Let a and b be irrational numbers such that their sum a + b = q is a rational number. What can be said about the product ab?

Introduction / Context:
This question explores properties of irrational and rational numbers. You are given two irrational numbers a and b whose sum is rational. You must determine the nature of their product ab. Understanding such relationships is a key part of number theory and algebra, especially in questions involving surds and special algebraic constructions.

Given Data / Assumptions:
- a and b are irrational numbers.
- Their sum a + b = q is rational.
- We are asked about the nature of the product ab, whether it is always rational, always irrational, or can be either depending on the specific values of a and b.

Concept / Approach:
There is no simple universal rule that says the product of two irrational numbers with a rational sum must be of a single fixed type. Instead, we should look at concrete examples. By choosing specific irrational numbers a and b that still satisfy the condition a + b rational, we can see whether ab can be rational in some cases and irrational in others. This example driven approach is often the best way to understand such abstract questions.

Step-by-Step Solution:
Step 1: Consider an example where the product is rational. Let a = sqrt(2) and b = -sqrt(2).Step 2: Both a and b are irrational because sqrt(2) is irrational and so is its negative.Step 3: Their sum is a + b = sqrt(2) - sqrt(2) = 0, which is rational.Step 4: Their product is ab = sqrt(2) * (-sqrt(2)) = - (sqrt(2))^2 = -2, which is rational.Step 5: This example shows that it is possible for ab to be rational.Step 6: Now consider an example where the product is irrational. Let a = sqrt(2) and b = 1 - sqrt(2).Step 7: The number sqrt(2) is irrational. The number 1 - sqrt(2) is also irrational because it differs from an irrational number by a rational amount.Step 8: Their sum is a + b = sqrt(2) + (1 - sqrt(2)) = 1, which is rational.Step 9: Their product is ab = sqrt(2) * (1 - sqrt(2)) = sqrt(2) - 2.Step 10: The expression sqrt(2) - 2 is irrational, since it is the sum of an irrational number sqrt(2) and a rational number -2.Step 11: This second example shows that it is also possible for ab to be irrational.

Verification / Alternative check:
In both examples, the condition a + b rational is satisfied. In the first, the sum is 0 and the product is -2 (rational). In the second, the sum is 1 and the product is sqrt(2) - 2 (irrational). There is no contradiction in these constructions, so they are valid examples that demonstrate the logical possibilities for ab. Because ab can be rational in some cases and irrational in others, there is no single fixed classification.

Why Other Options Are Wrong:
The statement that ab is always rational is false because of the second example where ab = sqrt(2) - 2 is irrational. The statement that ab is always irrational is false because in the first example ab = -2 is rational. The claim that ab is never rational is contradicted by the first example, and the claim that ab is always zero is unrealistic and not compatible with a wide variety of possible irrationals a and b that still give rational sums. Therefore these options are not correct.

Common Pitfalls:
One common misconception is to assume that irrational numbers always behave in a single consistent way under addition or multiplication. In reality, irrationals have very flexible combinations. Another pitfall is ignoring the condition a + b rational when constructing examples. Carefully choosing examples that satisfy the sum condition immediately reveals that the product can have different types, which is the essential insight for this problem.

Final Answer:
The product ab is sometimes rational and sometimes irrational, depending on the particular irrational numbers a and b chosen.