Create a problem where the sum of the two numbers is an irrational number.
Problem: You have a jar of marbles, and you are asked to pick two marbles at random without replacement. The jar contains an infinite number of marbles, half of which are red and half of which are blue. Each marble is labeled with a real number chosen at random from the interval [0, 1], where the probability of selecting any specific number is equal.
What is the probability that the sum of the two numbers on the marbles you pick is an irrational number?
(Note: A number is considered irrational if it cannot be expressed as the ratio of two integers.)
Problem: Find two numbers such that their sum is an irrational number.
Solution:
Step 1: Let's assume the two numbers to be x and y.
Step 2: To create a sum that is an irrational number, we can choose x to be a rational number and y to be an irrational number.
Step 3: Let x = 2 (a rational number) and y = √3 (an irrational number).
Step 4: Now, let's calculate the sum of x and y.
x + y = 2 + √3
Step 5: Since √3 is an irrational number, the sum of x and y, which is 2 + √3, is also an irrational number.
Therefore, the sum of the two numbers, 2 + √3, is an irrational number.