Final answer:
A proof by contradiction shows that if the sum of an irrational number and a rational number were rational, it would lead to the contradiction that an irrational number would be equal to a rational number. This contradiction means our initial assumption is false, and thus the sum is indeed irrational.
Step-by-step explanation:
Proof by Contradiction: Sum of Irrational and Rational Numbers Let's use a proof by contradiction to show that the sum of an irrational number and a rational number is irrational. Assume the contrary, that the sum is rational. Let a be irrational, and b be rational. Supposing that a + b is rational, we can write a + b = c where c is a rational number. Since the sum of two rational numbers is rational and b is rational, then c - b should also be a rational number. Subtracting b from both sides of the equation a + b = c gives us a = c - b. Thus, a, an irrational number, is equal to c - b, a supposed rational number, which is a contradiction because we defined a as being irrational. Since our assumption leads to a contradiction, it must be false; therefore, the sum of an irrational number and a rational number is irrational.