Step 1, Step 3, Step 2, Step 4
Here's the reasoning:
Step 1: Subtract, writing s as a fraction.
When we subtract r + s from both sides, we get t - r - s = 0. By rearranging this equation, we have t = r + s.
Step 3: Realize a contradiction.
Assume that t is rational, meaning it can be expressed as a fraction a/b, where a and b are integers. Now let's write r and s as fractions as well. We can write r as c/d and s as e/f, where c, d, e, and f are integers.
Step 2: For a contradiction, assume that t is rational, and write r and t as fractions.
Substituting the given expressions into the equation t = r + s, we get a/b = c/d + e/f. By combining the fractions on the right side, we have a/b = (cf + de) / (df).
Step 4: Find a common denominator.
To obtain a common denominator, we multiply both the numerator and denominator (cf + de) / (df) by (bd). This gives us a/b = (cf + de)(bd) / (df)(bd).
At this point, we realize a contradiction. If t is rational, then it can be expressed as a fraction. However, when multiplying the numerator and denominator by (bd), we get a fraction in which the denominator contains the prime factors d and f, which are not present in the denominators of r and s. This contradicts the assumption that r and s are both rational.
Therefore, we can conclude that if r is rational and s is rational, the sum t = r + s is irrational.