Why the sum of two rational numbers is rational

The sum of two rational numbers is rational because of the way rational numbers are defined. A rational number is any number that can be expressed as the quotient of two integers, i.e., in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \neq 0 \).

Let's break it down step by step:

### Given:
Two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \), where \( a, b, c, d \) are integers and \( b, d \neq 0 \).

### Step 1: Adding the two rational numbers:
\[
\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
\]
In this step, we find a common denominator (by multiplying the denominators \( b \) and \( d \)), and the numerators are adjusted accordingly.

### Step 2: Simplification:
\[
\frac{ad + bc}{bd}
\]
Here, the numerator is \( ad + bc \), and the denominator is \( bd \). Since \( a, b, c, d \) are integers, both \( ad + bc \) (the sum of two products of integers) and \( bd \) (the product of two integers) are also integers.

### Step 3: Conclusion:
Since the result can be written as the quotient of two integers, and the denominator \( bd \) is not zero (because neither \( b \) nor \( d \) is zero), the result is a rational number.

Thus, the sum of two rational numbers is rational.

Do you want further clarification on any part of this explanation?

Here are 5 related questions to explore this concept further:
1. What happens when you subtract two rational numbers?
2. How do you multiply two rational numbers, and why is the result rational?
3. What is the sum of a rational and an irrational number, and why is it irrational?
4. Can a rational number ever have a repeating decimal form?
5. How do rational and irrational numbers differ in terms of their decimal expansions?

**Tip:** When adding or subtracting fractions, always remember to find the least common denominator to simplify calculations.

Why is the sum of two rational numbers rational?

Understand why the sum of two rational numbers is always rational. This problem involves key number theory concepts and the properties of rational numbers, using fractions and basic algebra. Ideal for students in Grades 6-8.

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