Step 1: Let [tex]a[/tex] be a rational number and [tex]b[/tex] be an irrational number.
Step 2: Assume that [tex]a + b = x[/tex] and that [tex]x[/tex] is a rational number.
Step 3: By the subtraction property of equality, [tex]b = x - a[/tex].
Step 4: If [tex]b = x - a[/tex], then [tex]b = x + (-a)[/tex].
Step 5: [tex]x + (-a)[/tex] is a rational number because [....] However, it was stated that [tex]b[/tex] is an irrational number. This is a contradiction. Therefore, the assumption that [tex]x[/tex] is rational in the equation [tex]a + b = x[/tex] must be incorrect, and [tex]x[/tex] should be an irrational number. In conclusion, the sum of a rational number and an irrational number is an irrational number.

Which of the following best completes the proof?

A. It is the sum of two rational numbers.
B. It is the sum of two irrational numbers.
C. It represents a non-terminating, non-repeating decimal.
D. Its terms cannot be combined.



Answer :

Let's go through the solution step-by-step to understand which statement best completes the proof.

1. Define the Numbers:
Let's start by defining two numbers:
- Let [tex]\( a \)[/tex] be a rational number.
- Let [tex]\( b \)[/tex] be an irrational number.

2. Assume the Sum:
Assume that [tex]\( a + b = x \)[/tex], where [tex]\( x \)[/tex] is a rational number.

3. Subtracting to Identify [tex]\( b \)[/tex]:
By the subtraction property of equality, we get:
[tex]\[ b = x - a \][/tex]

4. Relating Rationals and Irrationals:
Now, since both [tex]\( x \)[/tex] and [tex]\( a \)[/tex] are rational numbers, their difference must also be a rational number (because the set of rational numbers is closed under subtraction). This implies that:
[tex]\[ b = x - a \quad \text{is rational} \][/tex]

5. Contradiction:
However, from our initial definition, [tex]\( b \)[/tex] is an irrational number. This leads to a contradiction because one number (b) cannot be both rational and irrational simultaneously. Therefore, the assumption that [tex]\( x \)[/tex] is rational must be incorrect.

6. Conclusion:
Since our initial assumption led to a contradiction, we conclude that the sum [tex]\( a + b \)[/tex] must not be rational. Hence, [tex]\( a + b \)[/tex] must be irrational.

In conclusion, the statement that best completes the proof is:

"it is the sum of two rational numbers."

This accurately identifies the nature of the original contradiction and provides a correct understanding of the sum of a rational and an irrational number.