To prove that the sum of two rational numbers is also a rational number, we can follow a step-by-step logical approach based on the definition of rational numbers.

Step 1: Definition of a Rational Number

A number is called rational if it can be expressed as the quotient of two integers, i.e., a number $r$ is rational if: $r = \frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.

Step 2: Let the Two Rational Numbers Be

Let $r_1$ and $r_2$ be two rational numbers. By definition of rational numbers, we can write them as: $r_1 = \frac{a_1}{b_1} \quad \text{and} \quad r_2 = \frac{a_2}{b_2}$ where $a_1, a_2$ are integers, and $b_1, b_2$ are non-zero integers.

Step 3: Sum of Two Rational Numbers

Now, consider the sum of these two rational numbers: $r_1 + r_2 = \frac{a_1}{b_1} + \frac{a_2}{b_2}$

Step 4: Simplify the Sum

To add these two fractions, we need a common denominator. The common denominator is $b_1 b_2$. So, we can rewrite the sum as: $r_1 + r_2 = \frac{a_1 b_2}{b_1 b_2} + \frac{a_2 b_1}{b_1 b_2} = \frac{a_1 b_2 + a_2 b_1}{b_1 b_2}$

Step 5: Verify the Result

In the expression $\frac{a_1 b_2 + a_2 b_1}{b_1 b_2}$:

• The numerator $a_1 b_2 + a_2 b_1$ is the sum of twoproducts of integers, which means it is also an integer because the sum and product of integers are always integers.
• The denominator $b_1 b_2$ is a product of two non-zero integers, which is also a non-zero integer.

Step 6: Conclusion

Thus, we can express the sum $r_1 + r_2$ as: $r_1 + r_2 = \frac{\text{integer}}{\text{non-zero integer}}$ which is again in the form of a rational number. Hence, the sum of two rational numbers is also a rational number.

Final Statement

Therefore, we have proven that the sum of two rational numbers is a rational number.

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Would you like any further details or clarifications? Here are some related questions you might find interesting:

1. What is the product of two rational numbers, and is it always rational?
2. How do we prove that the difference of two rational numbers is rational?
3. Can the sum of a rational and an irrational number ever be rational?
4. How can we prove that the set of rational numbers is closed under addition?
5. What is the geometric interpretation of the sum of rational numbers?

Tip: Remember, a property of rational numbers is that any arithmetic operation involving rational numbers (addition, subtraction, multiplication, and division, except by zero) results in a rational number.