Yes, the sum of a rational number and a rational number is indeed a rational number. To understand why, we first need to clarify what rational numbers are.
A rational number is defined as any number that can be expressed as the quotient of two integers, where the denominator is not zero. This means that any number that can be written in the form a/b, where a and b are integers and b ≠ 0, qualifies as a rational number.
Now, let’s take two rational numbers: r1 = a/b and r2 = c/d, where a, b, c, and d are all integers and b, d ≠ 0. To find the sum of these two rational numbers, we follow the formula for addition:
r1 + r2 = (a/b) + (c/d)
To add them together, we need a common denominator, which we can find by multiplying b and d. Thus, we can rewrite the sum as:
r1 + r2 = (a * d) / (b * d) + (c * b) / (d * b)
This simplifies to:
(a * d + c * b) / (b * d)
Here, ad + cb is an integer since it is the sum of two integers (as integers are closed under addition). The denominator bd is also an integer (since it is the product of two non-zero integers). Therefore, the entire expression can be written in the form of a fraction where both the numerator and denominator are integers, confirming that the sum is a rational number.
In conclusion, the sum of two rational numbers is still a rational number, as it can always be expressed in the required form.