are integers (with the denominator not equal to zero), the sum \( \frac{a}{b} + \frac{c}{d} \) is a rational number.
Therefore, we have proved that the sum of two rational numbers is also a rational number.
\[
\frac{a}{b} \quad \text{and} \quad \frac{c}{d}
\]
where \( a, b, c, \) and \( d \) are integers, and \( b \) and \( d \) are non-zero (since denominator cannot be zero).
We want to find the sum:
\[
\frac{a}{b} + \frac{c}{d}
\]
To add these two fractions, we need a common denominator. The common denominator of \( b \) and \( d \) is \( bd \). We express each fraction with this common denominator:
\[
\frac{a}{b} = \frac{a \cdot d}{b \cdot d} = \frac{ad}{bd}
\]
\[
\frac{c}{d} = \frac{c \cdot b}{d \cdot b} = \frac{cb}{bd}
\]
Now we can add the two fractions:
\[
\frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{cb}{bd} = \frac{ad + cb}{bd}
\]
The numerator \( ad + cb \) is an integer (since it is the sum of the products of integers), and the denominator \( bd \) is a non-zero integer (since \( b \neq 0 \) and \( d \neq 0 \)).
Thus, we can conclude that:
\[
\frac{ad + cb}{bd}
\]
is a rational number.
Since the sum of two rational numbers is in the form of a fraction where both the numerator and the denominator are integers (with the denominator not equal to zero), the sum \( \frac{a}{b} + \frac{c}{d} \) is a rational number.
Therefore, we have proved that the sum of two rational numbers is also a rational number.