The sum of two rational numbers will always be rational.
Let's break this down step-by-step:
1. Definition of Rational Numbers:
- A rational number is any number that can be expressed as the quotient or fraction [tex]\( \frac{a}{b} \)[/tex] of two integers [tex]\( a \)[/tex] and [tex]\( b \)[/tex], with [tex]\( b \neq 0 \)[/tex]. Examples include [tex]\( \frac{1}{2} \)[/tex], [tex]\( \frac{3}{4} \)[/tex], and even whole numbers like 5 (which can be written as [tex]\( \frac{5}{1} \)[/tex]).
2. Adding Rational Numbers:
- Suppose we have two rational numbers [tex]\( \frac{a}{b} \)[/tex] and [tex]\( \frac{c}{d} \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] are integers and [tex]\( b \)[/tex] and [tex]\( d \)[/tex] are not zero.
- To find their sum, we find a common denominator, which is typically the product of the two denominators [tex]\( b \)[/tex] and [tex]\( d \)[/tex].
- The sum can be expressed as:
[tex]\[
\frac{a}{b} + \frac{c}{d} = \frac{a \cdot d}{b \cdot d} + \frac{c \cdot b}{d \cdot b} = \frac{ad + cb}{bd}
\][/tex]
- The numerator [tex]\( ad + cb \)[/tex] and the denominator [tex]\( bd \)[/tex] are both integers because [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] are integers, and the operations of multiplication and addition of integers result in integers.
3. Conclusion:
- The fraction [tex]\( \frac{ad + cb}{bd} \)[/tex] is in the form of [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]. Therefore, this fraction is a rational number.
Thus, the sum of two rational numbers is always rational.
