Create a problem where the sum of the two numbers is an irrational number.

Answer 1

Problem:

Find two numbers such that their sum is an irrational number.

Solution:

Let's consider the equation x + y = π, where x and y are the two numbers we need to find.

To prove that the sum of x and y is an irrational number, we assume that x and y are rational numbers. Since the sum of two rational numbers is always rational, this would contradict our initial assumption.

Taking the contrapositive of that statement, we can conclude that if the sum of two numbers is an irrational number, then at least one of the numbers must be an irrational number.

Therefore, in our equation x + y = π, we can take x = π - y. By substituting it into the equation, we have (π - y) + y = π.

This simplifies to π = π, which is a true statement.

Thus, we have found a solution, where x = π - y and y can take any value we desire. As long as y is a rational number, the sum x + y will always be π, an irrational number.