Final answer:
To prove that the sum of an irrational number and a rational number is irrational, we can use proof by contradiction. Assume the opposite and show that it leads to a contradiction. By doing this, we can demonstrate that the sum of an irrational number and a rational number is indeed irrational.
Step-by-step explanation:
To prove that the sum of an irrational number and a rational number is irrational using proof by contradiction, we assume the opposite. So, let's assume that the sum of an irrational number and a rational number is rational.
Let's say we have an irrational number 'a' and a rational number 'b'. We assume that the sum of a + b is rational, which means there exists a rational number 'c' such that a + b = c.
Now, let's rearrange the equation to isolate the irrational number 'a': a = c - b.
Since 'c' and 'b' are both rational numbers and rational numbers are closed under subtraction, the difference c - b will also be a rational number. However, this contradicts our assumption that 'a' is irrational. Therefore, our initial assumption that the sum of an irrational number and a rational number is rational must be false.